WKB Approximation in Atomic Physics

WKB Approximation in Atomic Physics

Boris Mikhailovich Karnakov Vladimir Pavlovich Krainov

WKB Approximation in Atomic Physics

123

Boris Mikhailovich Karnakov Moscow Engineering Physical Institute Moscow Russia

ISBN 978-3-642-31557-2 DOI 10.1007/978-3-642-31558-9

Vladimir Pavlovich Krainov Moscow Institute of Physics and Technology Dolgoprudny Russia

ISBN 978-3-642-31558-9

(eBook)

Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012944377 Ó Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book has evolved from lectures of authors devoted to applications of the WKB (or quasi-classical) approximation and of the method of 1/N—expansion for solving of various problems in atomic and nuclear physics. The intent of this book is to help students and investigators in this field to extend and to make contemporary their knowledge of these important calculational methods of quantum mechanics. It is envisioned that both advanced students and active researchers in this field will find it useful. Much material is contained herein that is not to be found elsewhere. WKB approximation, while constituting a fundamental area in atomic physics, has not been the focus of many books. A novel method has been adopted for the presentation of the subject matter. The material is presented as a succession of problems. These problems are stated succinctly, solved using basic principles of quantum mechanics, and then the results are discussed in detail. It has been our experience that a possible initial discomfort with this unfamiliar structure gives way to an appreciation of its important advantages: First, different aspects of a single topic are treated in separate problems, which makes possible a progressive deepening of the understanding of the subject. Second, by considering limited cases of a general topic, it is possible to simplify the underlying mathematics so as to highlight the fundamental concepts. Third, although some of these problems build progressively on the results of those that precede it, there is also the possibility to enter into the subject matter at any point. Fourth, a very important feature is that the problem/ solution format reinforces in the reader the ability to analyze the content of a physical problem and to apply the suitable mathematical and physical tools to solve it. Finally, the qualitative discussion of the outcome of the solution aids in the development of physical intuition. It is presumed that the reader is already acquainted with the basics and mathematical apparatus of quantum mechanics in the frames of standard university courses described, for example, in the well-known book of Landau and Lifshitz [1].

v

vi

Preface

Detailed investigation of the chapter ‘‘Quasi-Classical Approximation’’ in this book is desirable. The authors wish to thank Professors V. D. Mur and V. S. Popov for many valuable discussions.

Contents

WKB Approximation in Quantum Mechanics . . . . . . . . 1.1 One-Dimensional Motion . . . . . . . . . . . . . . . . . . . . 1.2 WKB-Approximation for a Particle in Central Field. Langer Transformation . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........ ........

1 1

........ ........

16 30

2

1/N-Expansion in Quantum Mechanics . . . . . . . 2.1 1/N Expansion for Energy Levels of Binding 2.2 Wave Functions of 1/n Expansion . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

31 35 49 55

3

Rydberg States of Atomic Systems . . . . . . . 3.1 Unperturbed Rydberg States of Atoms . . 3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

................. .................

57 59

................. .................

75 104

1

..... States ..... .....

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

4

Penetrability of Potential Barriers and Quasi-Stationary States . 4.1 Quasi-Stationary States of One-Dimensional Systems . . . . . . 4.2 Quasi-Stationary States and Above-Barrier Reflection . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

105 107 115 153

5

Transitions and Ionization in Quantum Systems . . . . . . . . . . . . . . 5.1 Adiabatic Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Ionization of Quantum Systems. . . . . . . . . . . . . . . . . . . . . . . .

155 155 161

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

vii

Chapter 1

WKB Approximation in Quantum Mechanics

Wentzel–Kramers–Brillouin (WKB) or quasi-classical approximation is one of the most important approximation methods in quantum mechanics and mathematical physics. This is explained by several factors. Unlike the perturbation theory this approximation does not require smallness of the perturbation potential. Therefore, WKB approximation allows to investigate both qualitative and quantitative properties of quantum–mechanical systems with strong interaction between particles. WKB approximation gives, as a rule, sufficiently simple, obvious from the physical point of view, analytical expressions for investigated physical quantities (energy levels, wave functions, transition probabilities, and so on). This is a useful additional approach to the numerical calculations using modern computers. Though formally WKB approximation is valid for high-excited states of quantum systems, usually its applicability is extended up to the values of quantum numbers n ∼ 1 (including even the ground state) in the case of smooth potentials. Finally, WKB approximation allows to find the so-called correspondence principle between quantum–mechanical and classical quantities and relations.

1.1 One-Dimensional Motion WKB Wave Functions of Stationary States Let us remember the general statements of WKB approximation (see Refs. [1–3]). Stationary one-dimensional Schrödinger equation for the wave function is of the form: 2  ψ (x) + U (x)ψ E (x) = Eψ E (x), − (1.1) 2m E or

B. M. Karnakov and V. P. Krainov, WKB Approximation in Atomic Physics, DOI: 10.1007/978-3-642-31558-9_1, © Springer-Verlag Berlin Heidelberg 2013

1

2

1 WKB-Approximation in Quantum Mechanics

ψ E (x) +

1 2 p (x)ψ E (x) = 0, 2

(1.2)

where the linear momentum of a particle is p(x) =



2m (E − U (x)).

Here, E is the total energy, and U (x) is the potential energy, m is the mass of the particle. Let us introduce the quantity S (it is similar to classical action), which is connected with the wave function by relation:  ψ E (x) ≡ exp

 i S (x, E) . 

We obtain from (1.2) the nonlinear differential equation for this quantity: 

S  (x)

2

− p 2 (x) − iS  (x) = 0.

The solution of this equation is presented as a series in powers of the Planck constant (pure classical case corresponds to the limit  → 0): S = S0 + S1 + 2 S2 + · · ·

(1.3)

The first two terms of this expansion are of the simple form x S0 (x) = ±

p(x  )dx  ; S1 (x) =

i ln p(x). 2

(1.4)

Hence, in WKB approximation, two linearly independent solutions of Eq. (1.1) are ⎤ ⎡ x 1 i ψ± p(x  )dx  ⎦ . exp ⎣± E (x) = √  p(x)

(1.5)

x0

In the classically forbidden region the momentum p(x) is an imaginary quantity so that one of the solutions (1.5) increases exponentially, while the other decreases exponentially. The solutions (1.5) are applicable under the WKB condition  dλ d   p  (x) = = = m U (x)  1. p 3 (x) dx dx p(x) p 2 (x)

(1.6)

The general solution of the Schrödinger equation (1.1) is the superposition of two WKB solutions (1.5):

1.1 One-Dimensional Motion Fig. 1.1 The simplest potential energies

3

(a)

(b) U(x) U(x) E

E

x

x b

a

− ψ E (x) = C1 ψ + E (x) + C 2 ψ E (x).

(1.7)

However, usually the WKB condition (1.6) is violated in some regions of the variable x, for example, near the classical turning points. In this case the problem of matching of WKB solutions on both sides of such regions should be solved. Thus, we must connect the coefficients C1 , C2 in Eq. (1.7) for the different WKB regions of the particle motion on both sides of the turning point.

Matching of WKB Solutions. Kramers Relations We can use matching conditions of WKB solutions for the simplest potential energies depicted in Fig. 1.1a, b. The linear Taylor expansion of these potential energies is assumed to be valid in the vicinity of both classical turning points a, b: U (x) ≈ U (x0 ) + U  (x0 ) (x − x0 ) .

(1.8)

Here, x0 = a (Fig. 1.1a), or b (Fig. 1.1b). We assume also that this expansion is valid up to such values of x on both sides from x0 , at which WKB condition (1.6) is applicable. If we start from the exponentially decreasing solution at x → −∞ in the left classically forbidden region in Fig. 1.1a ⎡ 1 C exp ⎣− ψ E (x) = √  2 | p(x)|

a

⎤ p(x  ) dx  ⎦ , x < a,

(1.9)

x

then the wave function in classical region of the particle’s motion should be written in the form ⎡ x ⎤  1 C π ψ E (x) = √ p(x  )dx  + ⎦ , x > a. sin ⎣ (1.10)  4 p(x) a

4

1 WKB-Approximation in Quantum Mechanics

Fig. 1.2 The potential well

U(x) En

x a

b

Let us underline that both the above functions describe the same solution of Schrödinger equation (1.1), but for different values of the independent variable x (and not too close to the classical turning point x = a ). Equations (1.9) and (1.10) are called Kramers relations [4]. Analogously, if we start from exponentially decreasing solution at x → +∞ in the right classically forbidden region in Fig. 1.1b C

⎡

1 exp ⎣− ψ E (x) = √  2 | p(x)|

x

⎤ p(x  ) dx  ⎦ , x > b,

(1.11)

b

then the wave function in classical region of the particle’s motion can be written in the form ⎡ b ⎤  1 C π p(x  )dx  + ⎦ , x < b. sin ⎣ ψ E (x) = √ (1.12)  4 p(x) x

Bohr-Sommerfeld Quantization Rule Now we suggest that the particle is found in the potential well depicted in Fig. 1.2. Then the wave functions (1.10) and (1.12) correspond to the same binding state of the particle with the energy E. Hence, they must coincide with each other. Thus, the sum of phases of sine functions should be equal to (n + 1)π where n is the integer number (this is so-called quantum number n; it coincides with the number of zeros in the wave function and it determines the ordinal number of the level ). This results in the so-called well-known Bohr-Sommerfeld quantization rule: 1 

b a

here we have

1 pn (x)dx ≡ 



b  1 ; 2m (E n − U (x))dx = π n + 2 a

(1.13)

1.1 One-Dimensional Motion

5

p(a) = p(b) = 0; C  = (−1)n C, n = 0, 1, 2, . . . Let us underline that the value of 1/2 should be added to n in Eq. (1.13) just in the case when the matching conditions of Kramers based upon linear expansion (1.8) for the potential are fulfilled. This value should be changed for other matching conditions (see, for example, Problem 1.1). Usually Eq. (1.13) produces good results for energy eigenvalues E n in the case of smooth potentials not only for large values of the quantum number n 1, but also for n ∼ 1. It should be noted that various methods are used in the derivation of matching conditions for WKB-solutions, and in the derivation of the quantization rule, if linear expansion (1.8) for the potential energy is valid near the turning points. The so-called Zwaan method (see Refs. [1, 2]) is based on avoiding of non-WKB region by means of shifting to the complex plane of the independent variable x. Another method is based on the exact solution of Schrödinger equation (1.1) in non-WKB region. The latter approach is more general, and it is applicable, unlike the Zwaan method, when linear expansion (1.8) for the potential is invalid near the vicinity of the turning point. If Eq. (1.8) is valid, then the exact wave functions are so-called Airy functions (see Ref. [3] and also Problem 1.2). Finally, Problem 1.4 is devoted to the derivation of the quantization rule based on the analytical properties of solutions of Schrödinger equation. Such derivation allows to find also the high-order WKB-expansions with respect to the Planck constant.

Normalization of the WKB Wave Function of Binding State Let us remark that we can use a simpler expression for WKB wave function in order to calculate its normalization factor. Since this function decreases quickly in classically forbidden region, we can simplify its form: ψn (x) =

⎧ ⎨ √Cn ⎩

  x  1 π sin p dx + (x) 4 ,  p(x) a

0,

a < x < b;

(1.14)

x < a, x > b.

We can use Eqs. (1.9) and (1.11) in under-barrier regions x < a, x > b in case of need. However, the normalized coefficient Cn does not change in this case. In order to normalize this function by 1 we must choose

Cn2

2mω (E n ) 2π ; T (E n ) = = 2m = π ω (E n )

b a

dx . p (x, E n )

(1.15)

6

1 WKB-Approximation in Quantum Mechanics

Here, T and ω are the period and the frequency of the finite classical motion of the particle with the energy E n . It should be noted that such normalization gives correct value of the normalized coefficient Cn in the frames of WKB-approximation taking into account two terms in powers of the Planck constant [5]. Let us underline also that in the case of binding states expansion in the Planck constant reduces to expansion in the small parameter 1/n. Quantum–mechanical density of probability |ψn (x)|2 quickly oscillates as a function of x, since n 1. However, these oscillations disappear after averaging over small interval of x. Then we obtain |ψn (x)|2 =

2m , T (E n ) p (x)

that corresponds to classical probability dwclass (x) =

2m dx 2 dt = , a < x < b. T T p (x)

(1.16)

which is determined by time dt required for motion through the interval dx divided by the oscillation period (this interval has been gone twice during one period). Such averaging is achieved by the substitution of the expression sin2 {...} in the probability density by its mean value 1/2. Let us remark on the rules to calculate derivatives of WKB-functions (1.5) and (1.14). We should differentiate only the most quickly variating factor: exp(...) √ in (1.5), (1.9) or sin(...) in (1.10). Differentiation of the pre-exponential factor 1/ p(x) produces terms out of the applicability of the WKB-approximation used above [2].

Higher Corrections in Quantization Rule Finally, we write Bohr-Sommerfeld quantization rule which takes into account the next term of quasi-classical expansion in the Planck constant 1 

b a

⎧ ⎨

∂2

 1 pn (x) dx = π n + + ⎩ 2 24π ∂ E n2

b a

dU dx

2

⎫ dx ⎬ pn (x) ⎭

(1.17)

(see Problem 1.4). Below, we illustrate applications of WKB-approximation on some examples. More detailed consideration of applications of one-dimensional WKB-approximation can be found in Refs. [2, 6–8].

1.1 One-Dimensional Motion Fig. 1.3 The potential in problem 1.1

7 U(0) = ∞

U(x) En

0

b

x

Problems and Solutions Problem 1.1 Obtain the quantization rule for energy spectrum of a particle moving in the potential depicted in Fig. 1.3. Apply this rule to the case of the linear potential U (x) = F x at x > 0 and compare with the exact energy spectrum.

Solution We can use the matching conditions (1.11) and (1.12) for the WKB-solution in the vicinity of the right turning point x = b. However, now the WKB-solution is applicable up to the value of x = 0 ( this is the left turning point, but the momentum of the particle is nonzero in the origin). The boundary condition ψ (0) = 0 for WKB-solution (1.12) on the left infinite wall produces the quantization rule b



3 ; n = 0, 1, 2, . . . p (x) dx = π n + 4

(1.18)

a

This expression differs from the well-known Bohr-Sommerfeld rule (1.13) by a factor of 3/4 instead of 1/2 on the right side. In the case of the potential U (x) = F x we obtain from the above expression:  En =

 1/3

2 2 1/3 9 2 3 2  F π n+ ε0 ; ε0 ≡ . 8 4 m

This WKB-result differs very slightly from the exact solution as well as for small values of n. Indeed, in the case of the ground state (n = 0) we find E 0 = 1.842ε0 , while the exact numerical solution using the Airy functions produces quite near value E 0exact = 1.856ε0 . In the case of the first excited state (n = 1) we find E 1 = 3.240ε0 , while the exact numerical solution produces almost the same value E 1exact = 3.245ε0 .

8

1 WKB-Approximation in Quantum Mechanics

The relative error of the WKB-result decreases quickly with rise of n as ∼ 1/n 2 . It should be noted that we lose essentially the accuracy of calculations at n ∼ 1 by using the quantization rule of Bohr-Sommerfeld (1.13).

Problem 1.2 Calculate the probability to find the particle in the classically forbidden region for the stationary state of discrete energy spectrum. Apply the obtained result to the case of linear harmonic oscillator and compare with exact values for n = 0 and 1 (the most unfavorable cases for applicability of WKB approximation).

Solution The main contribution into the considered probability is given by non-WKB region nearly from the turning points. Therefore, we solve the Schrödinger equation in this region using linear approximation (1.8) for the potential energy (it should be noted that the value of the probability changes for other matching conditions). Nearly from the right turning point x = b we write this equation in the form ψ  (x) −

2m |F (b)| (x − b) ψ (x) = 0, 2

where the notation is introduced |F (b)| = U  (b) and E = U (b). Substituting the independent variable x by

z=

2m |F (b)| 2

1/3 (x − b) ,

we rewrite the Schrödinger equation in the simpler form ψ  (z) − zψ (z) = 0. The solution of this equation which decreases in the classically forbidden region is the so-called Airy function ψ (z) = C Ai (z), and its asymptotic expressions are (see Ref. [9]):  1 √ 1/4 exp (−ζ) , z 1;   (1.19) Ai (z) ≈ 2 πz 1 π √ −z 1. 1/4 sin ζ + 4 , π|z|

Here, the notation is introduced ζ ≡ 2 |z|3/2 /3. It should be noted that since

1.1 One-Dimensional Motion

9

x  p 2 (x) 1 , ζ = p (x) dx , |z| = 2/3  (2mF (b)) a

these asymptotic expressions coincide with the WKB-solutions of Schrödinger equation, and we obtain Kramers matching conditions (1.9)–(1.12) based on the linear expansion (1.8) for the potential in the vicinity of the turning point. Further, we find the normalized coefficient using (1.15) and (1.19): 2π C = T (E)

2

4m 2  |F (b)|

1/3 .

Hence, the probability to find the particle in the right classically forbidden region is ∞ wr =

2π |ψ (x)| dx ≈ T (E)

2

b

2m F 2 (b)

1/3 ∞ Ai 2 (z) dz. 0

Using the known expression for this integral (see the end of this Problem): ∞ Ai 2 (z) dz =

31/3 2  4π 2

 2 , 3

(1.20)

0

we obtain finally the considered probability including both turning points: 31/3 w = wl + wr = 2 2πT (E)

    

2 2m 1/3 2m 1/3 + . 3 F 2 (a) F 2 (b)

(1.21)

Here,  (z)—gamma-function. In particular, for the simple potential of harmonic oscillator U (x) = mω 2 x 2 /2 the energy spectrum is of the well-known form E n = ω (n + 1/2), and the classical period is T = 2π/ω. Then we find F 2 (a) = F 2 (b) = 2mω 3 (n + 1/2). According to Eq. (1.21) we obtain wn =

31/3  2 (2/3) 2π 2 (n

+ 1/2)

1/3

≈

0.134 (n + 1/2)1/3

.

(here we used the value of  (2/3) ≈ 1.354 . . .). It is seen that wn  1 as it should be. Its value decreases with the growing of the quantum number n. We can check that WKB-approximation is of good accuracy even for small values of quantum numbers n ∼ 1. Indeed, we find from the last expression that w0 = 0.169 for the ground state while the exact value is 0.157. Further, w1 = 0.117 while the exact value is 0.112.

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1 WKB-Approximation in Quantum Mechanics

Let us make two concluding remarks. First, since the WKB-expansion in Planck constant is equivalent to the expansion in 1/n for binding states, the dependence wn ∼

1 , n 1, n 1/3

which follows from (1.21), is a quite general result. Second, though the probability wn decreases very slowly with the growing of n, the value of normalized constant Cn derived by means of integration over the classical region only (see Eqs. (1.14), (1.15)), produces higher accuracy of calculation of Cn and ψ (x). The explanation is that neglection by the contribution of the classically forbidden region into Cn is compensated partially by the contribution of the internal part of the non-classical region nearly from the turning point. In order to prove the last statement, we consider the contribution of the classical region x < b nearly from the right turning point into the normalized coefficient Cn , besides the contribution of under-barrier region: ∞

b ψ (x) dx =

b−ε

∞ ψ (x) dx +

2

ψ 2 (x) dx.

2

b−ε

(1.22)

b

Here, the value of ε is chosen so that WKB-approximation is valid at x ≈ b − ε, but we can still use the linear expansion of the potential energy (1.8). Therefore, ψ (x) ≈ C Ai (z) not only at x > b, but also in the interval [b − ε, b]. In order to derive the integrals in Eq. (1.22), we multiply both parts of the equation for Airy function Ai  (z) − z Ai (z) = 0 by Ai  (z) and integrate over z in the interval [0, z]. Integrating by parts, we obtain z

 2  2 Ai 2 (z) dz = z Ai 2 (z) − Ai  (z) + Ai  (0) .

(1.23)

0

It should be noted that it follows from (1.23) the above used value of the integral in Eq. (1.20) at z = ∞, since A (0) = −31/6 (1/3)/2π. Substituting (1.23) into (1.22) we find

∞ ψ (x) dx = C 2

b−ε

2

2 2m |F (b)|

1/3 



Ai  (z)

2

Here, the notation is introduced

2m |F (b)| −z ≡ ε 2

1/3

1.

 − z Ai 2 (z) .

1.1 One-Dimensional Motion

11

Using the known asymptotic expansions for Airy functions (see [9]) including two subsequent terms (the expansion (1.19) is insufficient, since it produces zero result), we obtain

∞ ψ (x) dx = C 2

b−ε

=C

2

2 2m |F (b)|

2 (2m |F

1/3 √

(b)|)1/3 2π

−z = C2 π

b x

2 2m |F (b)|

1/6 √

ε π

dx . p (x)

Here, we have used the value of momentum of the particle nearly of the right turning point x = b  p (x) = 2m |F (b)| (b − x). These relations prove the above statement about the compensation of contributions produced by regions from different sides of the turning point.

Problem 1.3 Using the Bohr-Sommerfeld quantization rule, obtain in the first order WKBperturbation theory the energy shifts of the levels of a particle by small perturbative potential δU (x). Find the relation between obtained result and the result of the quantum–mechanical perturbation theory. What is the interpretation of the obtained result in the classical approach? Consider also, as an example, the shifts of levels for the linear harmonic oscillator due to small anharmonicity δU (x) = βx 4 and compare with the result of the first order quantum–mechanical perturbation theory.

Solution Let us denote E n and E n + δ E n the energies of the levels in the potentials U (x) and U (x) + δU (x), respectively. Expanding the integrand in the Bohr-Sommerfeld quantization rule b+b 





1 2m [E n + δ E n − U (x) − δU (x)]dx = π n + 2

a+a

in small quantities δ E n and δU (x), we obtain the energy shift of the level in the form

12

1 WKB-Approximation in Quantum Mechanics

2m δ En ≈ T (E n )

b

δU (x) (0)

a

pn (x)

dx ≡ δU (x),

(1.24)

where the unperturbed momentum of the particle is defined as pn(0) (x) =



2m [E n − U (x)].

The quantity T (E n ) is the period of motion of the particle in the unperturbed potential U (x). Thus, in the first order WKB-approximation the shift of the energy level is equal to the mean value of the perturbation potential over the period of unperturbed motion of the classical particle. This result coincides with the first order perturbation theory in quantum mechanics [1]: δ E n ≈ E n(1) = n |δU (x)| n . It can be obtained from this expression if we use WKB-expression (1.14) for wave function in the derivation of the matrix element of the perturbation potential and change the quickly oscillating factor sin2 (...) by its mean value of 1/2. Classical interpretation of Eq. (1.24) is based on the so-called adiabatic invariant: it describes variation in the energy of finite motion of a particle (this quantity is non-quantized in the classical approach) at the adiabatic  turn-on of the perturbation potential. As the adiabatic invariant is of the form I = p(x)dx = const (see [10]), so Eq. (1.24) follows from this relation in the case of small perturbation potential δU (x), analogously to the above consideration. According to (1.24) we find for the harmonic oscillator with small anharmonicity δU (x) = βx 4 the next result δ E n = βx 4 (t) =

3 β 2

 mω

2  1 n2 + n + . 4

This WKB-result differs from the exact value by only the last term on the right side which is independent on n. Its exact value is 1/30, instead of 1/4. Coincidence of the first and of the second terms at n 1 is in agreement with the accuracy of the WKBapproximation: it takes into account two orders of expansion in Planck constant. Finally, we mark that the obtained result agrees well with the exact calculation even at n ∼ 1 (excluding the case of the ground state).

Problem 1.4 Calculate the next correction for the Bohr-Sommerfeld quantization rule in Planck constant. Find also the corresponding correction for the energy of binding state in the form:

1.1 One-Dimensional Motion

13

Fig. 1.4 Contour C in complex plane of x

x

Re x a

x1 x2 … xn

b C



  dU 2 1 ∂2 2 E n = E n (B S) + E n ; E n = T (E) . 24m T (E) ∂ E 2 dx

(1.25)

Here, the quantity E n (B S) is determined by the usual Bohr-Sommerfeld rule (1.13), and we average the correction in (1.25) with the classical probability (1.16) at the E = E n (B S). Apply Eq. (1.25) to the case of linear harmonic oscillator and explain the obtained result.

Solution We use the approach which allows to derive corrections for Bohr-Sommerfeld quantization rule in all orders of Planck constant for analytic potentials. This approach is based on the investigation of the equation for logarithmic derivative of the wave function χ(x) = ψ  (x)/ψ(x). This nonlinear equation is obtained from the Schrödinger equation for wave function: 2m [U (x) − E n ] − χ2 (x). 2

χ (x) =

We rewrite this equation in the form  χ(x) = −

2m [U (x) − E n ] − χ (x). 2

(1.26)

(Choice of the sign “minus” is discussed below). Further, we integrate Eq. (1.26) over some contour C in complex plane of x shown in Fig. 1.4 and use the residue theorem (this contour includes zeros of the wave function, i.e., poles of χ (x)). Thus, we obtain χ(x)dx = −

1 



2m [U (x) − E n ] − 2 χ (x)dx = 2πin.

(1.27)

14

1 WKB-Approximation in Quantum Mechanics

Here, we took into account that zeros of wave function are poles of its logarithmic derivative (these poles are shown in Fig. 1.4; they are located between the turning points a and b) the number of poles is equal to n, and a residue for each pole is equal to 1. Such approach is applicable for analytic potentials U (x). It is valid for arbitrary choice of the integration contour C. However, first we choose this contour far from the real interval [a, b]. Then the term with derivative χ (x) in (1.27) is a small correction. Indeed, the wave function is of the form

  1 ψ (x) ∼ sin pdx + γ  nearly of this interval. Then the terms χ (x) and χ2 (x) are the quantities of the same order of magnitude, i.e., they are of the order of p 2 (x)/2 . But when we move the contour into the complex plane of x, the wave function ψ(x) increases exponentially. Then its logarithmic derivative can be written approximately as χ(x) = ψ  (x)/ψ(x) ≈ ±i p(x)/ and consequently

 dp dλ 1 dχ ∼ 2 ∼  1. 2 χ dx p dx dx

Hence, Eq. (1.26) can be solved by iteration procedure: χ = χ(0) + χ(1) + χ(2) + · · · . We find first three iteration terms: 1 dχ(0) ; χ(0) = −i pn (x)/; χ(1) = − (0) dx 2χ ⎡ ! "2 ⎤ 2 (1) 2 (0)  dχ 1  dχ ⎦ − χ(2) = χ(0) ⎣ 2 2 pn dx 2 2 pn2 dx

(1.28)

and so on. Here, the notation is introduced for momentum of the particle pn (x) = √ 2m [E n − U (x)]. It should be noted that the turning points a and b are branching singularities of pn (x) in the plane of complex variable x. We consider this momentum on the sheet of Riemann surface where the cut has been made along the path between branching points a and b. Thus, the quantities pn (x) and χ(x) are single-valued functions on contour C. We put also pn > 0 on the upper side of the cut; then pn < 0 on the lower side of the cut. On the interval x > b of the real axis of the variable x we have i pn = | pn |; here χ(x) < 0 for x > b as it should be. This is in agreement with the choice of the sign in Eq. (1.26). Substituting now the obtained expansion (1.28) into Eq. (1.27), we obtain:

1.1 One-Dimensional Motion

1 

b pn (x)dx ≡ −

15

1 2

pn (x)dx = πn +

i # 2

χ(s) (x)dx.

(1.29)

s=1 C

a

This relation presents, in principle, the solution of the problem of how to calculate the corrections of arbitrary orders in WKB-quantization rule for energies of discrete levels. Let us underline that the expansion (1.28) for χ(x) is valid if the integration contour C is sufficiently far from the interval [a, b]. However, after substitution of expansion for the function χ(x) into Eq. (1.29) we can again change the integration contour C arbitrarily (but without crossing the cut along the interval [a, b]). The values of integrals do not depend on the choice of the integration contour, and they remain quasi-classical smallness which takes place far from the cut. Usually the contour C can be chosen nearly from the real axis; it was used on the left side of Eq. (1.29). Now we consider the derivation of the right side in Eq. (1.29). We have for the first term with s = 1: i 2

χ(1) dx = −

i 4

dx χ(0) (x)

dχ(0) i =− dx 4

d ln pn (x) =

π . 2

This term together with the term nπ in (1.29) produces the well-known BohrSommerfeld quantization rule (1.13). It should be noted that we have used here the relation lnz = ln |z| + i arg z and took into account that the phase of the function pn (x) changes by 2π during the motion around the contour C (the phase π appears in the way around each of the two root branching points). The next term with s = 2 determines the correction to the quantization rule (1.13): 1 

b a

⎡  ∂2 1 pn (x)dx = π ⎣n + + 2 24π ∂ E n2

b a

dU dx

2

⎤ dx ⎦ . pn (x)

(1.30)

In order to obtain this formula we integrated the corresponding term with χ(2) in Eq. (1.29) by parts and then we used the relation

dU dx

2

dx 1 ∂2 = 3m 2 ∂ E 2 pn5

dU dx

2

dx 2 ∂2 =− 2 p(E, x) 3m ∂ E 2

b

dU dx

2

dx . p(E, x)

a

Further, the last term on the right side of Eq. (1.30) is a small correction. Therefore, we can write on the left side of Eq. (1.30) the energy in the form E n = E n (B S) + E n , where E n is the correction to the energy derived from Bohr-Sommerfeld quantization rule. Expanding the left side of Eq. (1.30) in small quantity E n and neglecting this quantity in other parts of Eq. (1.30), we obtain the required result (1.25) for the energy shift in the second order of expansion in Planck constant. It

16

1 WKB-Approximation in Quantum Mechanics

should be noted that in Eq. (1.25) we defined the quantity

dU T (E) dx

b

2 =2

dU dx

2

a

dx . √ 2m(E − U (x))

In the example of harmonic oscillator we have U (x) = mω 2 x 2 /2 and dU/dx = mω 2 x, so that (dU/dx)2 = mω 2 E. Since T = 2π/ω = const, according to Eq. (1.25) we obtain E n = 0. This result is obvious, since the Bohr-Sommerfeld quantization rule (1.13) gives exact values of the energies for harmonic oscillator; hence, all corrections must vanish.

1.2 WKB-Approximation for a Particle in Central Field. Langer Transformation Radial Schrödinger Equation The wave function of the stationary Schrödinger equation for a particle in a central potential U (r ) can be presented in the form  Elm (r) = R El (r )Ylm (n) due to commutativity of the Hamiltonian with the angular momentum operators l z and l2 . The radial equation is   2 1 d 2 2 l(l + 1) − r+ + U (r ) R El (r ) = E R El (r ). 2m r dr 2 2mr 2 Substitution R El (r ) = χ El (r )/r allows to obtain the usual one-dimensional Schrödinger equation (1.1): Hl χ El

  2 d 2 ≡ − + Ueff (r ) χ El = Eχ El . 2m dr 2

(1.31)

We define here the effective potential energy Ueff (r ) = U (r ) +

2 l(l + 1) . 2mr 2

(1.32)

The boundary condition in the origin is χ(0) = 0. However, direct application of one-dimensional WKB-approximation considered in the previous section is usually quite restricted. It is explained by the fact that at

1.2 WKB-Approximation for a Particle in Central Field. Langer Transformation

17

l ∼ 1 (but l = 0) the centrifugal potential in Eq. (1.32) is non-quasi-classical √ at small values of r where this potential dominates. Indeed, we have dλ/dr = 1/ l(l + 1) for this potential, so that the WKB-condition is fulfilled only at l 1. Thus, WKBapproximation produces incorrect form of the radial wave function nearly from the origin (r → 0): √ l(l+1)

χ El (r ) ∼ r 1/2+

instead of correct form χ El (r ) ∼ r l+1 .

Besides this, the Kramers matching conditions (1.9) and (1.10) for the left turning point nearly of the origin also become incorrect. Hence, in the Bohr-Sommerfeld quantization rule with the effective potential (1.32) the term 1/2 becomes incorrect. The correct value of this term is important at small and moderate values of n which are the most interesting in applications.

Langer Transformation A very nice method to avoid the above difficulties produced by non-quasi-classical centrifugal potential nearly of the origin was suggested by Langer [11]. Let us change the independent variable r and the function χ El (r ): r/d = exp(x); ψ(x) = exp(−x/2)χ El (r ). Here, d is the parameter with the dimension of the length. The value of this parameter is inessential for the suggested approach. Usually the quantity d is chosen as the radius of the potential U (r ). Then Eq. (1.31) can be rewritten in the form (we put below m =  = d = 1): d2 ψ(x) + P 2 (x)ψ (x) = 0; dx 2 $ P (x) ≡



  1 2 + 2 E − U (exp (x)) exp(2x). (1.33) − l+ 2

This equation is also a one-dimensional Schrödinger equation,  but with the total  energy −(2l + 1)2 /8 and with the new effective potential energy U (exp (x)) − E · exp (2x). The origin r = 0 is transformed into the infinite point x = −∞. It is very important that WKB-approximation is valid for large values of |x|. We obtain immediately that ψ(x)

∼

x→−∞

exp [(l + 1/2) x] and χ El (r ) ∼ r l+1 ,

as it should be in the exact solution.

r →0

18

1 WKB-Approximation in Quantum Mechanics

Usually in order to match WKB-solutions (1.33) in the vicinity of the left classical turning point, Kramers conditions (1.9) and (1.10) are used. Then the quantization rule (1.13) with p(x) = P(x) (see Eq. (1.33)) after returning to the initial independent variable r takes the form 1 

b



1 ; nr = 0, 1, 2, . . . p L (r )dr = π nr + 2

(1.34)

a

where the notation is introduced %   & & 2 (l + 21 )2 ' p L (r ) = 2m E nl − U (r ) − . 2mr 2

(1.35)

This is the so-called quantization rule with Langer correction. This correction consists in change of the factor l(l + 1) in the centrifugal potential by the factor (l + 1/2)2 . This factor appears as a result of correct mathematical operations. It is interesting that yet Kramers showed that formal application of WKB-approximation for Eq. (1.31) with such modification of the centrifugal potential deletes the above cited disadvantage of the WKB-approximation.

Disadvantages of Langer Method Strictly speaking, Langer approach is needed in some making more precise. It is incorrect to use the linear expansion of the potential (1.8) in the vicinity of the left turning point for Eq. (1.33) at l ∼ 1. The non-quasi-classical region is much more extensive. This statement can be checked in the example of the Coulomb potential. Substituting this potential U (r ) = −1/r = − exp(−x) into Eq. (1.33) and neglecting the term with total energy E ∼ n −2 → 0, we obtain the equation determining the left turning point x = A : √ (2l + 1)2 = 8 exp(A). If the linear expansion is valid, we find P(x) ≈ (l + 1/2) x − A and |x − A|  1. Hence, the WKB-condition (1.6) takes the form |x − A|3/2 (2l + 1)−1 . Both inequalities for the difference |x − A| can be fulfilled simultaneously in the case (2l + 1) 1 only. Thus, in the case l ∼ 1 Kramers matching conditions (1.9) and (1.10) of one-dimensional motion are violated for Eq. (1.33). Below, we obtain some generalizations of Kramers matching conditions (1.9) and (1.10) for WKB-solutions of Eq. (1.33) in the vicinity of the left turning point x = A (again m =  = d = 1):

1.2 WKB-Approximation for a Particle in Central Field. Langer Transformation

⎧   ⎪ x ⎪ C ⎪ P(x)dx + γ , ⎪ ⎨ √ P(x) sin A   ψ (x) = ⎪ A  ⎪ C ⎪ ⎪ ⎩ √|P(x)| exp − |P(x)| dx ,

19

x > A; (1.36) x < A.

x

We assume that the potential energy satisfies the condition r 2 U (r ) → 0 at r → 0.

(1.37)

Modified Matching Conditions for WKB-Solutions Matching of WKB-solutions (1.33) and (1.31) can be made if the exact solution is known in the non-quasi-classical region. Of course, it is impossible to obtain the general analytical solution. The situation is simplified at large values of the energy E, when the left turning point A → −∞ (and a → 0 in Eq. (1.31)). It is explained by the fact that the potential energy U (r ) can be considered as small perturbation under condition (1.37). Neglecting this term, we obtain the solution of Eq. (1.33) via Bessel functions: ) * 2E exp(x) . (1.38) ψ (0) (x) = c Jl+1/2 This solution allows to match asymptotic WKB-solutions (1.36). It should be noted that the solution (1.38) is corresponding to the solution (0)

χ El ∼

√ r Jl+1/2 (kr )

of Eq. (1.31) with U (r ) = 0 and E = k 2 /2. Using the well-known asymptotic expression for Bessel function  Jν (z) ≈

 πν π 2 sin z − + , z ν = l + 1/2 ∼ 1 πz 2 4

and finding that in Eq. (1.38) z≡ and simultaneously

√ 2E exp(x) ≈ P(x) at z ν

 √ 1 2E exp(x) ≈ P(x)dx + πν, 2 x

A

(1.39)

20

1 WKB-Approximation in Quantum Mechanics

we find that at z ν the solution (1.38) has the WKB-form for classically permitted region of motion: ⎧ x ⎫ ⎨ ⎬ 2 π sin ψ (0) (x) = c P(x)dx + . ⎩ π P(x) 4⎭ $

(1.40)

A

and the phase has its usual value, i.e., γ = π/4 as in (1.10). On the other hand, we find, using the expansion of the Bessel function at small values of its argument: Jν (z) ≈

) z *ν 1 ,  (ν + 1) 2

z→0

and the value of the integral is A |P(x)| dx = ν ln

−

)√ * 2E exp(x) + ν − ν ln(2ν), x → −∞,

x

so that the solution of Eq. (1.38) at x → −∞ is of the WKB-form for classically forbidden region: ⎫ ⎧ A ⎬ ⎨  ψ (0) (x) = c exp − |P(x)| dx . √ ⎭ ⎩  (ν + 1) |P(x)| ν ν+1/2 exp(−ν)

(1.41)

x

Here, |P(x)| ≈ l + 1/2. Comparing Eqs. (1.36), (1.40), and (1.41), we obtain the required matching condition of WKB-solutions of Eq. (1.33) at large values of the energy, i.e., E |U (a)|, √ where a = ν/ 2E : C 1 π = ξ (ν), γ = , (1.42) C 2 4 The notation is introduced here for the function ξ (ν) ≡

√ ν ν−1/2 exp(−ν) . 2π  (ν)

(1.43)

This matching condition is valid also for the solution χ El (r ) of the radial Schrödinger equation (1.31). In this case we change in Eq. (1.36) ψ by χ and the momentum P(x) by WKB-momentum p L (r ) (1.35) with Langer correction. It should be noted that the function (1.43) √ simpler √ can be approximated by the expression. Indeed, we have ξ (1/2) = 2/e = 0.8578 and ξ (1) = 2π/e = 0.9221. Further

1.2 WKB-Approximation for a Particle in Central Field. Langer Transformation

ξ (ν) = 1 −

21

1 1 + + ... at ν 1. 12ν 288ν 2

This expression allows to derive the values of ξ (ν) with the accuracy of 10−4 for ν ≥ 3. The obtained matching condition reduces to Kramers condition at l 1, as it should be. However, at moderate values of l just the conditions (1.42) are asymptotically exact at E → ∞. It gives higher precision of the wave function in classically forbidden region using WKB-approximation at the finite values of the energy E. Let us underline also that the former value of the phase γ = π/4 does not produce any modification of the quantization rule (1.34) with Langer correction, though Kramers matching condition is inapplicable. Illustration of the obtained results is given in Problem 1.5.

Matching Conditions for Long-Range Power Potentials Let us obtain the matching condition for the case of power attractive potentials of the form g g > 0, 0 < α < 2. U (r ) = − α ; r The case of the Coulomb potential is also included into this consideration (αC = 1). The number of levels in such potentials is infinite for any value of orbital quantum number l, because of their rapprochement at E n → 0. We can neglect the term with energy E in Schrödinger equations (1.31) and (1.33) for high-excited states at small values of r . Hence, the solution is of the form   2l + 1 1  (0) . 8g exp ((2 − α) x) ; μ ≡ ψ (x) = c Jμ 2+α 2−α Analogously to the previous case we find the matching condition C 1 = ξ (μ) ; C 2

γ=

π . 4

Here, the function ξ (μ) is given by Eq. (1.43). In particular, we have C  /C = + 1) for the Coulomb potential. It should be noted also that the value of the phase γ again does not change. 1 2 ξ (2l

22

1 WKB-Approximation in Quantum Mechanics

The Case of l = 0 Langer method is applicable for any value of the orbital quantum number l, in particular, for l = 0. However, this case (absence of the centrifugal barrier) is often better to be considered using the radial Eq. (1.31). If the potential U (r ) is satisfied for WKB-conditions at small values of r , then the WKB-energy spectrum of s-states can be derived from the quantization rule 1 

b 



3 2m [E n0 − U (r )]dr = π nr + 4

(1.44)

0

(compare with the solution of Problem 1.1). With the same accuracy of the order of 1/nr2  1 this energy spectrum can be derived using the quantization rule (1.34) with Langer correction for l = 0. The reader can check that in the case of spherical harmonic oscillator (the potential is U (r ) = mω 2 r 2 /2 ) both approaches give exact results for s-levels: 

3 . E nr l=0 = ω 2nr + 2 However, if the potential U (r ) is non-quasi-classical at small distances r (for example, Coulomb potential U (r ) = −g/r ), then Eq. (1.44) is inapplicable. In such cases we must find the exact solution of the radial Schrödinger equation at small distances. Langer approach for s-states is valid also for such potentials. Various modifications of quantization rules and their applications are considered, for example, in Refs. [7, 8]. Here, we restrict ourselves to discussion of two problems only. The first problem presents some illustration of the above modifications of matching conditions. The second problem is devoted to falling of the particle to the origin (see [1], Sect. 35).

Problems and Solutions Problem 1.5 Find the WKB-expression for the asymptotic coefficient Cnl in the expansion of the normalized radial wave function in the origin: χnr l (r ) = Cnl r l+1 + · · · , r → 0. for binding states of the particle in the central potential. Illustrate the obtained results on the examples of Coulomb potential and of the potential of spherical harmonic oscillator.

1.2 WKB-Approximation for a Particle in Central Field. Langer Transformation

23

Solution Using the Langer approach considered above, we can write normalized wave function in the classical region and in the classically forbidden region nearly of the origin (below  = m = 1): ⎧ r ⎫ $ ⎨ 2ωr π⎬ sin p L (r ) dr + , a < r < b; χnr l (r ) = ⎩ π p L (r ) 4⎭ a

$ Al χnr l (r ) = 2

⎫ ⎧ a ⎬ ⎨  2ωr exp − | p L (r )| dr , r < a. ⎭ ⎩ π | p L (r )|

(1.45)

r

Here, a, b are the left and right radial classical turning points, respectively; p L (r )—is the WKB momentum of the particle with Langer correction determined by Eq. (1.35); further, the frequency of classical radial oscillations of the particle is given by 2π π = b . ωr = dr Tr a p L (r )

We have used in Eq. (1.45) the value of γ = π/4 for phase; therefore the energy levels are determined from the quantization rule (1.34). Finally, we have introduced into the wave function in Eq. (1.45) the factor Al which can appear due to change of Kramers matching conditions. Nearly of the origin r → 0 we have | p L (r )| ≈ (l + 1/2) /r. Then we can write the exponent in Eq. (1.45) in the form: a | p L (r )| dr = r

, a + a l + 1/2 | p L (r )| − dr + (l + 1/2) ln . r r r q

WKB . Here, Then we obtain WKB-expression for the coefficient Cnl = Al Cnl

 WKB = Cnl

⎫ ⎧ a  ⎬ ⎨   1 ωr l + 1/2 | p L (r )| − dr exp − ⎭ ⎩ π (2l + 1) a l+1/2 r

(1.46)

0

is the value of this coefficient derived by using Kramers matching conditions. Let us illustrate the obtained results on the examples of known exact analytical solutions. In the case of spherical harmonic oscillator U (r ) = ω 2 r 2 /2, the quantization rule taking into account Langer correction gives the energy spectrum E N = (N + 3/2) ω, where N = 2nr +l. This formula coincides with the exact solution. We find ωr = ∂ E N /∂nr = 2ω. Further, the correcting factor Al = ξ (l + 1/2)

24

1 WKB-Approximation in Quantum Mechanics

WKB Table 1.1 ηnl

l l l l

=0 =1 =2 =5

nr = 0

nr = 1

nr = 2

nr = 5

nr = ∞

1.1469 1.0293 1.0039 0.9826

1.1620 1.0493 1.0251 1.0045

1.1642 1.0531 1.0295 1.0095

1.1655 1.0555 1.0325 1.0133

1.1658 1.0563 1.0337 1.0153

nr = 0

nr = 1

nr = 2

nr = 5

nr = ∞

0.9838 0.9744 0.9711 0.9678

0.9967 0.9934 0.9917 0.9895

0.9986 0.9969 0.9959 0.9943

0.9997 0.9992 0.9988 0.9981

1 1 1 1

q

Table 1.2 ηnl l l l l

=0 =1 =2 =5

according to Eq. (1.42). In order to calculate the integrals in Eq. (1.46) (see [12]), we substitute first the lower limit 0 by small value ε > 0 and then make limit ε → 0. Thus, we obtain + WKB Cnl

=

n n exp (ν) 1 π (nr + 1/2)nr +1/2 ν 2(l+1)

,1/2

ω (2l+3)/4 ,

where n = nr + l + 1 is the principal quantum number and ν = l + 1/2, as above. Exact wave function of spherical oscillator is of the form * ) exact l+1 r exp(−ωr 2 /2) · F −nr , l + 3/2, ωr 2 , χnr l (r ) = Cnl where F (α, β, z) is the degenerate hypergeometric function, and the exact value of the coefficient is exact Cnl

ω (2l+3)/4 =  (l + 3/2)

+

, 2 3 1/2  nr + l + . nr ! 2

Here,  (x) is the gamma function. Comparison of the values of C exact , C q and C WKB is given in Tables 1.1 and 1.2. The ratios q q WKB WKB exact exact = Cnl /Cnl and ηnl = Cnl /Cnl (1.47) ηnl are shown in Tables 1.1 and 1.2, respectively. It is seen that only with the use of the correcting factor Al WKB-approximation allows to obtain asymptotically exact result at nr → ∞. Besides this, the accuracy of WKB-calculation is improved also for nr ∼ 1.

1.2 WKB-Approximation for a Particle in Central Field. Langer Transformation

25

WKB Table 1.3 ηnl

l=0 l=1 l=2

nr = 0

nr = 1

nr = 2

nr = 5

nr = ∞

1.0602 0.9974 0.9844

1.0787 1.0189 1.0063

1.0819 1.0235 1.0112

1.0838 1.0267 1.0150

1.0844 1.0281 1.0162

nr = 0

nr = 1

nr = 2

nr = 5

nr = ∞

0.9776 0.9702 0.9681

0.9947 0.9911 0.9897

0.9977 0.9955 0.9945

0.9994 0.9987 0.9982

1 1 1

q

Table 1.4 ηnl l=0 l=1 l=2

Analogous conclusions occur in the case of Coulomb potential U (r ) = −g/r. WKB-quantization rule with the Langer correction results in exact Balmer formula E n = −g 2 /2n 2 ; n = nr +l +1 is the principal quantum number; hence, ωr = g 2 /n 3 . We obtain according to Eq. (1.46): WKB = Cnl

ν 2l+3/2 gl+3/2 exp (ν) (n + ν)(n+ν)/2 . √ l+1 l+2 π2 n (n − ν)(n−ν)/2

The exact radial wave function is given below by Eq. (1.2) so that the exact value of the asymptotic coefficient is exact Cnl

2l+1 gl+3/2 = l+2 n (2l + 1)!

$

(n + l)! . nr !

We have seen above that for Coulomb potential the correcting factor is Al = WKB = C WKB /C exact is given in Table 1.3 and the ratio ξ (2l + 1). The ratio ηnl nl nl q q exact ηnl = Cnl /Cnl is given in Table 1.4. It should be noted that in Problem 3.5 another expression is obtained for the coefq ficient Cnl . In that problem we do not use matching of WKB-solutions with Langer correction, but solve exactly the Schrödinger equation for radial wave function at small values of the radial variable r in all non-classical regions. Though the obtained expression in Problem 3.5 is asymptotically exact at nr → ∞, it has much more q inaccuracy at nr ∼ 1 and l = 0, compared with the above derived quantity Cnl . The reader can compare the results of this problem with the derivation of the coefficient Cnl for nr = 0 by the method of 1/n-expansion (see Problem 2.7). This coefficient is an important parameter for systems with two potentials whose radii differ strongly each from other, for example, Coulomb and nuclear potentials for hadron atoms.

26

1 WKB-Approximation in Quantum Mechanics

Problem 1.6 Investigate the falling of a particle to the origin in WKB-approximation for singular attractive potentials U (r ) = −g/r α at r → 0, α > 2.

Solution The specific property of such potentials is that both linearly independent solutions of radial Eq. (1.31) for any value of l have the same form at small values of r : χ El

⎧ r ⎫ ⎨1  ⎬ C =√ p(r )dr + γl , sin ⎩ ⎭ p(r )

(1.48)

r0

√ whereas sin(…) oscillates strongly at r → 0. Since 1/ p(r ) ∼ r α/4 , then χ2El ∼ r α/2 → 0, and normalized integral is convergent at small values of r → 0. These properties differ strongly from properties of radial wave functions for regular potentials having α < 2. In the latter case one of the independent solutions is equal to zero, χ1l ∼ r l+1 , while the other is of the form χ2l ∼ r −l at r → 0, i.e., it is divergent in the case of l ≥ 1. Therefore, the second solution is excluded from consideration due to physical reasons. In the case of the singular potentials considered we can always obtain the exponential decreasing of the radial wave function in the classically forbidden region at r → ∞ by the choice of the parameter γl in Eq. (1.48). Then we could make the conclusion, at first glance, that the energy spectrum of a particle is continuous as well as at E < 0. Of course, the energy spectrum at E < 0 is of a discrete form in the case considered. In order to understand the physical peculiarities of this problem, we cut the potential on small, but finite value of r0 . Then we can go to the limit r0 → 0. For the sake of simplicity we take such cutting in the form of non-penetrable sphere of the radius r0 . Then the boundary condition is χ El (r0 ) = 0,and in Eq. (1.48) we should put γl = 0. Thus, we obtain the quantization rule 

 b $  3 g 2 (l + 1/2)2 dr = π n + . 2m E nr l + α − r 2mr 2 4

(1.49)

r0

Here, we assume that U (r ) = −g/r α for all values of r > r0 . The energy spectrum E nr l derived from Eq. (1.49), depends on the choice of the cutting parameter r0 . When the value of r0 diminishes the level with the fixed value of nr is going down, and E nr l → −∞ at r0 → 0 for this level. Further, inside any fixed energy interval [E, E + E] with E < 0 we find new levels with larger values of nr , and nr → ∞ at r0 → 0 for these levels. Though the energies of these levels

1.2 WKB-Approximation for a Particle in Central Field. Langer Transformation

27

depend on r0 , and the limit of the energies at r0 → 0 does not exist, the interval between the neighboring levels is finite; it is equal to ω (E) =

π 2π ∂ E nr l = b ; and ω (E) = . mdr ∂nr T (E) 0 p(r )

Thus, the energy spectrum at r0 → 0 is a discrete one, but the energies of the levels depend on the details of the limiting procedure. Therefore, additional conditions are needed for single-valued determination of the energies of levels. We will find that if one level is fixed only for each value of l, then other levels are determined singlevalued. Indeed, we should write the equation which is analogous to Eq. (1.49), but with other values nr of quantum number nr . Taking the difference of these equations, we obtain the relation where the limit r0 → 0 is possible: b $

  b $  g 2 (l + 1/2)2 g 2 (l + 1/2)2 dr − dr 2m E + − 2m E nl + α − 0l r 2mr 2 rα 2mr 2 

r0

r0

= πn at r0 → 0.

(1.50)

Here, E nr l is substituted by E nl , the notation is introduced n ≡ nr − nr and the energy E 0l = E n  l determines the position of one of the levels for each value of l. r The quantity E 0l < 0 is a free parameter which can be chosen arbitrarily. When this quantity is determined, energies of all other levels are found single-valued (by means of the quantization rule (1.50) in WKB-approximation). The number of levels is infinite, and E nl → −∞ at n → ∞. It should be noted that the number of levels lying above E 0l is finite in the case of α > 2. Let us underline that now quantum number nr is absent, since nr = ∞ due to rapprochement of zeros of radial wave function (1.48) at r → 0. The above considered physical approach for quantization of energy levels in the case of singular potentials has correct mathematical explanation. This approach is connected with the specific boundary condition for solutions of radial Schrödinger equation. In order to obtain such condition, we find first the phase of sine function in radial wave function (1.48) at r → 0 : ⎧ r ⎫  ⎨1  ⎬ 2mg 1 2 pdr + γl ≈ − + γ˜ l . (1.51) lim ⎭ r →0 ⎩  α − 2 2 r α/2−1 r0

√ We take into account here that p (r ) ≈ −2mU (r ) at r → 0. Now we check the Hermite property of the Hamiltonian. Integrating by parts and using (1.51) we obtain

28

1 WKB-Approximation in Quantum Mechanics

∞

χ∗2 Hˆ l χ1 dr

=

0

∞ )

Hˆ l χ2

*∗

χ1 dr −

 ∞ 2  ∗  χ χ − χ1 χ∗ 2 r →0 . 2m 2 1

χ1 dr +

 ∗   sin γ˜ l2 − γ˜ l1 . 2m

0

=

∞ )

Hˆ l χ2

*∗

0

The operator Hl is determined in Eq. (1.31). The condition of hermicity of this oper ∗ − γ˜ l1 = 0. It follows from this relation that the phase ator requires that sin γ˜ l2 γ˜ l in Eq. (1.51) should be the same real quantity for all radial wave functions with the given orbital quantum number l: γ˜ l = const (it does not depend on E). This is just the required boundary condition. From the mathematical point of view, this additional boundary condition determines so-called self-conjugated generalization of the operator Hˆ l in the case of singular potentials (see Ref. [7] for details). From the physical point of view, such generalization corresponds to the fixed position of one of the levels. Let us illustrate the above approach on the example of the potential U (r ) = −g/r 2 . In this case the Schrödinger equation has the exact solution: √ χ El = C r K iνl (κr ), where E < 0, K ν (z) is the McDonald function, and  κ=

$ 2m E − 2 ; 

νl =



2mg 1 2 − l + -real quantity 2 2

(in the opposite case binding states are absent). For small values of r we have   √ κr + arg  (−iνl ) . χ El = C | (iνl )| r cos νl ln r →0 2

(1.52)

Since such radial dependence of the wave function must be the same for all levels, i.e., it does not depend on κ, we obtain the exact quantization rule in the form 

κn 2πn ; n = 0, ±1, ±2, . . . = −πn; or E nl = E 0l exp − νl ln κ0 νl

(1.53)

All levels of the discrete spectrum are determined single-valued by the value E 0l of one of these levels for each value of the orbital quantum number l. Let us underline that the number of levels is infinite both because of possibility to fall to the origin, n → −∞, and due to usual condensation of levels E n = 0 at n → +∞, connected with slow depletion of the potential U (r ) = −g/r 2 on large distances. It should be noted that the quantization rule (1.50) with Langer correction gives the exact expression for the energy spectrum in the case of α = 2.

1.2 WKB-Approximation for a Particle in Central Field. Langer Transformation

29

In conclusion, let us underline that the above condition of self-conjugated generalization of Hamiltonian under the condition of falling of the particle to the origin is valid not only for bound states, but also for continuum states (at E > 0). We should take into account this circumstance in the derivation of phase shifts for scattering of particles. For example, in the case of potential U (r ) = −g/r 2 the solution of Schrödinger equation for the continuum state with the energy E = 2 k 2 /2m and orbital quantum number l, satisfying the boundary condition (1.52) at r → 0 with κ = κ0l , is of the form + , ) κ *−iνl √ ) κ0l *iνl 0l Jiνl (kr ) − J−iνl (kr ) . χkl (r ) = C r k κ Here, Jν (z) is the Bessel function. The asymptotic limit of this expression at r → ∞ should coincide with general representation of the radial wave function of the scattering problem ([1], Sect. 123):   χkl (r ) = D (−1)l+1 exp(−ikr ) + Sl exp(ikr ) ,

r → ∞.

Here, Sl = exp(2iδl ) is the element of S−matrix for l-th partial wave. Comparison of both expressions produces the result Sl (k) = i(−1)l+1 exp(iηl )

1 − exp (−πνl − iηl ) . 1 − exp (−πνl + iηl )

(1.54)

Here, the notation is introduced ηl ≡ 2νl ln (κ0l /k). Let us make some concluding remarks. First, it is seen that |Sl | = 1. Thus, absorption is absent in the considered system, i.e., particles return back after falling to the origin. Second, the function Sl (E) has poles on the physical sheet of the complex plane of the energy E. These poles E nl are located on the semi-axis of real negative values of E: 2 κ2nl ; κnl > 0, E nl = − 2m k = iκ and lnk = ln κ + iπ/2. The condition Sl (E nl ) = ∞ requires vanishing of denominator in Eq. (1.54). This condition gives the relation πνl − iηl = 2πin; it follows from this relation Eq. (1.53) for the energy spectrum. Thus, locations of poles of S−matrix coincide with the energies of bound levels, as it should be. Finally, we can mark the peculiarities of the function Sl (k) and of the scattering phases δl (k) in the cases of slow particles (k → 0) and rapid particles (k → ∞). These peculiarities are connected with the singular property of the considered potential.

30

1 WKB-Approximation in Quantum Mechanics

References 1. L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd edn. (Pergamon, Oxford, 1977) 2. A.B. Migdal, Qualitative Methods in Quantum Theory (Benjamin, New York, 1977) 3. L.I. Schiff, Quantum Mechanics, 2nd edn. (McGraw-Hill, New York, 1955) 4. H.A. Kramers, Z. Phys. 39, 828 (1926) 5. W.H. Furry, Phys. Rev. 71, 360 (1947) 6. A.B. Migdal, V.P. Krainov, Approximation Methods in Quantum Mechanics (Benjamin, New York, 1969) 7. V.M. Galitsky, B.M. Karnakov, V.I. Kogan, Problems in Quantum Mechanics, 2nd edn. (Nauka, Moscow 1992, in Russian) 8. B.M. Karnakov, V.P. Krainov, Quasi-Classical Approximation in Quantum Mechanics (WKBMethod) (MEPhI Publishing, Moscow 1992, in Russian) 9. M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions (National Bureau of Standards, Washington 1964) 10. L.D. Landau, E.M. Lifshitz, Mechanics (Pergamon, Oxford, 1977) 11. R.E. Langer, Phys. Rev. 51, 669 (1937) 12. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965)

Chapter 2

1/N-Expansion in Quantum Mechanics

The method of 1/N -expansion has been developed recently. It is used in different branches of theoretical physics. The idea of this method is the seeking of some parameter N  1 for the considered system. The solution of the problem is achieved by the expansion in small parameter 1/N . Sometimes the applicability of the solution is good even up to values of N ∼ 1. For example, WKB-approximation produces often result with high accuracy and at the quantum numbers n ∼ 1, though formally the condition of its applicability is n  1. Choice of parameter N is usually connected with the increasing of number of states, or degrees of freedom of the considered system, dimension of space, and so on. Below we consider some simple examples of such approach for the solution of single-particle Schrödinger equation. 1/N Expansion for Deep One-Dimensional Potential Well Let us consider the discrete states of a particle in one-dimensional potential well U (x) = U0 v(x/d) with the width d and with the one minimum in the point x0 (see Fig. 1.2). The function v determines the form of the potential well. The properties of such system depend on the value of the dimensionless parameter  N=

md 2 U0 . 2

(2.1)

The number of bound states increases with the growing of this parameter. In the case of N  1 wave functions of low-lying levels are located nearly of the minimum of the potential, where U  (x0 ) = 0. We can expand the potential in powers of (x −x0 )/d (below we put in this Section m = d =  = 1): 1 U (x) = U (x0 ) + ω 2 (x − x0 )2 + α (x − x0 )3 + β (x − x0 )4 + · · · 2

B. M. Karnakov and V. P. Krainov, WKB Approximation in Atomic Physics, DOI: 10.1007/978-3-642-31558-9_2, © Springer-Verlag Berlin Heidelberg 2013

(2.2)

31

32

2 1/N-Expansion in Quantum Mechanics

where ω=

 U  (x0 );

α=

1  U (x0 ); 6

β=

1 (I V ) . U 24

(2.3)

Here, the third and next terms of expansion are small perturbations of the harmonic oscillator with the frequency ω. The perturbation series for the energy E n in powers of anharmonicity coefficients is (see [1], Sect. 38)       15 α2 3 β 1 11 1 2 2 n n − + +··· E n = U (x0 ) + ω n + + n + + n + 2 4 ω4 30 2 ω2 2 (2.4) This is typical example of 1/N-expansion for the energy levels in the considered problem. We have in this expansion: U (x0 ) ∼ N 2 ; ω ∼ N ,

α2 β ∼ 2 ∼ 1 = N 0 and so on. 4 ω ω

Such expansion is asymptotically exact at N → ∞. However it has high accuracy as well as at N ∼ 1. In order to increase the accuracy we should take into account next terms in expansion (2.4). However, we should have in mind that this expansion is an asymptotic (divergent) series in 1/N . Therefore, corrections of high orders increase, beginning with some number k, which depends on N . Usually, it occurs for sufficiently large values of k. 1/N Expansion for a Particle in Central Potential Well In order to investigate binding states of a particle in the smooth central potential well U (r ) = U0 v(r/R) we choose the expansion parameter N ≡ n = l + nr + 1.

(2.5)

In the case of Coulomb potential the quantity n is the principal quantum number. Radial quantum number nr is supposed to be fixed, and orbital quantum number l → ∞. Further, we substitute U0 ≡

n 2 g2 . m R2

Let us write the effective potential (1.32) as an expansion in 1/n:   1 2nr + 1 nr (nr + 1) . Ueff (r ) = n 2 gv(r ) + 2 − + 2r 2nr 2 2n 2 r 2

(2.6)

(2.7)

Here and below in this section we put  = m = R = 1. Analogously, we write the energy of some level:

2 1/N-Expansion in Quantum Mechanics

33

  1 1 (2) E nnr = n 2 ε(0) + ε(1) + ε + · · · . n nr n 2 nr

(2.8)

At n → ∞ the particle is localized near to the minimum of the effective potential r0 (this the point of classical equilibrium). Hence, gr03 v  (r0 ) = 1, ε(0) =

1 + gv(r0 ). 2r02

(2.9)

Further, we restrict ourselves in Eq. (2.7) by the square term in powers of (r − r0 ). Then Eq. (1.31) for radial wave function χ(r ) takes the form of Schrödinger equation for harmonic oscillator :  n(2nr + 1)  2 2 2 (1) − 2nεnr χnnr (r ) = 0. (2.10) χnnr (r ) − n ω (r − r0 ) − r02 Here the notation is introduced  ω≡

gv  (r0 ) +

3 . r04

(2.11)

In this approximation we obtain  

 (1) 1 = ω − r 2 nr + 21 ; εn r 0

1/4 

(0) nω χnnr (r ) = Hnr (ξ) exp −ξ 2 /2 . 2 2nr π2

(2.12)

(nr !)

√ Here once more notation is introduced ξ ≡ nω (r − r0 ) . According to Eq. (2.10), the unperturbed harmonic oscillator has the frequency ω0 = nω. In the region of localization of a particle the typical deviations of this particle from the equilibrium point are small: −1/2

(r − r0 ) ∼ ω0

∼ n −1/2 1.

Analogous estimates are valid for deviations of radial coordinate in anharmonic terms which can be considered as small perturbation of the harmonic oscillator. We write this perturbation in the form of sum of two terms: one term is even function of the variable x ≡ r − r0 while the other term is odd function of this variable V = V (+) + V (−) ; V (+) (x) = 2r12 nr (nr + 1) − 23 n (2nr + 1) 0

V (−) (x) = (2nr + 1) n rx3 + n 2 αx 3 . 0

x2 r04

+ n 2 βx 4 + · · · ;

(2.13)

34

2 1/N-Expansion in Quantum Mechanics

Here the notations are introduced α=−

2 1 5 1 + gv  (r0 ); β = 6 + gv (I V ) (r0 ). 5 6 24 2r0 r0

(2.14)

The perturbation V (+) is a small quantity of the magnitude of 1/n compared to the √ (0) Ueff , while the perturbation V (−) is a small quantity of the magnitude of 1/ n. However, this distinction is essential in corrections to the wave functions only. Contributions to the energy shifts from even and odd parts of the perturbation are of the same order of magnitude. We obtain ε(2) nr

   15 3 22 = − s+ A2 + (s + 2) B 16 15 8    s 3 1 3/2 2 − (s + 1) 6AC + 3C + C + C ω. 4 2 8

(2.15)

(compared with Eq. (2.4)). Here the notations are introduced s = nr (nr + 1) and A=

α ; ω 5/2

B=

β 1 ; C= . ω3 ωr02

(2.16)

The further developments of 1/N -expansion method are considered in the problems of this chapter. Parameter N as Dimension of the Space Finally, we consider the case when the parameter N is the dimension of the space (i.e. N = 3 for real systems). The Laplacian for spherical symmetric functions in N −dimensional space is  f (r ) = f  +

 N −1  f ; r= xi2 . r N

i=1

This result can be obtained using the relation  = div grad, and also grad f (r ) = f  r/r and div r = N . Therefore, the Schrödinger equation for s-state in the central

2

1/N-Expansion in Quantum Mechanics

35

potential U (r ) is reduced to the usual one-dimensional wave equation by substitution  = r −(N −1)/2 χ(r ): 1 (N − 1) (N − 3) − χ + [Ueff (r ) − E] χ = 0; Ueff (r ) = U (r ) + 2 8r 2

(2.17)

(here  = m = 1 ). This effective potential differs from U (r ) by centrifugal potential for N-dimensional space (compare with the Eq. (1.32)). Therefore, 1/N-expansion can be made analogous to 1/n-expansion, see Problem 1.6. It should be noted that the method of 1/N-expansion is applicable also for manydimensional systems with non-separable variables, including many-body systems.

2.1 1/N Expansion for Energy Levels of Binding States Problems and Solutions Problem 2.1 Find the energy spectrum of binding states using 1/N expansion where N is given by Eq. (2.1). Consider the one-dimensional potentials: U0 ; cosh2 (x/a) a x 2 − ; 0 < x < ∞; (b) U (x) = U0   x a  x  2x − b exp − . (c) U (x) = U0 exp − a a

(a) U (x) = −

Here the parameters U0 , a, b are positive numbers. Compare the obtained results with corresponding exact solutions.

Solution All calculations are based on Eq. (2.4). Below we put  = m = a = 1, N = (a) We have ω=

√ 2 2N , α = 0, β = − N 2 3

√ U0 .

36

2 1/N-Expansion in Quantum Mechanics

Table 2.1 Energies of the ground term n = 0 at different values of N √ N = U0 1 1.5 2

3

4

−E 0 (1/N ) −E 0exact q −E 0

7.129 7.114 7.004

13.422 13.411 13.297

0.543 0.500 0.418

1.439 1.410 1.314

2.836 2.814 2.711

Table 2.2 Energies of the forst excited term n = 1 at different values of N √ N = U0 1.5 2 3

4

−E 1 (1/N ) −E 1exact q −E 1

8.765 8.732 8.640

0.318 0.231 0.193

1.007 0.942 0.882

3.886 3.842 3.761

and       √ 1 1 1 1 − n2 + n + +O E n (1/N ) = −N 2 + N 2 n + 2 2 2 N      1 1 1 − n2 + n + . (2.18) ≈ −U0 + 2U0 n + 2 2 2 Let us compare this spectrum with exact solution (see [1], Sect. 23): E nexact = −

2 1  1 + 8U0 − (2n + 1) , 8

q

(2.19)

and with the WKB-solution E n which differs from Eq. (2.19) by change 1 + 8U0 → 8U0 . In the case of the ground term n = 0, we obtain at different values of N the results presented in Table 2.1: It is seen that according to√Eq. (2.19) the total number of levels in the considered potential is approximately N 2 at N  1. In the case of the first excited term n = 1 (this level appears at N = 1) analogous results are presented in Table 2.2: These results allow to make a conclusion that 1/N expansion with terms up to the second order in 1/N is applicable even at N ≥ 1. Bad accuracy occurs for shallow levels only. (b) In this case we obtain   1  1 8U0 + + ... E n (1/N ) = n + 2 8 The exact expression for energy levels is E nexact

=



   1 1  8U0 n + + 8U0 + 1 − 8U0 . 2 4

2.1

1/N Expansion for Energy Levels of Binding States

37

The comparison of both expressions shows that 1/N-expansion has high accuracy also at N ≥ 1; for example, we obtain for the ground level (n = 0) and N = 1 that E 0 (1/N )/E 0exact = 1.026. (c) The first three terms of 1/N-expansion give the exact value of the energy spectrum (see [1], Sect. 23):      1 1 1 2 1 1 U0 b 2 n + − n+ E n (1/N ) = E nexact = − U0 b2 + 4 2 2 2 2 2 1  =− 2U0 b − (2n + 1) . 8 Obviously, the higher terms of 1/N-expansion vanish in the considered case. We can conclude that in all considered cases 1/N-expansion has high numerical accuracy even on the boundary of its applicability. This method presents useful supplement to the WKB-approximation for low-lying energy levels.

Problem 2.2 Find the energy spectrum for power attractive potentials U (r ) ∼ r ν using 1/nexpansion (see Eq. (2.6)). Compare with exact values of the energy E nnr .

Solution Let us write the power potential in the form 2 1  r ν 1 . U (r ) = G r ν ≡ gn 2 ν m R2 ν R

(2.20)

( G > 0 for attractive potential). The notation is introduced here  R=

n 2 g2 mG

1  2+ν

.

We restore in Eq. (2.8) the dimensional factor 2 /m R 2 and put g = 1 in the sakes of simplicity. Then we obtain 1/n-expansion for energy levels of a particle in the power potential:

38

2 1/N-Expansion in Quantum Mechanics

E nnr

  2 1 (1) 1 (2) (0) ε + εn r + 2 εn r + · · · = =n m R2 n n ν   ν+2  2 2  n  2/ν 1 1 (2) ε(0) + ε(1) G = + ε + · · · . m n nr n 2 nr 2

(2.21)

(k)

The expansion coefficients εnr are determined by Eqs. (2.9), (2.12) and (2.15). In the considered case we have v(r ) = r ν /ν and g = 1. √ Elementary derivations give the results r0 = 1, ω = 2 + ν and    2+ν 1 √ = n + 2+ν−1 . (2.22) ε(0) = ; ε(1) r nr 2ν 2 More cumbersome derivations result in the expression for the second order of expansion ε(2) nr =

√ 2−ν ν 2 − 15ν − 52 + 36 2 + ν + 6nr (nr + 1) 144 (2 + ν)   √ × ν 2 − 9ν − 34 + 24 2 + ν

(2.23)

It should be noted that in the case ν < −2 the falling on the origin occurs (see Problem 1.6), and the frequency ω becomes imaginary quantity. Below we illustrate the accuracy of the 1/n-expansion at n ∼ 1 on the examples of some values of the power ν. (1) Coulomb and oscillator potentials First of all, we note that in the case of Coulomb potential (ν = −1) and oscillator potential (ν = 2) the first two terms ( 2.21) and ( 2.22) exactly reproduce the energy (2) spectrum. The coefficient εnr = 0 in both cases, as it should be. (2) Linear potential In the case of ν = 1 we obtain according to Eqs. (2.22–2.23)

ε

(0)

ε(2) nr

   3 1 √ (1) = ; εn r = n r + 3−1 ; 2 2  √ √ 11 − 6 3 7 − 4 3 + nr (nr + 1) . =− 72 12

(2.24)

Below we compare in several tables the results of derivation of energy levels according to Eqs. (2.21–2.24) with exact values of the energy  exact E nn ≡ r

2 G 2 m

1/3 εexact nnr

2.1

1/N Expansion for Energy Levels of Binding States

39

Table 2.3 Energies of levels with nr = 0 l

0

1

2

εexact n0 (0) δn0 (1) δn0 (2) δn0

1.85576 −0.19 5.5·10−3 9.9·10−4

2.66783 −0.11 1.4·10−3 1.6·10−4

3.37178 −0.075 6.3·10−4 5.3·10−5

Table 2.4 Energies of levels with nr = 1 l

0

1

2

εexact n1 (0) δn1 (1) δn1 (2) δn1

3.24461 −0.27 2.5·10−3 −1.9·10−5

3.87679 −0.20 1.2·10−3 −4.9·10−6

4.46830 −0.15 7.2·10−4 −1.9·10−6

which are derived from numerical solution of radial Schrödinger equation. In those tables the quantity E nnr (1/n) (0,1,2) = −1 (2.25) δnn r exact E nn r is given for nr = 0 and 1. The indexes 0, 1, and 2 in Eq. (2.25) correspond to the order of the terms taken into account in 1/n-expansion in Eq. (2.21). Besides of this, in these tables the exact values of εexact nnr are given. It is seen that even at l = 0 1/n-expansion for the energy E nn r has high accuracy already in the first order on 1/n. If we take into account the term ∼ 1/n 2 then accuracy of derivations increases strongly; errors of approximation decrease when l rises. Therefore, we restrict ourselves by the small values of the orbital quantum number l in order to illustrate applicability of 1/n-expansion in the most unfavorable case. We restricted ourselves also by small values of the radial quantum number nr , but due to other reason. Small values of nr are most favorable for applicability of 1/nexpansion. The localization region of a particle in the potential increases with growing of nr (at the fixed value of l). Therefore effects of anharmonicity are more essential in such cases. Hence, the accuracy of 1/n-expansion diminishes (in particular, at l ∼ 1). It is seen from the dependence of the coefficients of 1/n-expansion on nr (see Eqs. (2.22) and (2.23)). (3) High values of the power ν Let us illustrate the accuracy of 1/n-expansion at larger values of the power ν. We restrict ourselves by terms on the order of 1/n in expansion (2.21). In Table 2.5 we give exact values of energies and errors of their calculations using 1/n-expansion for some values of the power ν and of the quantum numbers (l, nr ).

40

2 1/N-Expansion in Quantum Mechanics

Table 2.5 High values of the power ν l

nr

ν=4

ν=6

ν=8

0 0 0 1 1 1

0 1 2 0 1 2

1.50790; −2.2·10−2 4.62122; 1.7·10−3 8.42845; 5.1·10−3 2.82099; −6.4·10−3 6.36257; 2.9·10−3 10.4570; 5.5·10−3

1.64831; −4.1·10−2 5.67414; 1.6·10−2 11.1315; 2.2·10−2 3.21619; −1.2·10−2 8.08082; 1.7·10−2 14.1499; 2.3·10−2

1.80214; −5.3·10−2 6.57180; 3.5·10−2 13.4512; 4.6·10−2 3.58452; −1.4·10−2 9.54346; 3.7·10−2 17.3455; 4.7·10−2

The first number for each values of l, nr is the exact value of εexact nnr , and the (1)

second number is the quantity δnnr (see Eq. (2.25)). The energy of the level is given by relation  exact E nn = r

2 m

 G 2/ν

ν ν+2

εexact nnr

(2.26)

so that according to Eq. (2.21) the 1/n-expansion for the quantity εnnr is of the form   2ν 1 1 (2) + ε + · · · . εnnr = n ν+2 ε(0) + ε(1) n nr n 2 nr It is seen that the accuracy of 1/n-expansion decreases with growing of ν. This is explained by the fact that the considered potential is of the sharper form at large ν. Indeed, at ν → ∞ the power potential is transformed into the infinitely deep spherical potential well.

Problem 2.3 Apply 1/n-expansion for calculation of energy levels of a particle in short-range Hulthén potential U (r ) = −

U0 . exp(r/R) − 1

Solution We determine v(r ) = −

1 . exp(r ) − 1

2.1

1/N Expansion for Energy Levels of Binding States

41

According to Eqs. (2.7, 2.9, 2.12) of 1/n-expansion we find the energy spectrum in parametric form: g(r0 ) ≡

m R 2 U0 (exp(r0 ) − 1)2 = ; 2 2  n r03 exp(r0 )

(2.27)

    2 2 1 1 1− exp(−r0 ) 1 1 n − + 2 + nr + ω (r0 ) − 2 + · · · , m R2 2 r03 2r0 n r0 (2.28) where the notation is introduced (1/n) E nnr =



3 1 ω (r0 ) = 4 − 3 r0 r0

 1+

2 exp (r0 ) − 1

 1/2 .

(2.29)

The parameter r0 determines the point of stable classical equilibrium of a particle. According to Eq. (2.27) we have two values of r0 for the given value of g (at g < g∗ , see Fig. 2.1). We choose the smaller value of r0 , since the larger values corresponds to unstable equilibrium: the maximum of the effective potential. Equation (2.28) is asymptotically exact at n → ∞ and fixed values of r0 and nr . However, at n ∼ 1 this expression also has a good accuracy. As an example, we consider the most unfavorable case l = 0 for 1/n-expansion. The exact energy spectrum of a particle in Hulthén potential with l = 0 is given by exact =− E nn r

1 2 2 8m R (nr + 1)2



2m R 2 U0 − (nr + 1)2 2

2 , l = 0.

(2.30)

Using Eq. (2.27), we obtain exact E nn r

 2 2 2 2 2 (exp(r0 ) − 1) =− −1 . (nr + 1) 8m R 2 r03 exp(r0 )

(2.31)

Here n = nr + 1 for states with l = 0. We compare in Table 2.6 the results of calculations of energies for ground 1s−state (nr = 0) using Eqs. (2.28) and (2.31) at the various values of potential parameters. We present the values of the quantities δ (0,1) =

E nnr (1/n) − 1, E nnr

(2.32)

where indexes 0 and 1 correspond to the order of 1/n-expansion in Eq. (2.28) (compare with results of the previous Problem).

42

2 1/N-Expansion in Quantum Mechanics

U (0)eff (r)

2

1

E (0)nn(2)

 (0) Fig. 2.1 Dependence of the effective potential Ueff = n 2 gv (r ) + 1/2r 2 (see Eq. (2.7)) at the different values of the coupling constant g.Curve 1: g = gcr where the level with the energy E nnr = 0 appears; curve 2: g = g∗ Table 2.6 Energies of the ground 1s-state at the various values of potential parameters r0

0.1

0.2

0.5

0.8

1.0

1.3

1.5

g/gcr δ (0) δ (1)

20 −10−3 10−6

10 −4 · 10−3 2·10−5

4.1 −3.5 · 10−2 7·10−4

2.6 −0.12 6·10−3

2.1 −0.23 2·10−2

1.8 −0.52 7·10−2

1.6 −0.82 0.16

It is seen that 1/n-expansion is applicable as well as at n = 1. Its convergency is slower with increase of r0 , but essential errors appear only at r0 ∼ 1.5. The depth of the potential well diminishes with increase of r0 ; the ratio g/gcr illustrates this fact. The quantity gcr is corresponding to the exact value of g when E nnr = 0, , i.e., when a new level appears (see Fig. 2.1). According to Eq. (2.30) we have gcr = 0.5 for the ground level. 1/n-expansion is inapplicable for shallow s-levels, since the wave function of s−level is localized far outside of the short-range potential at E → 0. Indeed, at r → ∞ we have χ → const = 0. However, the basics of 1/n-expansion is the assumption about localization of a particle nearly of the equilibrium point. Similar situation is realized for shallow levels in one-dimensional potentials (see Problem 2.1).

2.1

1/N Expansion for Energy Levels of Binding States

43

Above conclusions are confirmed by the fact that at n → ∞ according to Eq. (2.28) we find the critical value r0,cr as a solution of the equation 2 (1 − exp(−r0 )) = 1. r0

(2.33)

It is equal to r0,cr = 1.5936. Hence, according to Eq. (2.27) we obtain the asymp(0) totic value of gcr (1/n) = 0.7721. More exact expressions of 1/n-expansion for calculation of the parameter gcr at the finite values of n are given in the Problem 2.4. Oppositely, at g → ∞ and r0 → 0 the exactness of 1/n-expansion rises sharply. Indeed, Hulthén potential at small values of r is similar to Coulomb potential. But just small values of r are important only at g → ∞. We have seen in the solution of the Problem 2.2 that zero approximation of 1/n-expansion gives exact values of Coulomb energies for any values of l and nr . In conclusion we consider shortly the case of l = 0. The wave function is localized even when a new level appears due to centrifugal potential barrier: χl ∼ r −l at r → ∞. Therefore, no difficulties occur for application of 1/n-expansion even at g → gcr , when E nnr → 0. Moreover, 1/n-expansion is applicable as well as at g < gcr , when E nnr > 0 (but the effective potential still has the minimum at the point r0 ). In this case the value of E nnr determines the energy of the quasi-stationary state. Its width nnr is very small at n → ∞. This width is determined by the penetrability of the potential barrier. However, when the quantity g diminishes further, points of stable and unstable equilibrium of a particle in the effective potential coincide with each other (see curve 2 in Fig. 2.1). The corresponding value of g = g∗ is determined from the equation ω (g∗ ) = 0.

(2.34)

In the case of Hulthén potential we find from this equation and Eq. (2.29) that r∗ = (0) (0) (0) 2.576 and g∗ = 0.6564. Then the ratio gcr /g∗ = 1.176. Of course, the 1/nexpansion is inapplicable at g → g∗ . Indeed, according to Eq. (2.15) we have (2) εnr → −∞. The effective potential does not have a minimum at g < g∗ , so that finite motion in the classical theory is impossible. Equilibrium points are moving to the (k) complex plane in this case, and coefficients of expansion εnr are complex quantities. It is interesting that the formulas of 1/n-expansion are valid in this case when the value of g is far from g∗ in quantum mechanics. They describe the quasi-stationary states, which present above-barrier resonance states (see Chap. 4). Imaginary part of the energy E nnr determines the width of such resonance states. Finally, let us underline that the general peculiarities of application of 1/n−expansion for Hulthén potential are correct also for other smooth shortrange potentials (Yukawa potential, exponential potential of the form U (r ) = −U0 exp(−r/R) and so on).

44

2 1/N-Expansion in Quantum Mechanics

Problem 2.4 Obtain 1/n-expansion for a particle in the short-range central potential U (r ) = U0 v(r/R) for critical value ξnnr ,cr of the parameter ξ≡

m R 2 U0 2

which corresponds to appearing of the level with quantum numbers n, l. Illustrate the obtained result on some examples. Solution Above we have obtained Eqs. (2.9), (2.12) and (2.15) of 1/n-expansion for the energy of nl-level at the fixed value of the parameter ξ ≡ n 2 g. Critical values of this parameter are determined by 1/n-expansion of the form ξnnr ,cr = n 2 gnnr ,cr ; (1/n)

gnnr ,cr = g0 +

g1 n

+

g2 n2

+ ···

(2.35)

Coefficients of expansion are found from the condition that the energy E nnr = 0 when the level appears. Accordingly, we have ε(k) nr = 0 in all orders of expansion. Using the expansion ( 2.35) for the parameter g in the effective potential (2.7), we obtain the next results. Equation (2.9) is applicable in zero order of 1/n-expansion. However, we should substitute g by g0 and put ε(0) = 0. Hence, we obtain g0 = −

1 ; r0 v  (r0 ) = −2v(r0 ). 2r02 v(r0 )

(2.36) (1)

The first approximation can be found from Eq. (2.12) by formal substitution εnr → −g1 v(r0 ). Taking into account Eq. (2.36), we obtain     2nr + 1 1 g1 = − ω − 2 = (2nr + 1) ωr02 − 1 g0 . 2v(r0 ) r0

(2.37)

Here, the frequency ω is given by Eq. (2.11), but with substitution g → g0 . It should be noted in the derivations of second-order terms in 1/n that besides substitution ε(2) nr → −g1 v(r0 ), additional terms occur in the perturbative potential:   1 δV = ng1 v  (r0 ) (r − r0 ) + v  (r0 ) (r − r0 )2 , 2

2.1

1/N Expansion for Energy Levels of Binding States

45

(compare with (2.13)). Taking it into account, we obtain    15 22 ω 3 g2 = + s A2 − (2 + s) B v (r0 ) 16 15 8   √ 1+s 1 s 6A C + 5C + 1 − . − C+ 8 4 C

(2.38)

Here the coefficients A, B, C, and s are determined by Eq. (2.16) with substitution g by g0 . Equations (2.36–2.38) present solutions of the problem for calculation of ξnnr ,cr with accuracy up to terms of the order of 1/n 2 . Now we consider some applications of these expressions. We begin with the Tietz potential U (r ) = −U0

R3 r (r + R)2

.

In this case we have v(r ) = −r −1 (r + 1)−2 . According to above formulas we obtain: 3 23 1 , α=− , β= , 2 8 32 1 g0 = 2, g1 = − (2nr + 1) , g2 = nr (nr + 1) . 2 r0 = 1, ω =

It follows from Eq. (2.35) that ξnnr ,cr =

1 (2n − nr − 1) (2n − nr ) . 2

(2.39)

This result of 1/n-expansion coincides with exact solution at all values of quantum numbers l and nr . It should be noted that in the case of nr = 0 the wave function with zero energy is of a simple form χ(0) n0 (r ) = N

r l+1 (r + R)2l+1

.

Next we consider the Yukawa potential U (r ) = −U0

 r R exp − . r R

46

2 1/N-Expansion in Quantum Mechanics

Table 2.7 Accuracy of calculation of energies with nr = 0 l

0

1

2

5

exact ξnn r ,cr (0) δ δ (1) δ (2)

0.8399 0.62 0.14 0.055

4.5410 0.20 0.022 5.4·10−3

10.947 0.12 8.3·10−3 1.5·10−3

46.459 0.053 1.8·10−3 1.5·10−4

Table 2.8 Accuracy of calculation of energies with nr = 1 l

0

1

2

5

exact ξnn r ,cr δ (0)

3.2236 0.69 −0.054 0.030

8.8723 0.38 −0.025 5.5·10−3

17.210 0.26 −0.014 1.8·10−3

58.496 0.14 −5.6·10−3 2.4·10−4

δ (1) δ (2)

In this case we have v(r ) = − r1 exp(−r ), and parameters of 1/n-expansion are: 55 1 e 2 , g0 = , r0 = 1, ω = √ , α = − , β = 3 48 2 2 √   √ √ e 2 − 1 (2nr + 1) e 43 2 − 72 37 2−48 g1 = − + nr (nr +1) . , g2 = √ √ 288 48 2 2 2 Here e = 2.718... . Two Tables 2.7 and 2.8 illustrate the accuracy of calculation of the parameter ξnnr ,cr with the help of 1/n-expansion. In above tables the values of the quantities δ (0,1,2) =

(1/n)

gnnr ,cr −1 exact gnn r ,cr

(2.40)

exact are the exact values of the critical parameter ξ, which are are given. Here gnn r ,cr derived by means of the numerical solution of the Schrödinger equation. Indexes 0, 1, and 2 are corresponding to the order of the terms taken into account in 1/nexpansion in Eq. (2.35). Finally, let us consider the Hulthén potential. In the first order of 1/n−expansion we find    2nr + 1  (1/n) 2 1 − r0,cr − 1 , gnnr ,cr = g0 n 1 − n

where g0 = 0.7721 and r0,cr = 1.5936 (see the previous Problem). Comparison of results of 1/n-expansion with exact calculations is presented in Table 2.9.

2.1

1/N Expansion for Energy Levels of Binding States

47

Table 2.9 Comparison of results of 1/n-expansion with exact calculations l

0

0

0

1

1

1

nr exact gnn r (0) δnnr (1) δnnr

0 0.5 0.54 0.19

1 2 0.54 0.012

2 4.5 0.54 −0.047

0 2.7486 0.12 −0.0054

1 5.3623 0.30 −0.0016

2 9.0505 0.36 −0.027

It follows from these Tables that though zero approximation has a great error at l ∼ 1, calculations taking into account terms of the first and the second order in 1/n increase the accuracy of calculations essentially. Analogous situation occurs also for other smooth short-range potentials, as a rule.

Problem 2.5 Find the energy spectrum of bound s-states for a particle in the central potential U (r ) using 1/N-expansion. Choice of the parameter N is based on the dimension of the space. Illustrate the exactness of results on the examples of power potentials.

Solution Analogous to the derivations in 1/n-expansion method, we present the potential energy in the form U (r ) = U0 v

r  R

≡

r  2 . N 2 gv 2 mR R

Then 1/N -expansion gives: Ueff E nnr

    r R2 2 R2 3R 2 2 , + 2− = N gv + m R2 R 8r 2Nr 2 8N 2 r 2   2 1 1 (2) = N 2 ε(o) + ε(1) ε + ··· . nr + 2 mR N N 2 nr

(see Eq. (2.17)). Further we put R = 1. In the point of classical equilibrium of a particle we have 1 1 gr03 v  (r0 ) = , ε(0) = 2 + gv (r0 ) . 4 8r0

48

2 1/N-Expansion in Quantum Mechanics

The next, oscillator, approximation of 1/N-expansion gives: ε(1) nr

  1 1 − 2; = ω nr + 2 2r0

 ω=

3 + gv  (r0 ) 4r04

1/2 .

Corrections of the second order on 1/N can be derived by the same method as in the case of 1/n-expansion (see Eq. (2.15)). In the case of s−states for the power potential  ν G 2 2 r U (r ) = r ν ≡ N , ν m R2 R

 where

R=

2 N 2 mG

1  ν+2

,

(g = 1 here and thereafter, in the sakes of simplicity) we obtain according to above formulas: 2 2−ν ω = 2 2+ν (ν + 2)1/2 . r0 = 2− ν+2 , Hence, the 1/N-expansion for the energy takes the form ν  ν+2 2 N 2 2/ν (4G) m      ν+2 1 1 nr + × + (ν + 2)1/2 − 1 + · · · . (2.41) 8ν 2N 2

 E nnr (1/N ) =

In the case of ν = 2 and G = mω02 we obtain from (2.41) that E nnr

  N , = ω0 2nr + 2

which coincides with exact expression for energies of s−states of N − dimensional harmonic oscillator (it is clear without further derivations that corrections of next orders, ε(k) with k ≥ 2 vanish ). In the case of ν = 1 we obtain from Eq. (2.41) for ground state, nr = 0:  2/3      1 4 2 1 4 20 1+ + 2 + · · · − √ − √ N N 3 3 3 3 27  

3 N G 2/3 = 1 − 0.17863N −1 + 0.02906N −2 + · · · . (2.42) 2 2

E 0 (1/N ) =

3 2



NG 2

(2)

Here  = m = 1 and the correction εnr is taken into account. In the case of N = 3 we obtain from Eq. (2.42) that

2.1 1/N Expansion for Energy Levels of Binding States

49

E 0 = 1.8549G 2/3 . This value differs by 0.05 % only from the exact value of the energy of ground state. It should be noted that in the case of one-dimensional potential, U (x) = G |x|, Eq. (2.42) gives (N = 1) the energy E 0 = 0.8036G 2/3 , that is also near to the exact value of 0.8086G 2/3 . Finally, let us discuss shortly the case of Coulomb potential, ν = −1. In the case of ground state we obtain the next expansion:   2 3 2mG 2 + 2 + ··· . E 0 (1/N ) = − 2 2 1 +  N N N

(2.43)

Convergence of 1/N -expansion in this case is, obviously, poor compared to the previous case. It is explained by the fact that in one-dimensional Coulomb potential (N = 1) the falling of a particle to the origin occurs. Therefore, the case of N = 1 is the singularity for the function E 0 (1/N ), so that E 0 (1) = −∞. Therefore, it is useful to substitute the parameter of expansion N → N − 1. It is interesting that after this substitution according to Eq. (2.43) we obtain: E0 = −

2mG 2 2 (N − 1)2

.

This expression coincides with the exact value of the energy of the ground state for N −dimensional Coulomb potential.

2.2 Wave Functions of 1/n-Expansion Problems and Solutions Problem 2.6 Find the next correction to the unperturbed wave function (2.12) with respect to 1/nexpansion for a particle in a central potential. Investigate the accuracy of the obtained result on the examples of solvable potentials at n ∼ 1. Solution The perturbed wave function is calculated according to general formulas of the perturbation theory (see [1], Sect. 38). The perturbation is given by Eq. (2.13). Taking into account the well-known values of dipole matrix elements for linear oscillator, we obtain that the most important√is the odd part V (−) of the potential (2.13). It results in terms of the order of 1/ n, while the even part V (+) results in terms of the order of 1/n.

50

2 1/N-Expansion in Quantum Mechanics

We restrict ourselves by states with nr = 0. Non-zero matrix elements for the harmonic oscillator with the frequency ω0 are (here  = m = 1): √         3 1 6  3  3 1 |x| 0 = √ , 1 x  0 = , 3 x  0 = . 3/2 2ω0 (2ω0 ) (2ω0 )3/2 In our case ω0 = nω, where the frequency ω is given by Eq. (2.11). In the first order perturbation theory the wave function is [1]:    (−)    0 (0) k V (0) χnk . (2.44) χn0 = χn0 + −kω k

Using the explicit form (2.12) of unperturbed wave functions, we obtain: χn0 (r ) =

 nω 1/4 π 



α 1 × 1− +

3/2 ω 3/2 ωr02   nω × exp − (r − r0 )2 . 2



√ nα ω (r − r0 ) + (r − r0 )3 3ω



(2.45)

The parameter α is determined by Eq. (2.14). At n  1 the wave function (2.45) describes a particle in a region |r − r0 | ∼ (nω)−1/2 where this function is localized. √ The correction in square brackets of Eq. (2.45) is of the order of magnitude 1/ n. Equation (2.45) is asymptotically exact one at n → ∞. Its accuracy is ∼ 1/n. We illustrate its accuracy on the example of Coulomb potential U (r ) = −n 2 /r in the most unfavorable cases for 1/n-expansion: n = 1 (l = 0) and n = 2 (l = 1). We obtain r0 = 1, ω = 1, α = 1. Equation (2.45) takes the form: χn0 (r ) =

 n 1/4 π



 n n exp − (r − 1)2 1 + (r − 1)3 . 2 3

Exact wave functions of 1s− and 2 p−states are of the form  2 2 r exp(−2r ) χ1s (r ) = 2r exp(−r ); χ2 p (r ) = 8 3

(2.46)

(2.47)

(it should be noted that the Bohr radius is a B = n −2 ). In order to compare wave functions (2.46) with (2.47), we give two Tables (2.10) and (2.11) In these Tables the values of the quantities are presented: δ

(0,1)

(0,1)

χ (r ) − 1. (r ) = n0 χn0 (r )

2.2 Wave Functions of 1/n-Expansion

51

Table 2.10 Comparison of wave functions (2.46) with (2.47) for n = 1 (l = 0), nr = 0 r

0.5

1.0

1.5

δ (0) δ (1)

0.099 0.047

0.021 0.021

−0.010 0.032

Table 2.11 Comparison of wave functions (2.46) with (2.47) for n = 2 (l = 1), nr = 0 r

0.5

1.0

1.5

δ (0) δ (1)

0.16 0.061

0.010 0.010

−0.049 0.030

(0,1)

Here χn0 (r ) is the exact wave function (2.47), while χn0 (r ) is the wave function in the approximation of 1/n-expansion (zero and first approximation, respectively). It is seen that in nodeless states ( nr = 0) 1/n-expansion describes quite well the wave functions even at n ∼ 1, in the vicinity of their maxima ( r ≈ 1). It should be noted that the position of maximum, i.e., r0 = 1 is the same for exact wave functions and for the wave functions of 1/n-expansion.

Problem 2.7 Using 1/n-expansion, find the wave function of nl−level in classically forbidden region. Calculate the asymptotic coefficient Cnl in the origin r → 0 and investigate its accuracy. Solution In classically forbidden region the parabolic expansion of effective potential is incorrect when r is far from the equilibrium point r0 . Then, the Eq. (2.12) for the wave function is invalid (see also the previous Problem). Instead of this, the wave function can be found using WKB-approximation. Radial Schrödinger equation taking into account the expansions (2.7) and (2.8) is of the form   1 1  2nr + 1 1 (1) χ(r ) = 0. χ (r ) − n 2 gv(r ) + 2 − ε(0) − ε − 2 2r 2nr 2 n nr

(2.48)

We neglected here the terms ∼ 1/n 2 . The mathematical form of Eq. (2.48) is similar to Eq. (1.1), but the Planck constant is substituted by the parameter 1/n. The limit n → ∞ is corresponding to the WKB-

52

2 1/N-Expansion in Quantum Mechanics

limit  → 0. This consideration allows to obtain the solution of Eq. (2.48) in the form of WKB-solutions, Eq. (1.5). In particular, in the left classically forbidden region (from the equilibrium point r0 ) the radial wave function is of the form (we take into account the boundary condition χ(0) = 0): ⎧ ⎫ ⎨ "a ⎬ C χ(r ) ≈ √ exp −n q(r  )dr  ⎩ ⎭ q(r ) ⎧ r a  ⎫  " ⎨ ⎬ C 2nr + 1 1 1 ≈ √ − 2 +ω dr  . exp −n q0 (r  ) −  2 ⎩ ⎭ 2nq0 (r ) r q0 (r ) r0 r

(2.49) Here the notations are introduced 1/2   2 2nr + 1 (1) + ε ; q(r ) = q02 (r ) − nr n 2r 2 1/2   1 q0 (r ) = 2 gv(r ) + 2 − ε(0) . 2r

(2.50)

(1)

We used here expression (2.12) for εnr and expand over 1/n. We omit in Eq. (2.49) terms of the order of 1/n, since they are outside of accuracy of our approximation (in order to take them into account, we should use more exact WKB-solutions, see Ref.[1], Sect. 46). We should determine now the normalized coefficient C in Eq. (2.49), which depends on the choice of a. The wave function (2.49) should be matched with the normalized oscillator wave function (2.12) in common region of their applicability: 1

√

nω |r − r0 |

√ n.

(2.51)

Here, the left side of the inequality corresponds the condition of applicability of WKB-approximation (at E = 0) in the region where the harmonic oscillator potential occurs. Then we have (2.52) q0 (r ) ≈ ω |r − r0 | and (see Eq. (1.6))

  1  dq0  1. nq02  dr 

Right side of the inequality (2.51) corresponds to the condition of applicability of parabolic expansion of the effective potential ( i.e., condition of 1/n-expansion).

2.2 Wave Functions of 1/n-Expansion

53

Let us discuss how to match the solutions. It is impossible to put a = r0 in Eq. (2.49) directly, since the latter integral in Eq. (2.49) diverges when a → r0 . This divergence does not depend on r ; therefore, it can be compensated by renormalization of the coefficient C. We pick out from the integrand the divergent part: according to (2.52) it is equal to |r − r0 |−1 . Then this integral takes the form "a r

P0 (r ) dr = q0 (r )

"a  r

P0 (r ) 1 − q0 (r ) r0 − r



    a r − ln 1 − . (2.53) dr + ln 1 − r0 r0

Here the notation is introduced P0 (r ) =

1 1 − 2 + ω. 2 r r0

In the integral expression of right side of Eq. (2.53), we can put a = r0 . The contribution of the term ln(1 − a/ro ) can be included into still undetermined coefficient C, since it does not depend on the variable r . Then the wave function, Eq. (2.49) takes the form ⎧   ⎨ "r0  2nr + 1 P0 (r ) C 1 dr q0 (r ) − χ(r ) = √ exp −n − ⎩ 2n q0 (r ) r0 − r q0 (r ) r   r 2nr + 1 . (2.54) ln 1 − + 2 r0 In the matching region determined by the conditions (2.51) we obtain 1 χ(r ) ≈ C  √ ωr0



r − r0 r0

nr

  1 exp − nω (r − r0 )2 , 2

(2.55)

while in this region the harmonic oscillator wave function (2.12) takes the form   & √ 'nr 1 nr 2 . nω − r nω 2 − r exp − (r (r ) (−1) ) 0 0 2 22nr π (nr !)2 (2.56) Wave functions (2.55) and (2.56) should coincide each one other. Then we can calculate the coefficient C  : χ(0) nnr (r ) ≈



1/4

nω

 

C = (−1)

nr

nω 3r02 π (nr !)2

1/4

 nr /2 2nωr02 .

(2.57)

54

2 1/N-Expansion in Quantum Mechanics

Table 2.12 The values of J0 and J1 ν

0

1

4

8

J0 J1

−0.67202 0.28381

−0.55843 0.34245

−0.43983 0.30358

−0.38972 0.16622

Expressions (2.54) and (2.57) determine the normalized radial wave function in the underbarrier region to the left from the point r = r0 . If r → 0, we obtain q0 (r ) ≈ 1/r and according to Eq. (2.54) we conclude that the radial wave function has correct limiting form near the origin: χ(r ) ≈ Cnl r l+1 . Further, we can pick out divergent terms of Eq. (2.54) at r → 0 by the method described just above. Then we can derive the asymptotic coefficient in the origin: (1/n)

Cnl

−l−1/2

= C  r0

exp {− [n J0 + (2nr + 1) J1 ]} ,

(2.58)

where the notations are introduced  "r0  1 J0 = dr ; q0 (r ) − r

1 J1 = 2

0

"r0  0

 P0 (r ) r0 − dr. r (r0 − r ) q0 (r )

(2.59)

Equation. (2.58) is asymptotically exact at n → ∞ and at fixed value of nr . Its accuracy is of the order of 1/n. For nodeless states the Eq. (2.58) is valid with a good accuracy also at n ∼ 1. As an example, we consider the case of power potentials U (r ) = r ν /ν. Simple numerical derivation according to Eq. (2.59) gives the values of parameters J0 (ν) and J1 (ν) presented in Table 2.12. The case of ν = 0 corresponds to the logarithmic potential U (r ) =lnr. The integrals in Eq. (2.59) are derived analytically in the cases of Coulomb potential (ν = −1) and of the harmonic oscillator potential (ν = 2): 1 ln 2. 2 (2.60) (see also Problem 2.2 where parameters of 1/n-expansion are given for power potentials). After derivation of parameters J0 , J1 final value of the asymptotic coefficient can be found elementary using Eq. (2.58). Accuracy of such calculations for nodeless states in the case of power potentials is illustrated in Table 2.13. In this Table, the ratios J0 (ν = −1) = −1;

J1 (ν = −1) = 0;

1 J0 (ν = 2) = − ; 2

J1 (ν = 2) =

2.2 Wave Functions of 1/n-Expansion

55

Table 2.13 Values of the ratio (η) according to Eq. (2.61) ν l=0 1 2 5

−1 1.0209 1.0104 1.0070 1.0035

0 1.0280 1.0146 1.0098 1.0050

1 0.9952 0.9976 0.9984 0.9992

2 0.9803 0.9898 0.9931 0.9965

4 0.9900 0.9984 1.0002 1.0009

8 1.0862 1.0731 1.0598 1.0378

(1/n)

η=

Cnl exact ; n = l + 1 Cnl

(2.61)

exact are calculated numerically in the case of power potentials U (r ) = r ν /ν. Here Cnl (1/n) is the exact value of the asymptotic coefficient, and Cnl is derived using Eqs. (2.58– 2.59). It is shown from the Table 2.13 that maximum error occurs for Cnl at ν = 8. It is explained by the fact that the power potential looses its smoothness at the increasing of ν. It is transformed into the infinitely deep spherical potential well in the limit of ν → ∞.

Reference 1. L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd edn. (Pergamon, Oxford, 1977)

Chapter 3

Rydberg States of Atomic Systems

Let us consider multielectron atomic systems which are found in the definite stationary states. Most of the calculations of the properties of such systems and the classification of their quantum states are based on the single-particle approximation; according to this approximation each of the atomic electrons is found in the definite single-particle state. We define so-called Rydberg states of an atomic system (i.e., of a neutral atom, or of a positive ion) by the requirement that one of the electrons is found in a highly excited state , which is determined by the large value of the principal quantum number n = nr + l + 1  1. Here, nr,l—are the radial quantum number and the orbital quantum number, respectively. The energy of this state is given by [1] E nl = −

 2 m e Z e2 22 (n − δl )2

.

(3.1)

Here, Z e > 0 is the charge of the atomic core (not a charge of the atomic nucleus), Z = 1 for neutral atoms. The quantity δl is a so-called quantum defect; the Rydberg correction is defined as l = −δl . It should be noted that if the orbital (L) and spin (S) momenta of atomic core are nonzero, then the quantum defect depends also on the orbital and spin momenta of the considered atom. The energy defined by Eq. (3.1) is counted off from the energy of the ground state of the unperturbed atomic core. The quantum defect appears due to distortion of the pure Coulomb potential UC (r ) = −Z e2 /r by the electrons of atomic core. Usually we have δl ∼ 1 for the states with small values of the orbital quantum number l = 0, 1, 2 only; quantum defect diminishes quickly with the rise of l. This is explained by decreasing of penetration probability of a highly excited electron inside the atomic core due to centrifugal barrier.

B. M. Karnakov and V. P. Krainov, WKB Approximation in Atomic Physics, DOI: 10.1007/978-3-642-31558-9_3, © Springer-Verlag Berlin Heidelberg 2013

57

58

3 Rydberg States of Atomic Systems

We use below often the atomic system of units (a.u.) where the electron mass, elementary charge, and Planck constant are equal to unity: m e = e =  = 1. Often we can use unperturbed Coulomb wave functions as zero approximation for wave functions of the high excited electron states in the investigations of properties of Rydberg states (see Ref. [1], Sect. 36). They are appropriate, in particular, for states with l  1, since in this case δl  1. Thus, nlm (r) = Rnl (r )Ylm (n) ;  2Z 3/2 (n + l)! Rnl (r ) = (2Zr )l l+2 (2l + 1)!n (n − l − 1)!     2Zr Zr F −n + l + 1, 2l + 2, . × exp − n n

(3.2)

Here, Ylm (n) is the normalized spherical harmonic, the confluent hypergeometric function F coincides the Lagerre polynomial except the constant factor. We neglect the influence of atomic core in Eq. (3.2); the radius of atomic core is of the order of Bohr radius a B = 2 /m e e2 , while the typical radius of orbit for Rydberg electron is much more, ∼n 2 a B . WKB radial wave function of high excited atomic electron outside of the atomic core (where WKB approximation is applicable) is given by the expression ⎧ r ⎫ ⎨

⎬ ˜ C π nl (q) Rnl (r ) = √ p L (r  )dr  + + πδl , a < r < b; sin ⎩ ⎭ 4 r p L (r ) a    2ω 2Z 2 2Z (l + 1/2)2 r ˜ nl = − = , C . p L (r ) = 2E nl + r r2 π πn 3∗

(3.3)

Here, a is the left turning point, b is the right turning point, p L (r ) is the radial momentum of an electron, C˜ nl is the normalized constant, ωr is the classical oscillator frequency, and n ∗ = n − δl is the effective principal quantum number. We have used in Eq. (3.3) Langer correction: this is substitution l (l + 1) → (l + 1/2)2 in the centrifugal potential (see Sect. 1.2). The argument of sine function in Eq. (3.3) is chosen so that quantum defect δl vanishes in the case of pure Coulomb potential U (r ) = −Z e2 /r for all values of r . Let us discuss now the applicability of WKB approximation for a Coulomb potential U (r ) = −Z e2 /r. In the classical region of electron’s motion determined by relation U (r ) ∼ E nl , the condition for applicability of WKB approximation (1.6) takes the form m e Z 2 e4 v |E nl |  , or  1. (3.4) 2  Z e2

3

Rydberg States of Atomic Systems

59

Thus, the velocity v of the Rydberg electron should be small compared to the velocity of an electron in the ground atomic state (i.e., on the first Bohr orbit). The condition of applicability of WKB approximation is fulfilled for Rydberg states on the radial dimensions 2 r  aB = . (3.5) m e Z e2 The inequality (3.4) for electron energy allows to neglect the dependence of the quantum defect on the principal quantum number n.

3.1 Unperturbed Rydberg States of Atoms Problems and Solutions Problem 3.1 Obtain the energy spectrum, Eq. (3.1) of Rydberg states of atoms using WKB wave function, Eq. (3.3).

Solution Incorporation of the quantum defect δl into the WKB wave function (3.3) corresponds to taking into account the boundary condition at r → 0 in the exact radial wave function, where Eq. (3.3) is inapplicable. The boundary condition in the infinity, r → ∞, is Rnl (r ) → 0. It follows from the matching condition for WKB wave function in the right classical turning point r = b that ⎧ r ⎫ ⎨  ⎬  C (q) Rnl (r ) = √ exp −  p L (r  ) dr  , r > b; ⎩ ⎭ 2r | p L (r )|

(3.6)

b

and (q) Rnl (r )

⎧ b ⎫ ⎨

⎬ C π = √ p L (r  )dr  + sin , r < b. ⎩ 4⎭ r p L (r )

(3.7)

r

The WKB wave functions (3.3) and (3.7) are the same solutions of radial Schrödinger equation; therefore we should equalize these functions. Then we obtain the quantization condition

60

3 Rydberg States of Atomic Systems

b

  1 . p L (r )dr + πδl = π nr + 2

(3.8)

a

Here, we took into account that the sum of arguments of sine functions in (3.3) and (3.7) should be integer number multiplied by the number π. Hence, C = (−1)nr C˜ nl . Substituting p L from (3.3) into (3.8), and using the value of the integral

b 

(r − a) (b − r )

√  dr π a + b − 2 ab , a < r < b, = r 2

a

we obtain the energy spectrum of the considered Rydberg atomic states: E nl = −

Z2 , n ∗ = n − δl , 2n 2∗

(3.9)

that coincides with (3.1).

Problem 3.2 Using WKB wave functions (3.3), obtain the recurrent relation  2 (s − 1) E nl

1 rs



 + (2s − 1) q

Z r s+1

 q

    1 2 1 −s l + = 0. 2 r s+2 q

(3.10)

Here the notation 1/r s q is introduced for mean WKB values of the quantities 1/r s in Rydberg states of the valence electron. Calculate also the values of 1/r s q in the cases s = 1, 2, 3,and 4, and compare these values with exact expressions for Coulomb potential UC = −Z /r.

Solution The WKB expression for the considered mean values is of the form 

1 rs

 q

Z2 = πn 3∗

b 

a

dr

r s 2E nl +

2Z r

−

ν2 r2

≡

Z2 Is , πn 3∗

(3.11)

3.1

Unperturbed Rydberg States of Atoms

61

where the notations are introduced 1 n ∗ = n − δl , ν = l + , 2

E nl = −

Z2 . 2n 2∗

Integration over radial coordinate r in (3.11) is restricted by the classical range [a, b] only, is the usual procedure in WKB calculations. Square of sine functions is replaced by its mean value of 1/2. Elementary integration of (3.11) in the cases s = 1 and s = 2 gives results     1 Z 1 Z2 = 2, = . 2 r q n∗ r q (l + 1/2) n 3∗

(3.12)

In the case of pure Coulomb potential we have δl = 0, and n ∗ = n. Then these WKB results coincide exact quantum-mechanical mean values (see Ref. [1], Sect. 36). It should be noted that it follows from (3.12) the relation −

  Z = 2E nl ; r q

this is the generalization of the virial theorem for Coulomb potential to the case of Rydberg atomic states (strictly speaking, this theorem is inapplicable for distorted Coulomb potential since in the latter case U (r ) = − Z /r ). In order to obtain the recurrent relation (3.10), we use two obvious expressions

b

p L (r )dr = rs

a

b a

  2E nl + (2Z /r ) − ν 2 /r 2 dr = 2E nl Is + 2Z Is+1 − ν 2 Is+2 , r s p L (r ) (3.13)

and

b 0= a

d dr



 

b 

b Z dr p L (r ) p L (r ) ν2 dr ≡ − + (1 − s) − dr. r s−1 r s+1 r s+2 p L (r ) rs a

a

(3.14) Here the notation is introduced p L (r ) =



2E nl + 2Z /r − ν 2 /r 2 .

We took into account in derivation of (3.14) that p L (a) = p L (b) = 0. Substituting (3.13) into (3.14) we obtain the recurrent relation 2 (s − 1) E nl + (2s − 1) Z Is+1 − s (l + 1/2)2 Is+2 = 0, which coincides (3.10).

62

3 Rydberg States of Atomic Systems

Table 3.1 ηs l

1

2

3

4

5

6

η3 η4

0.889 0.494

0.960 0.806

0.980 0.900

0.988 0.939

0.992 0.959

0.994 0.971

It should be noted that the analogous recurrent relation takes place in the case of pure Coulomb potential U = −Z e2 /r for exact quantum-mechanical mean values (see the interesting remarks in Ref. [2]):         1 1 1 s2 s 2 1 2 2 2E n (s − 1) s + Z e (2s − 1) s+1 + = 0. − l+ r r me 4 2 r s+2 (3.15) In this relation we contain the Planck constant in order to illustrate the difference with WKB relation (3.10) in the term which is proportional to 2 . Here we restore the usual system of units. It is seen that in WKB expression (3.10) there is no term ∼ 2 s 2 /4 (this term vanishes in the limit of  → 0), but there is the term ∼ 2 (l + 1/2)2 (which is equal to M2 in classical limit). Let us compare now WKB and exact mean values each with other in the cases of s = 3 and 4. In the case of s = 3 we find     1 Z3 1 Z3 = ; = . (3.16) r3 q r 3 exact l (l + 1) (l + 1/2) n 3 (l + 1/2)3 n 3∗ 

In the case of s = 4 we obtain   1 3Z 4 = ; r4 q 2 (l + 1/2)5 n 3∗     l (l + 1) 1 3Z 4   1 − . (3.17) = r 4 exact 3n 2 2l (l + 1) (l + 1/2) (l + 1/2)2 − 1 n 3 In order to compare exact and WKB mean values we present in Table 3.1 the ratio  ηs =

1 rs



 :

q

1 rs

 . exact

in the cases of s = 3 and 4 (in calculation for this table we put n ∗ = n and neglect the term ∼n −5 in the quantity r −4 exact ): It is seen that WKB results at large values of s and small values of l differ strongly from exact results, though at large values of l → ∞ they coincide with each other:

3.1

Unperturbed Rydberg States of Atoms

η3 = 1 −

1 (2l + 1)

2

63

; η4 = 1 −

5 (2l + 1)

2

+

4 (2l + 1)4

.

Poor quality of WKB results is explained by the specific property of the considered quantity 1/r s , namely, by its divergence in the origin which becomes more important with the growing of s. This reflects the divergence of the exact mean values in the cases of s = 3 and 4 at l = 0 (let us remember that the radial wave function Rnl (r ) ∼ r l at r → 0). Thus, the small values of r are essential in the exact derivation of mean values

1/r s  at small values of l. However, in the WKB approximation small values of r are excluded from consideration, since the integration is performed only between the classical turning points. Moreover, taking into account Langer correction, we can formally exclude small values of r even for s-states. This consideration explains convergence of WKB expressions for 1/r s q for any values of l and s.

Problem 3.3 Calculate the contribution into the quantum defect due to polarization of atomic core produced by a Rydberg electron.

Solution We use the atomic system of units in this problem. Polarization potential Upol (r ) = −α/2r 4 is produced by the polarization of the atomic core by an external electron which is found at a large distance from this core. Here, α is the static core polarizability. This potential should be added to Coulomb potential UC (r ) = −1/r (we restrict ourselves by the case of a neutral atom, i.e., Z = 1). Let us remember that the polarizability α is found from the expression for mean dipole moment of a system induced by the external electric field strength F: d=αF (in the linear approximation on F ). The energy of the atomic core is shifted by the value of −αF 2 /2. Analogously, the polarization potential produces small (at n  1) shift of the energy of the Rydberg electron. We calculate this shift in the first order perturbation theory using WKB approximation     1 α pol δ E nl = − nl  4  nl . 2 r In the derivation of this diagonal matrix element we use the expression (3.3) for radial WKB wave function and substitute square of sine function in the integrand by its mean value of 1/2:

b dr α pol δ E nl = − . (3.18) 3 4 2πn r p L (r ) a

64

3 Rydberg States of Atomic Systems

The energy of the Rydberg atomic state can be written in the form E nl = −

1

≈−

2 (n − δl )2

1 δl − 3. 2n 2 n

(3.19)

Comparing (3.18) and (3.19) with each other we find the contribution of core polarization into the value of the quantum defect δl : pol δl

α = 2π

b a

α dr ≈ 4 r p L (r ) 2π

∞ 

a

dr

r 3 2r − (l + 1/2)2

.

(3.20)

Here, we have used Eq. (3.3) for the electron momentum p L (r ) and omitted in that (0) equation the small term including the energy E n = −1/2n 2 . It is seen that in the considered approximation the quantum defect does not depend on the principal quantum number n. Elementary calculation of the integral in (3.20) can be fulfilled using the substitution of the integrand variable 2r = [(l + 1/2) / sin ϕ]2 . The angle ϕ is varied in the interval [0,2π]. Then we obtain the final result pol

δl

=

3α 4 (l + 1/2)5

.

(3.21)

Let us make some concluding remarks: (1) It is seen from (3.20) that the typical values of r for which the polarization potential is important , are of the order of magnitude of (l + 1/2)2 a.u. These values should be much larger than the radius of the atomic core; therefore the final result (3.21) is valid under the condition l  1. (2) We can neglect under this condition l  1 the penetration of the Rydberg electron into the atomic core due to smallness of the radial wave function Rnl (r ) ∼ r l produced by the centrifugal potential. Thus, we confirm the applicability of the perturbation theory in the derivation of the energy shift δ E nl . The contribution of the polarization potential into the quantum defect is the most important though the value (3.21) decreases quickly with growing of l. The contribution of other effects (for example, of the exchange effects) decreases still more strongly. (3) Though the final result (3.21) is of a qualitative form at l ∼ 1 (but l = 0) this expression gives correct dependence of the quantum defect on the orbital quantum number l (at least, for atoms with moderate charge Z˜ of atomic nucleus). For example, in the case of helium atom we can use the well-known expression for the polarizability of the ground state of a hydrogenlike atom with the charge Z˜ : α = 9/2 Z˜ 4 . Then from (3.21) we obtain the values of the quantum defect (compare with Ref. [1], Sect. 68) presented in Table 3.2. More details about the polarization potential can be found in the book by Bethe and Salpeter [3].

3.1

Unperturbed Rydberg States of Atoms

Table 3.2 Polarization quantum defects

65

l

1

2

3

pol δl

0.03

0.002

0.0004

Problem 3.4 Calculate the value of spin–orbit splitting for atomic Rydberg states with l = 0 in the case when the atomic core is found in 1 S-state. Compare the result with the interval between the neighboring Rydberg levels.

Solution The operator of spin–orbit interaction of an electron moving in central electrostatic field with potential ϕ(r ) is of the form [1], Sect. 72: Uls (r ) = −

2 e ∂ϕ (ls) . 2m 2 c2 r ∂r

(3.22)

Here l, s—are operators of the electron orbital and spin momenta. Further in this problem we use the atomic system of units e = m =  = 1, and speed of the light is c = 1/α = 137. Each unperturbed level with l = 0 is degenerated having 2(2l + 1) states (we took into account also the electron’s spin). This level is splitted due to spin–orbit interaction upon two sublevels with good value of the total angular momentum j = l ± 1/2. The operator (ls) has also good value in these states, namely:    3 1 l/2, j ( j + 1) − l (l + 1) − = (ls) = − (l + 1) /2, 2 4

j = l + 1/2 j = l − 1/2

(3.23)

  This result follows from the relation (ls) = 21 j2 − l2 − s2 . Substituting (3.23) into (3.22), we obtain that the operator of spin–orbit interaction is of the form of the usual central potential in the states with the given value of j. This potential is a small perturbation of a system. In the first order perturbation theory we find splittings of atomic levels:      1 dU  2l + 1 j=l+1/2 j=l−1/2   (3.24) nl − δ E nl = E nl = δ E nl  r dr  nl . 4c2 Neglecting the penetration of the Rydberg electron into the atomic core we can approximate the potential U (r ) by −1/r (again we consider the case of neutral atoms only, so that Z = 1) and use the WKB expression (3.3) for unperturbed radial

66

3 Rydberg States of Atomic Systems

wave functions. Then the diagonal matrix element in (3.24) can be written in the form    

b

b  1 dU  1 dr 1   nl = nl  ≈ r dr  πn 3∗ r 3 p L (r ) πn 3∗ a

0

dr  . r 2 2r − (l + 1/2)2

(3.25)

This derivation can be compared to the analogous calculation in Problem 3.3 for the polarization potential and in Problem 3.2 for mean values of the 1/r s  . Elementary derivation of this integral using the substitution 2r = [(l + 1/2)/ sin ϕ]2 , gives the spin–orbit splitting of nl-level: 1

E nl =

2c2 (l

+ 1/2)2 n 3∗

.

(3.26)

It should be noted that in the considered approximation the sublevel with the less value of j is placed lower. It is seen also that the splitting (3.26) has the same dependence on the principal quantum number n (∼n −3 ) as the interval between the neighboring Rydberg levels. However, the value of this splitting is much less (104 − 105 times) compared to the interval between the neighboring levels due to the factor c−2 ∼ 5 · 10−5 . Let us make some remarks concerning Eq. (3.26) and its applications. This result is valid when the penetration of the Rydberg electron into the atomic core can be neglected. Such condition is realized when the quantum defect of the considered term (nl) is small. In the opposite case δl ∼ 1, Eq. (3.26) is inapplicable, since small values of r are essential in the integrand of the diagonal matrix element (3.24) where the unperturbed potential of the atomic core differs strongly from the Coulomb potential. Nevertheless, the dependence of the matrix element (3.24) on the principal quantum number n is contained in the form n −3 ∗ (see also the result of the next Problem 3.5). Of course, the numerical coefficient in such dependence differs from (3.26). Thus, we can generalize the result (3.26) in the form E nl · n 3∗ = K l = const, n  1.

(3.27)

where the quantity K l does not depend on n , and in the limit of l → ∞ we have (q)

Kl → Kl

=

2 (2l + 1)2 c2

.

Let us illustrate the obtained results on the examples of atoms of Li and Na. Their atomic cores are described by the states (1s)2 1 S and (1s)2 (2s)2 (2 p)6 1 S, respectively. In Table 3.3 we present a comparison of the experimental values of E nl · n 3∗

3.1

Unperturbed Rydberg States of Atoms

67

Table 3.3 Comparison of the experimental energies with WKB result, Eq. (3.26) l (q)

K l · 10−5 K l · 10−5 , Li K l · 10−5 , Na

1

2

3

4

0.779 – – 53.7 (0.855)

0.280 0.29 (0.002) −0.92 (0.016)

0.143 0.144 (8·10−5 ) 0.14 (0.00145)

0.0865 0.09 – – –

with the WKB result (3.26). In the brackets the values of the quantum defect are given. The values of (3.26) are given in MHz (1 a.u. = 27.21 eV = 6.58 · 109 MHz) [4]. It is seen that the role of the atomic core is negligible for states of Li atom with l ≥ 2. However, the situation for p- and d-states of Na atom is quite different: the experimental value of the quantity K l is much more than the WKB expression ! It is interesting that in the case of p-states the result (3.27) is valid even for the lowest excited states: for example, K exp · 10−5 (3 p) = 51.0 MHz, and K exp · 10−5 (4 p) = 52.5 MHz. However, in the case of d-states the value of K depends on the principal quantum number n and even changes its sign; for example, K exp · 10−5 (3d) = 0.39 MHz.

Problem 3.5 Find the dependence of the normalized wave functions of Rydberg electron states inside the atomic core on the principal quantum number n. In particular, determine the dependence of the quantity ns (0) on n.

Solution WKB expression (3.3) for the normalized wave function of the Rydberg state is valid in the classical region of electron’s motion r ∼ n 2 a B and it is inapplicable inside the atomic core r < a B . The Coulomb potential UC = −Z /r is strongly distorted inside the atomic core. Therefore, it is impossible to obtain the explicit expression for radial wave function Rnl (r ). However, we can determine the dependence of the wave function on the principal quantum number n. The energy E nl = −Z 2 /2n 2∗ of the considered Rydberg state with n  1 is small; we can omit it in the Schrödinger equation inside the atomic core. Then the radial wave function takes the form Rnl (r ) = Anl Rl (r ), r ≤ 1/Z ,

(3.28)

68

3 Rydberg States of Atomic Systems

where the function Rl (r ) does not depend on the principal quantum number n. Thus, the whole dependence on n is contained in the normalized factor Anl only. This expression coincides, the WKB wave function (3.3) at the values of r outside of the atomic core. In the matching region of these solutions (l + 1/2)2 /Z  r  n 2 /Z we can simplify the expression for the electron momentum p L (r ) in Eq. (3.3) putting E nl = 0. Then we obtain 1 Rnl (r ) ≈ r



2r π 2 n 6∗

1/4

    √ 1 + πδl sin 8Zr − π l + 4

(3.29)

(the integral in argument of sine function is derived by substitution 2Zr = x 2 + ν 2 , ν = l + 1/2 ). It follows from (3.28) and (3.29) that Anl =

Cl 3/2

n∗

.

(3.30)

where the quantity Cl does not depend on n. The expressions (3.28) and (3.30) determine the dependence of the radial wave function on the principal quantum number n inside the atomic core. In many cases this result gives the dependence on n for contribution of the values of r < a B into the considered physical quantity, though the form of the function Rnl (r ) is unknown (for example, see the derivation of spin–orbital splitting of nl-term in the previous problem). It should be noted that in the general case of an arbitrary regular potential (r 2 U (r ) → 0 at r → 0 ) the normalized radial wave function of the binding state with the orbital quantum number l at r → 0 is of the form (see also Problem 1.5): Rnl (r ) = Cnl r l + · · · , r → 0

(3.31)

Here the quantity Cnl is the so-called asymptotic coefficient in the origin. It is the important parameter of the considered quantum system. The dependence of this parameter on the principal quantum number n for Rydberg electron states is described by the expression which is analogous to Eq. (3.30); in particular, we have for ns-states 2 (0) = ns

1 2 1 C ∼ 3. 4π n0 n∗

The Coulomb potential U = −Z /r is not distorted at the r → 0 for a singleelectron ionic state. Then Eq. (3.28) takes the form √  1 8Zr Rnl (r ) = Anl J2l+1 r

(3.32)

3.1

Unperturbed Rydberg States of Atoms

69

where J2l+1 (z) is the Bessel function (see Ref. [1], Sect. 36). Now we obtain from comparison of (3.32) with expressions (3.29) and (3.31):  Anl = Z

2 2l+1 Z l+3/2 (q) ; C = nl n3 (2l + 1)!n 3/2

(3.33)

(here, of course, we have δl = 0). It should be noted that WKB expression (3.33) coincides with the exact value of the asymptotic coefficient for s-states. However, when l increases at the fixed value of nr ∼ 1, error of Eq. (3.33) rises quickly (though Eq. (3.33) is asymptotically exact at nr → ∞). A Different expression for this asymptotic coefficient was obtained in Problem 1.5 using another approach; the latter is also asymptotically exact at nr → ∞ , but it is of a high accuracy as well as at nr ∼ 1. The difference of two WKB-results is explained by different corrections for the same main term of asymptotic series at nr → ∞. (q) Finally, we determine the dependence of the coefficient Cnl on l at l  1. Using the Stirling formula for n! we find that (q) Cnl

 √  1 Z l+3/2 l 2 ≈ √ exp −2l · ln . 2 πl 3 n 3 e (q)

The exponential decreasing of Cnl on l  1 is explained by the centrifugal barrier which prevents the penetration of an electron inside the region of small r → 0. This circumstance results in sharp decreasing of contribution into the quantum defect produced by distortion of the Coulomb potential at r → 0 with growing of the orbital quantum number l. This conclusion explains the leading role of the polarization interaction in such problems.

Problem 3.6 Find the asymptotic form of the radial wave function of Rydberg atomic state at large values of r → ∞.

Solution The continuation of the WKB wave function, Eq. (3.3), into the under-barrier region r > b is

70

3 Rydberg States of Atomic Systems

Table 3.4 η l\nr

0

1

2

3

∞

0 1 2 ∞

0.951 0.959 0.961 0.965

0.978 0.982 0.983 0.987

0.986 0.988 0.989 0.992

0.990 0.991 0.992 0.994

1 1 1 1

 (q)

Rnl (r ) = (−1)nr

⎧ r ⎫ ⎨  ⎬  Z 1  p L (r  ) dr  , exp − √ ⎩ ⎭ 2πn 3∗ r p L (r )

(3.34)

b

where the electron momentum is given by  1 | p L (r )| = r



Z n∗

2 r 2 − 2Zr + ν 2 , ν = l +

1 2

(compare with derivation in Problem 3.1). Derivation of the integral in Eq. (3.34) (see Ref. [5], Eq. 2.267) at r → ∞ gives the result (q)

(q)

Rnl (r ) = (−1)nr anl x n ∗ −1 exp(−x); x = Zr/n ∗ ;  Z3 (2e)n ∗ (q) . anl = 4 −ν)/2 2πn ∗ (n ∗ − ν)(n ∗ (n ∗ + ν)(n ∗ +ν)/2

(3.35)

(here e = 2.718 . . .). It should be noted that at fixed value of l and n → ∞ we can neglect the centrifugal potential in Eq. (3.34); this circumstance simplifies essentially (q) the derivation of the integral and of the expression for the quantity anl . However, such simplification is too crude for the finite values of the principal quantum number n. It is interesting to compare this WKB result with the exact solution for pure Coulomb potential U (r ) = −Z /r (for all values of the radial coordinate r ). The asymptotic representation of the exact solution (see Eq. (3.2)) has the same radial dependence as Eq. (3.35) (with n ∗ = n), but with the different value of the coefficient anl : 2n Z 3/2 exact anl . = 2√ n (n + l)! (n − l − 1)! (q)

exact for various values of the orbital In Table 3.4 we present the ratio η = anl /anl quantum number l and of the radial quantum number nr : It is seen that the WKB result is correct even for radial quantum number nr ∼ 1 at the arbitrary orbital quantum numbers l. It should be noted that no restrictions for values of l exist if we use the WKB approximation with Langer correction.

3.1

Unperturbed Rydberg States of Atoms

71

Problem 3.7 Find the relation between the distortion of the Coulomb scattering phase by the atomic core in the collision of a slow electron (ka B  1) with a positive ion A+ and the quantum defect of the corresponding Rydberg state of an atom A.

Solution (1) Asymptotic representation at r → ∞ for the radial wave function of an electron with the energy E = 2 k 2 /2m and angular momentum l scattered by Coulomb potential U (r ) = −e2 /r is  (0) χ El (r )

= r R El (r ) =

  ln (2kr ) πl 2 sin kr + + σlC (k) , − π ka B 2

(3.36)

where the phase of the Coulomb scattering is (see Ref. [1], Sects. 36 and 135):   i . σlC (k) = arg  l + 1 − ka B

(3.37)

The distortion of Coulomb potential at small distances by the field of the atomic core results in modification of the radial wave function. The modification of its asymptotic representation reduces to the variation of the Coulomb phase in Eq. (3.36) only: (3.38) σlC (k) → σlC (k) + δlcs (k). In order to connect the value of δlcs (k) with the quantum defect δl of the Rydberg state, it should be noted that WKB expression (3.3) for the radial wave function is valid also for small values of energy E > 0 (the right classical turning point r = b is absent now, unlike the case of E < 0). Calculating the integral in Eq. (3.3) (see Ref. [5], Eq. (2.67)) we obtain the WKB representation of radial wave function at r → ∞, E > 0:      ln (ka B ) ln (2kr ) πl C 1 π 1 χ El (r ) = √ sin kr + − + − + l+ + πδl . ka B 2 ka B ka B 2 2 k (3.39) The term in square brackets of (3.39) coincides with the asymptotic expansion of the Coulomb scattering phase in Eq. (3.36) at ka B  1: σlC (k)

  1 1 1 π 1  (l + 1 − i/ka B ) l+ + ··· = = ln (ka B ) + − ln 2i  (l + 1 + i/ka B ) ka B ka B 2 2 (3.40)

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3 Rydberg States of Atomic Systems

We have used here the well-known asymptotic representation of gamma-function  (z) at z → ∞. Comparing Eqs. (3.36)–(3.40) with each other we obtain the final result (3.41) δlcs (k) = πδl , ka B  1. Let us make some remarks concerning the obtained expression (3.41). This expression is important, since we can predict the phase shifts for the scattering of low electrons using the experimental values of the quantum defects for discrete levels with E < 0 by means of extrapolation to the positive values of energy. Since the quantity δl depends slowly on the energy of the level, we can perform the linear extrapolation (usually δl is determined as a constant quantity for each value of l at nr → ∞ , or E n → 0). We apply Eq. (3.41) for the case of s-scattering (l = 0) of a slow electron on the He+ -ion in the ground 1s-state. The total spin of both electrons can be equal to S = 1 (the triplet state), or to S = 0 (the singlet state). Now Eq. (3.41) allows to express the scattering phases in these states via quantum defects for ortho-states 1sns 3 S and para-states 1sns 1 S of helium atom, respectively. In the case of orthostates the experimental values of quantum defects derived according to Eq. (3.1) for all n 3 S levels of helium atom (n ≥ 2) are described by linear dependence with a high accuracy; it follows from Ref. [6] that (3)

πδ0 (E) = 0.932 − 0.252E. Here the energy E is measured in atomic units. The Dotted line in Fig. 3.1 presents this dependence for E < 0; according to Eq. (3.41) its linear interpolation into the region of E > 0 allows to find the values of triplet phases for e−He+ (1s) scattering at small electron’s energies. Such approach is applied also in the case of complex positive ions and their atoms. The interesting peculiarity of the phase shift δlcs (k) is that it is nonzero at k → 0. In the case of the short-range potential there is the known dependence for phase shifts (see Ref. [1], Sect. 132) δlshort (k) ∼ k 2l+1 . This distinction can be explained from the physical point of view: in the case of slow particles the long-range Coulomb attraction increases strongly the probability to find a particle on small distances, r ≤ a B , where the Coulomb potential is distorted (compared to the case of a free particle). It is seen from the expansion of the radial √ Coulomb wave function of the continuum state that R CEl (r ) ∼ kr l at r, k → 0, l while for a free particle the expansion takes the form R free El (r ) ∼ k (kr ) at r → 0 [1]. This effect occurs in the inelastic collisions of slow electrons with the charged particles (see Ref. [1], Sects. 143 and 147, and also Problem 3.16). (2) Further, we make some comments about hadron atoms . These are systems containing two hadrons connected by Coulomb attraction. Examples are π − p (pionproton), p p¯ (proton-antiproton) atoms, etc. Unperturbed Coulomb energies of such

3.1

Unperturbed Rydberg States of Atoms

73

n=2 3 δcs0,3(E)

4 5

n=∞ πδ0(3)(E)

Fig. 3.1 Experimental values of quantum defects δ0(3) (E) for n 3 S levels of He atom (circles in the cs (E) for elastic e−He+ left side) and phase δ0,3 (1s) scattering in triplet state with l = 0 (right side)

2  systems are E n(0) = −m Z 1 Z 2 e2 /22 n 2 , where m is the reduced mass of the system, Z 1,2 e are the hadron charges, and Z 1 Z 2 < 0. The Bohr radius of this system is a B = 2 /(|Z 1 Z 2 | me2 ). The spectrum of hadron atoms differs from the above spectrum because of the distortion of the Coulomb potential at the small distances, r < r N by the field of (strong) nuclear forces (r N is the radius of nuclear forces). This distortion is similar to the distortion of Rydberg atomic states by the field of atomic core in usual atoms. Therefore the spectrum of hadron atoms is described by Eq. (3.1). Equation (3.41) is valid as well. Peculiarities of distortion of Coulomb spectrum in lightest hadron atoms are determined by small radius of nuclear forces, r N  a B (for example, a B = 57.6 fm, and ¯ Rydberg correction is small in this case, |δl |  1 (see below r N ∼ 2 fm for p p-atom). Eq. (3.43)). It should be noted that, besides phase δlcs , so-called nuclear-Coulomb (cs) is used; the scattering phase δlcs can be expressed via this scattering length al length at ka B  1 1 1 . (3.42) cot δlcs ≈ − (l!)2 a 2l+1 B (cs) 2π a l

This is the consequence of expansion of the effective radius [7]. Equation (3.42) is the generalization of the well-known relation ([1], Sects. 132–133) from the theory of low-energy scattering by short-range potentials k 2l+1 cot δlshort ≈ −

1 alshort

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3 Rydberg States of Atomic Systems

to the case of systems with Coulomb attraction at large distances. It follows from Eqs. (3.41) and (3.42) that ! (cs) al 1 δl ≈ − arctan 2π . π (l!)2 a 2l+1 B

(3.43)

(cs)

The estimate al ≤ r N2l+1 is valid when there is no resonance in nuclear interac(cs) tion between hadrons (it should be noted that al has the dimension of length for s-wave only). In this case the argument of arctangent function in this expression and Rydberg correction are small. Hence, shifts of Coulomb levels are small compared to the interval between the neighboring levels: 2  m Z 1 Z 2 e2 22 2 2 (cs) δl ≈ a −2l−3 al(cs) = al . E nr l ≈ − [(2l + 1)!!]2 Cnl B 2 2 3 3  n 2m m (l!) n (3.44) We have used in this expression the WKB approximation for the asymptotic coefficient in the origin Cnl derived for Coulomb potential (see Problems 1.5 and 3.5). The obtained expression (3.44) for E nr l is quite a general result, independent on the WKB approximation. It gives the shift of the level in the case of an arbitrary longrange potential U L (here it is the Coulomb potential) produced by non-resonance short-range potential U S (this is nuclear interaction in the case of hadron atoms). This approximation corresponds to so-called perturbation theory with respect to the scattering length. (3) We obtain this expression by the simple method (further applications can be found in Ref. [8]). First, we consider the potential U S as the small perturbation; then we obtain the shift of the level in the first order usual perturbation theory

E nr l =

∞ U S (r ) |nl (r)| dr ≈ 2

U S (r )r 2l+2 dr

2 Cnl

(3.45)

0

(here we have used the smallness of the radius of the potential U S and Eq. (3.31)). The last integral in this expression differs from the l-wave scattering length alBS calculated for the potential U S (r ) in the Born approximation by the factor only. We have alBS

m = 2 2 [(2l + 1)!!] 2

∞ U S (r )r 2l+2 dr.

(3.46)

0

Substitution of Born scattering length in the expression (3.45) for E nr l by the exact scattering length al S gives the result which is similar to the above result (3.44) for E nr l : [(2l + 1)!!]2 2 2 Cnl al S . (3.47) E nr l = 2m

3.1 Unperturbed Rydberg States of Atoms

75

(cs)

It should be noted that the relation al ≈ al S is valid in the non-resonance case, i.e. Coulomb renormalization of the scattering length is small. In the resonance case, when there is shallow level in the nuclear system (real, virtual, or quasi-stationary level at l = 0) with the energy E N  2 /mr N2 , the scattering length is large: al(cs)  r N2l+1 . Then the large shifts of atomic levels are possible (i.e., Coulomb spectrum is reorganized). This effect for energy spectrum of electrons in the impurity semiconductors was first found by Ya.B. Zel’dovich. Peculiarities of this effect depend strongly on the value of the orbital momentum l of the resonance wave. In the case of s-levels the obtained results are valid at the (cs) energies E ∼ 2 /ma 2B . Therefore, if δ0 ∼ 1, all atomic levels have the large shifts simultaneously. The situation changes at l = 0. We should take into account the energy dependence the right side of Eq. (3.42) at E ∼ 2 /ma 2B . Then we find that the distortion of the atomic spectrum does not occur, and the nuclear level (if it exists) is found among the weakly distorted Coulomb levels. This circumstance has the simple explanation: the centrifugal barrier separating nuclear and Coulomb attractive regions has small penetrability. In conclusion, let us remark that inelastic processes are possible in nuclear subsystems if open channels present (for example, the annigilation into π-mesons in the case of p p-atoms). ¯ Then the scattering length becomes a complex quantity, and the atomic levels have both shifts and widths: E = Er − i/2. The width of the level  determines the decay rate w = / for the considered state. Rydberg states of usual atoms have the analogous properties, when the atomic core is found in the excited state. The excitation energy is transferred to the outer electron during the quantum transition of the atomic core to the low-lying level. This electron can be ejected from an atom. Such states of atomic systems with two (and more) excited electrons are called autoionization states.

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation Quantum theory of radiation and absorption of photons is based on the so-called representation of population numbers for a photon subsystem (this is the method of secondary quantization). The interaction Vint between the charged particles and the field of electromagnetic radiation describes a process of radiation, or absorption of one photon only. This process is determined by a small dimensionless parameter— fine structure constant α = e2 /c ≈ 1/137. Therefore, usually the transition probabilities can be derived in the frames of perturbation theory. Let us remember the principal results of quantum theory of radiation in dipole approximation and also the analogous results of classical theory of radiation. The dipole interaction is (3.48) Vint = −dFrad (0).

76

3 Rydberg States of Atomic Systems

Here the dipole moment of an atomic system is d =

Frad (r) = i

# kσ



" a

ea ra , and

% 2πωk $ + ∗ aˆ kσ ekσ exp (ikr) − aˆ kσ ekσ exp (−ikr) V

(3.49)

is the operator of the electric field strength in Schrödinger representation. It corresponds to the field of a free electromagnetic radiation (together with the operator of magnetic field strength). The quantities k, ωk = ck, and ekσ are the wave vector, frequency and polarization vector of a photon, respectively. Since the electromagnetic + wave are transverse, the condition kekσ = 0 takes place. The quantities aˆ kσ and aˆ kσ are the operators of creation and destruction of a photon in the given quantum state. These operators change the population numbers of states:  + |. . . , n kσ , . . . = n kσ + 1 |. . . , n kσ + 1, . . . ; aˆ kσ √ aˆ kσ |. . . , n kσ , . . . = n kσ |. . . , n kσ − 1, . . . . The considered system is placed into large volume V . Then the number dν of independent states of the photon with a given polarization σ and with a given interval of photon moments d 3 p (here p = k) is equal to dν = V d 3 p/ (2π)3 . The radial dependence in Eq. (3.48) is absent since we put r = 0 and exp(ikr) = 1 in the dipole approximation. The wavelength of the emitted photon is λ = 1/k. Thus, dipole approximation is applicable under the condition λ  a; v ∼ ωa ∼ ca/λ  c.

(3.50)

where a is the dimension of the radiated system, and v is the typical velocity of its particles. The single-photon rate is calculated to the well-known first order perturbation theory [1]   2π | f |Vint | i|2 δ E i − E f ∓ ω dν (3.51) dw =  (here dν is the number of final states; the same indexes i, f are referred to whole system and to the states of only one emitter). The signs − and + correspond to the radiation and absorption of a photon with the frequency ω > 0, respectively. Using this expression for the rate of spontaneous emission of a photon in the transition between the binding states we obtain: dwωnσ =

2 ω 3  ∗  dn . e d f i kσ 2πc3

(3.52)

  Here ω = E i − E f / is the frequency of the emitted photon, E i, f are the energies of the radiative system, ekσ and n are the polarization vector, and the direction of its emission, respectively. Thus, k = kn and

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

dfi = f |

#

ea ra | i

77

(3.53)

a

is the matrix element of the dipole moment of the considered radiation transition. Summing Eq. (3.52) over two independent polarizations of a photon we obtain dwωn =

#

dwωnσ =

σ=1,2

2 ω 3  n, d f i  dn . 3 2πc

(3.54)

Then the integration over directions of its emission determines the total emission rate of a photon for the considered transition

wi→ f =

dwωn =

4ω 3  2 dfi . 3c3

(3.55)

Multiplying this expression by the photon energy, ω, we find the radiation intensity for the considered transition Ii→ f = ω · wi→ f =

4ω 4  2 dfi . 3c3

(3.56)

This expression does not contain the Planck constant, that allows to make some analogy with classical theory of dipole radiation for periodical motion of particles according to the so-called correspondence principle. The emission occurs for such classical systems at the frequencies ωn = nω0 , where ω0 = 2π/T and T is the period of motion, n = 1, 2, . . . ˙. The classical radiation intensity is given by expression [9]

I  =

∞ # n=1

In ;

In =

∞ # 4ωn4 2 |d | , d(t) = dn exp (−inω0 t) . n 3c3 n=−∞

(3.57)

Here, I  is the average value of the intensity on the period of radiation, and dn is the Fourier component of dipole moment; it is seen that d−n = dn∗ and, consequently, |d−n | = |dn |. Thus, the quantum expression (3.56) can be obtained from the classical formula (3.57) by formal substitution of the classical Fourier component of dipole moment dn by quantum-mechanical dipole matrix element d f i . The wave functions of the initial and final states of a radiative system are simplified for single-particle transition in the central potential: nlm (r) = Rnl (r ) Ylm (n) . The nonzero matrix elements of the radius-vector are given by expressions 

(l + 1)2 − m 2

n l + 1 |r | nl; (3.58) (2l + 1) (2l + 3)  (l ± m + 1) (l ± m + 2)  

n l + 1 |r | nl.

n l + 1m ± 1 |x ± i y| nlm = ± (2l + 1) (2l + 3)

n l + 1m |z| nlm =

78

3 Rydberg States of Atomic Systems

Hermite conjugated dipole matrix elements for Eq. (3.58) are, of course, also nonzero. It should be noted that we use the same expression for the spherical functions Ylm (n) that in the textbook [1], but without the phase factor i l . The quantity

∞

 

n l = l ± 1 |r | nl =

r 3 Rn  l  (r )Rnl (r )dr

(3.59)

0

is called the radial dipole matrix element (after multiplying by the charge of a particle e). Matrix elements (3.58) are nonzero under the conditions l = l  − l = ±1; m = m  − m = 0, ±1.

(3.60)

These conditions are the selection rules for dipole transitions of a single particle. Using the relation  #   n l  = l ± 1m  |r| nlm2 = l + l + 1 n l  |r | nl2 , 2 (2l + 1) 

(3.61)

m

we can find the photon emission rate for the transition nl → n l  , which is summed over the magnetic quantum numbers m  of the final state: wnl→n  l  =

3  2e2 ωnn  l +l +1

n l  |r | nl2 . 3c3 2l + 1

(3.62)

Of course, this quantity does not depend on the magnetic quantum number m of the initial state of the considered particle. The radial matrix element (3.59) for transitions between nearby states of a particle in the central potential U (r ) is described by the WKB expression 1

n l |r | nl = Tr  

T

r /2

r (t) exp {i [n · ωr t − l · ϕ(t)]} dt, −Tr /2 

n ≡ n − n ; l = l − l  = ±1.

(3.63)

Here r (t) and ϕ(t) are the polar coordinates of a particle on the classical orbit [10], which correspond to the average energy and average angular momentum of this particle:   1 l + l + 1 . E = (E n + E n  ) , M = L = 2 2 Further, the quantity Tr = 2π/ωr is the period of the classical radial motion. The value of the polar angle ϕ(0) = 0 corresponds to the direction to the nearest point

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

79

of the classical orbit; the point r (0) = a is the left classical turning point for radial motion (see Problem 3.11 for details).

Problems and Solutions Problem 3.8 Find the spectral expansion of the intensity of classical dipole radiation for a finite motion of a charged particle in the central potential U (r ) (see also the next problem).

Solution The finite trajectory of a particle moving in the arbitrary central potential, generally speaking, is not closed. The exceptions are the cases of the Coulomb potential U (r ) = −Z e2 /r and of the spherical oscillator potential U (r ) = mω 2 r 2 /2 where, on the contrary, all finite trajectories are closed curves. This property of classical orbits corresponds to the quantum-mechanical accidental degeneration of the levels in above potentials. Therefore, the dependence of the classical dipole moment d(t) = er(t) on the time t is usually an a periodical function. Hence, the expression ωn = nω0 for the spectrum of the dipole radiation (see Eq. (3.57)) is inapplicable. Nevertheless, the radiation spectrum is also a discrete one, though the spectral form of the radiation frequencies changes. Let us consider the classical motion of a particle (see Ref. [10], Sect. 14). Its orbit is placed in the plane x y including the origin. This plane is normal to the conserving angular momentum M directed along the z axis. We introduce the perihelium point x(0) = a, and y(0) = 0 (see Fig. 3.2). The period of the radial motion is 2π = 2m Tr (E, M) = ωr (E, M)

b a

dr , pr (r )

(3.64)

where the radial momentum is    pr (r ) = 2m E − U (r ) − M 2 /2mr 2 . The considered particle is found on the maximum distance b from the origin of the potential after the time t = Tr /2. The values of r = a and r = b are the classical turning points for the radial motion of the particle. The radius-vector of the particle turns on the angle

80

3 Rydberg States of Atomic Systems

Fig. 3.2 The classical motion of a particle in the central potential U(r)

y

r=b

π − ϕ0

a

x

0 r=b

b ϕ ≡ 2ϕ0 (E, M) = 2M a

dr

(3.65)

r 2 pr (r )

after one radial period. Now, we express the Fourier component of dipole moment with the arbitrary values of frequency ω via the integral over one radial period [using above considerations and the relation x + i y = r exp(iϕ)]: xω(0)

+ i yω(0)

1 ≡ 2π (0)

τ [x (t) + i y (t)] exp (iωt) dt, τ = Tr /2. −τ (0)

It should be noted that xω and yω are real quantities because of the symmetry of (0) (0) the orbit; besides of this, xω is the even function of the frequency ω while yω is the odd function of ω. The integral over the next period can be expressed via the previous integral by means of simple relation 1 2π

τ +Tr

  [x (t) + i y (t)] exp (iωt) dt = xω(0) + i yω(0) exp (2iϕ0 + iωTr ) .

τ

The analogous procedure can be made in the integrals over subsequent time periods. Thus, we obtain: 1 2π

T −T

N  #  qk, [x (t) + i y (t)] exp (iωt) dt = xω(0) + i yω(0) k=−N

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

81

where the notations are introduced   1 1 Tr ; q ≡ exp (2iβ) ; β ≡ ϕ0 + ωTr . T = N Tr + τ = N + 2 2 After summation over k we find 1 2π

T −T

 sin [(2N + 1) β]  . [x (t) + i y (t)] exp (iωt) dt = xω(0) + i yω(0) sin β

(3.66)

In order to calculate the Fourier components xω and yω which correspond to the expansion of the functions x (t) and y (t) in the Fourier integral, we should go to the limit T → ∞. Then, we have also N → ∞. It is seen from the right side of Eq. (3.66) that this limit does not exist, from the mathematical point of view. The explanation is that the functions xω and yω are so-called generalized functions. They can be expressed via Dirac δ-function. Let us remember that lim

N →∞

sin N x = πδ (x) . x

In order to derive this formula we take into account that

∞ −∞

sin N x dx = f (x) x

∞ f −∞

 z  sin z dz. N z

For any regular function f (x) in the limit of N → ∞ we can substitute this function by its constant value of f (0). Then, we obtain that this integral is equal to π f (0) . This is the proof of the above relation. The function sin(2N + 1) x/ sin x in Eq. (3.66) is restricted in any interval which does not contain the values xn = πn, and this function oscillates infinitely at N → ∞. Further, this function should be equal to zero for x = xn , since the integration of product of this function by any regular function results in zero. We have nearly of the values of x = xn : sin [(2N + 1) (x − xn )] sin (2N + 1) x ≈ , sin x (x − xn ) so that we obtain the function πδ (x − xn ) at N → ∞. Using these transformations, we find

82

3 Rydberg States of Atomic Systems

1 xω + i yω = 2π =

∞ [x (t) + i y (t)] exp (iωt) dt −∞

∞  #  & ϕ0 ' 2π  (0) xω + i yω(0) δ ω − ωr n − Tr π n=−∞

Here, we took into account that δ (ax) = |a|−1 δ (x) . It follows from this relation that the radiation spectrum contains discrete frequencies, since the radiation probability is determined by Fourier components xω and yω . The radiation spectrum contains two groups of equidistant frequencies:   ϕ0  ϕ0  (2) ω ωr . = n − ; ω = n + ωn(1) 1 r 2 n 1 2 π π

(3.67)

Minimum values of the integers n 1 , n 2 are determined by the condition that ωn > 0. It should be noted that if the turning angle ϕ = 2πs/k, where s, k are integers, i.e., if this angle is the rational function of 2π, then the orbit becomes the closed ( = kTr . In this curve, and the motion of a particle is periodical with the period T case, the radiation spectrum (3.67) contains only frequencies which are multiple to ( = ωr /k. However, a part of frequencies is absent the basic frequency ω0 = 2π/T in the spectrum. It is explained by the fact that many Fourier components vanish due to the definite symmetry of the considered orbit. The obtained property of the dipole radiation spectrum has the simple and more physically clear interpretation in quantum theory of radiation transitions between WKB states. In the latter case, it follows from the quantization rule for energy spectrum and the selection rule |l| = 1 for single particle dipole transitions. Each of two groups of frequencies in Eq. (3.67) corresponds to the fixed value of l ≡ l  − l which is equal to +1, or −1, respectively (see the next problem). In conclusion, we discuss how to derive the radiation intensities from above δ-function dependencies for Fourier components. The total radiation energy is determined by the integral:

∞ εrad = −∞

4π I (t) dt = 3 3c

∞ ω 4 |dω |2 dω. −∞

Let us substitute |dω | ∼ δ (ω − ωn ) into the integrand of this expression. Then we obtain the infinite value of the radiation energy εrad ∼ δ (0) = ∞. The point is that in such situation we should determine the discrete spectral expansion of the mean value of the radiation intensity, instead of spectral distribution of the total radiation energy [9]. The connection between these approaches can be found in the considered case from the interpretation of delta-function δ (0) based on its expansion in Fourier integral:

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

1 δ (ω) = lim T →∞ 2π

T /2 exp (iωt) dt; δ (0) = lim

T

T →∞ 2π

−T /2

83

.

The time T  Tr determines here the duration of the radiation pulse, so that εrad =

I  T. In order to illustrate this relation, we consider the case of periodical dependence of d(t) when the expansion in Fourier series is used in the limits of one period, as a rule (see (3.57)). The expansion in Fourier integral can be used as well as in this case, from the formal point of view. Thus,

∞ d(t) =

dω exp (−iωt) dω, where dω = −∞

∞ #

dn δ (ω − nω0 ) .

n=−∞

Now simple derivation of the radiation energy εrad according to above formula results in usual expression (3.57) for the radiation intensity I .

Problem 3.9 Find the dipole radiative spectrum of a charged particle in the central potential U (r ) for transitions between WKB states with n  n and l  1. How this spectrum is related with the radiation spectrum of a classical particle (see the previous problem).

Solution   The frequencies of emitted photons are ω = E nr l − E nr l  /. According to the dipole selection rules we have l  = l ± 1. The energy spectrum of the initial and final WKB states is determined from Bohr-Sommerfeld quantization rule ) !  

b * 2 (l + 1/2)2 1 *  1 +2m E − U (r ) − , dr = π nr + nr l  2mr 2 2 a

) !  

b˜ * 2 (l  + 1/2)2 1 *  1 +2m E   − U (r ) −  . + dr = π n nr l r  2mr 2 2 a˜

(3.68)

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3 Rydberg States of Atomic Systems

In Eqs. (3.68) next substitutions can be done:  1 1 E nr l + E nr l  ; E nr l(nr l  ) = E¯ ± ω, E¯ = 2 2    1 1  L = L¯ + l − l , L  = L¯ + l −l ; 2 2

 1 L + L ; L¯ = 2

where L = l + 1/2 and L  = l  + 1/2. According to the conditions of this problem ¯ Therefore, we can expand Eq. (3.68) over we have ω  E¯ and l − l   = 1  L. these small quantities (ω and l). Thus, we find the WKB-spectrum of emitted photons ⎤ ⎡

b ¯ L 2π ⎣ l − l ω= dr ⎦ . nr − nr  + Tr π r 2 pr

(3.69)

a

Here, pr and Tr are usual WKB radial momentum and period of radial oscillations, ¯ respectively, for a particle with the energy E¯ and with the angular momentum L. Equation (3.69) coincides with the corresponding result of classical electrodynamics for spectral expansion of dipole radiation which is described by Eq. (3.67). Let us underline that the values of the physical quantities (energy and momentum) determining the motion of a classical particle are found to be equal to their mean values for the corresponding quantum transitions. Usually, such substitution allows to apply WKB results for small values of the principal quantum number n ∼ 1.

Problem 3.10 Find the WKB expression for the radial dipole matrix element nl ± 1 |r | nl which corresponds to transition between components of fine structure in the case of hydrogen atom with n = n  . Compare the obtained result with the exact value.

Solution In the calculation of the dipole matrix element we use the radial WKB wave functions, Eq. (3.3) (without quantum defect). We contain in product of sine functions in integrand only the term with the difference of WKB-phases (see, for example, Refs. [1, 11]). The contribution of the term with the sum of phases is negligibly small due to its strong oscillation. Thus,

nl  |r | nl =

1 πn 3

b a

⎧ r ⎫

⎨ ⎬   r  pr − pr dr  dr. cos ⎩ ⎭ pr pr a

(3.70)

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

85

The expressions for radial momenta pr , and pr differ only by values of orbital quantum numbers in the centrifugal energy. We introduce notations        1 1 2 1 2  1 + ; = l¯ + + l − l  l¯ + L 21 ≡ l + 2 2 2 4  2  2     1 1 1 1 L 22 ≡ l  + = l¯ + − l − l  l¯ + + ; 2 2 2 4

(3.71)

Here, we determined the mean value of the orbital quantum number for the considered transition as  1 l + l  ; l − l  = ±1. l¯ ≡ 2 The expansion of difference of radial momenta in Eq. (3.70) according to Eq. (3.71) allows to write the dipole matrix element in the form

nl ± 1 |r | nl =

1 πn 3

b a

⎧ r ⎫

¯ ⎨ r l + 1/2  ⎬ cos dr dr, ⎩ r 2 p¯r (r  ) ⎭ p¯r (r )

(3.72)

a

where the notation is introduced )  *  2  * l¯ + 1/2 1 + p¯r (r ) = 2 E n + − , r 2r 2

En = −

1 . 2n 2

(we omit the term 1/4 in the right side of Eq. (3.71)). The integral in the argument of cosine function in Eq. (3.72) has the simple explanation in classical mechanics [10]: this integral determines the radial dependence of polar angle  ϕ (r ) of a particle with the energy E n and with the angular momentum M =  l¯ + 1/2 . The radius-vector of this particle is directed from the field origin to the nearest point of the orbit, where ϕ = 0 and r = a. Therefore, the relation (3.72) can be rewritten taking into account the symmetry of the orbit with respect to the large semi-axis of elliptic Coulomb motion

nl ± 1 |r | nl = n0 rclass (t; E n , M)

(3.73)

Here, n0 is the unit ort-vector along the radius-vector with ϕ = 0, and rclass (t; E n , M) is the mean value of the radius-vector of a particle for the cycle of its elliptic motion. The period of this motion is T = 2πn 3 . According to the classical Newton laws for Coulomb motion we have [10] 1 cos ϕ (r ) = ε



  M2 − 1 , ε ≡ 1 + 2E n M 2 r

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3 Rydberg States of Atomic Systems

(ε is the eccentricity of the ellipse). After derivation of the simple integral in Eq. (3.72) we find   2 l¯ + 1/2 3ε 3 2

nl ± 1 |r | nl = =− n 1− . (3.74) 4E n 2 n2 Here, sign “−” in the right side corresponds to the fact that the vector rclass (t; E n , M) is directed along the larger semi-axis of the ellipse to the to the most remote point of the orbit with ϕ = π. The obtained WKB expression (3.74) coincides the exact value of the radial matrix element for arbitrary values of quantum numbers n and l. It should be noted the importance of choice of a mean value of angular momentum as a mean value for the considered transition. Such approach usually gives better applicability of WKB approximation for the moderate values of quantum numbers.

Problem 3.11 Find the value of WKB radial dipole matrix element n 2 l2 |r | n 1l1  for hydrogen atom, where l2 = l1 ± 1 and n ≡ |n 1 − n 2 |  n 1,2 , using classical Fourier components of electron coordinates for the elliptic trajectory.

Solution (1) We write the expression for the WKB radial dipole matrix element from the solution of the previous problem 1

n 2 l2 |r | n 1 l1  = π(n 1 n 2 )3/2

b √ a

r pr 1 pr 2

⎧ r ⎨

cos

⎩

a

( pr 1 − pr 2 ) dr 

⎫ ⎬ ⎭

dr.

Let us expand here the argument of cosine function in Taylor series on small values of n and l:   ! l¯ + 1/2 l ωn ¯ − pr 1(2) = p¯r (r ) ± ; p¯r (r ) r 2 p¯r (r ) where the notations are introduced √ 1 (n 1 + n 2 ) ≈ n 1 n 2 ; 2 1 1 1 E¯ = (E n1 + E n2 ) ≈ − 2 ; ω¯ ≈ 3 . 2 2n¯ n¯

n ≡ n 1 − n 2 ; l = l1 − l2 = ∓1; n¯ = 1 l¯ = (l1 + l2 ) ; 2

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

87

The radial momentum p¯r (r ) is determined by the mean values of energy E¯ and orbital quantum number l¯ for the considered transition. Thus, the WKB radial matrix element takes the form: ⎧ r ⎫   !

b ⎨

l¯ + 1/2 r 2 dr  ⎬

n 2 l2 |r | n 1 l1  = l ωn ¯ − cos dr, ⎩ T p¯r (r ) r 2 p¯r (r  ) ⎭ a

a

(3.75) where T = 2π/ω¯ is the period of elliptic motion. Taking into account the dependence of the classical radius-vector on time, we obtain the relation between the WKB radial matrix element and classical Fourier components of electron’s coordinates:

n 2 l2 |r | n 1l1  = xn − i yn l, l = l1 − l2 = ∓1. Here, we defined classical Fourier components of electron’s coordinates as 1 xs = T

0

2 x (t) exp (is ωt) ¯ dt = T

b r cos ϕ cos (s ωt) ¯ dt; a

1 ys = T

0

2 y (t) exp (is ωt) ¯ dt = i T

b r sin ϕ sin (s ωt) ¯ dt; a

(compare with the solution of the previous problem). We suppose that the electron’s trajectory is found in (x y)-plane, and the axis X is directed along the angle ϕ = 0. It should be noted the symmetry relations xs = x−s and ys = −y−s . In particular, we have y0 = 0. These Fourier components are calculated in the textbook of Landau and Lifshitz ([9], Sect. 70). Using these values, we find 1  

n 2 l2 |r | n 1 l1  = 2n  E¯ 



√  Jn

(εn) + l

 1 − ε2 Jn (εn) . ε

(3.76)

  2 Here Jn (z) is the Bessel function, and the quantity ε = 1 + 2 E¯ l¯ + 1/2 is the eccentricity of the ellipse. The applicability of the obtained results is extended, as usually, up to values of quantum numbers n, l of order of 1. The accuracy of calculations according to Eq. (3.76) is of order of 1–3 % even for the transitions 2 p–1s, 3 p–2s and 3s–2 p (with n 1,2 ∼ 1). This is seen from the Table 3.5 for WKB and exact (from Ref. [12]) radial dipole matrix elements for the transitions between low-excited hydrogen states. (2) Let us consider the case 1  n  n in more detail. It follows from properties of Bessel functions that the values of the radial dipole matrix element, Eq. (3.76),

88

3 Rydberg States of Atomic Systems

decrease quickly with the increasing of the eccentricity ε at the fixed value of n. Therefore, we restrict ourselves by the case of strongly stretched elliptic orbits having (1 − ε)  1. In this case the Bessel function and its derivative can be simplified:  Jn (nε) ≈ Jn

2 |n|

2ε (nε) ≈ − n



1/3 Ai 

|n| 2

|n| 2

1/3 Ai



2/3  

|n| 2

1 − ε2

! 

;

! 2/3   2 1−ε .

(compare to Ref. [9]). Here we have     1 − ε2 = 2  E¯  (l + 1/2)  1; |n|  1. It should be noted that the Airy function Ai (z) can be expressed via the√ function  (z) used in the standard textbooks [1, 9] by relation Ai (z) =  (z) / π. The Airy function has positive values at z ≥ 0 (see Ref. [13]), and this function decreases monotonically with the increasing of z > 0. Hence, Ai  (z) < 0 at z > 0. Substituting these asymptotic expressions for Bessel functions into Eq. (3.76), we obtain: 1

n 2 l2 |r | n 1l1  =    4 E¯ 



2 |n|

5/3    l · n    2 −Ai t + t Ai t 2 , |n|

(3.77)

where 1  n  n and the notation is introduced       1 2 |n| 2/3 . t 2 ≡ 2  E¯  l¯ + 2 2 Now we summarize the properties of the WKB radial dipole matrix elements (3.76) (or (3.77)). They decrease with rise of the orbital quantum number l at the fixed values of the principal quantum numbers n 1 and n 2 . Such decreasing is of the exponential form in the limit l¯ ≥ n 1(2) / |n|1/3 . This follows from the well-known asymptotic representation of the Airy function   2 3/2 . Ai (z) ∼ exp − z z→∞ 3 In the case of moderate values of the orbital quantum number l¯  n 1(2) / |n|1/3 ¯ Only the first term in the right the WKB matrix element (3.77) does not depend on l. side of (3.77) should be taken into account. Then we obtain the simple approximation for WKB radial dipole matrix element:

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

89

Table 3.5 Radial matrix elements of transitions between some hydrogen states Transition

2 p − 1s

3 p − 2s

3d − 2 p

3s − 2 p

3 p − 1s

WKB Eq. (3.76) Exact

1.29 1.29

3.03 3.07

4.76 4.75

0.91 0.94

0.41 0.52

n 2 l2 |r | n 1l1  ≈

1   4 · 31/3  (1/3)  E¯ 



2 |n|

5/3 .

Here we used the numerical value for Ai  (0) = −

1 ≈ −0.2588. 31/3  (1/3)

Finally, it should be noted that it follows from Eq. (3.77) that the radial dipole matrix elements for transitions with the variation of n and l in the same direction (i.e., n > 0 at l = +1 and n < 0 at l = −1) should be larger than the values of these matrix elements in the case of variation of n and l with opposite signs (n · l < 0). This statement is explained by the fact that in the first case both terms in Eq. (3.77) have the same signs while in the second case they have opposite signs. This property (so-called Bethe’s rule) takes place as well as for transitions between low-excited states: compare the matrix elements for transitions 3 p–2s and 3s–2 p in above cited in Table 3.5. Simple explanation can be seen from Eq. (3.75). Quick oscillations of the cosine function diminish the value of the initial integral for WKB matrix element at opposite signs of n and l.

Problem 3.12 Generalize the results of the previous problem to the case of dipole transitions with n ∼ n 1(2) . Consider, in particular, the case of n 2  n 1 .

Solution Usually, the derivation of matrix elements with strongly different quantum numbers is very cumbersome due to fast oscillations of product of wave functions in the integrand. However, the calculation is more simpler in the case of Coulomb potential. Indeed, radial wave functions of high-excited states with different values of the principal quantum numbers n 1  1, n 2  1 and with the same value of the orbital quantum number l have the same form in sufficiently broad region nearly of the field origin r → 0. According to solution of Problem 3.5 we have

90

3 Rydberg States of Atomic Systems

√  1 Rnl (r ) ∼ √ J2l+1 8r . r This region gives the main contribution into the value of the integral for matrix element n 2 l2 |r | n 1l1  in the case of large principal quantum numbers n 1  1, n 2  1 and l ≈ l  . Therefore, the WKB expression for the radial matrix element is of the same form as in the case of |n|  n 1,2 :

b

1

n 2 l2 |r | n 1 l1  = π(n 1 n 2 )3/2

√ a

r pr 1 pr 2

⎧ r ⎨

cos

⎩

( pr 1 − pr 2 ) dr 

a

⎫ ⎬ ⎭

dr

(3.78)

(again the contribution of the term with ⎧ r ⎨

cos

⎩

( pr 1 + pr 2 ) dr 

a

⎫ ⎬ ⎭

is negligibly small due to rapid oscillations of the integrand). Terms with energy in WKB-momenta are small corrections in the essential region of integration in Eq. (3.78). Therefore, we can expand these momenta in Taylor series: pr 1(2)

  l · l¯ + 1/2 E n1(2) ∓ ≈ p¯r (r ) + . p¯r (r ) 2r 2 p¯r (r )

Here the notations are introduced   2 l¯ + 1/2 2 1 p¯r (r ) = − ; l¯ = (l1 + l2 ) ; l = l1 − l2 = ∓1. 2 r r 2 Then Eq. (3.78) takes the form 1

n 2 l2 |r | n 1l1  = π(n 1 n 2 )3/2

∞ a

⎧ r ⎫  ⎨ dr   r dr l¯ + 1/2 ⎬ cos ω − l ⎩ p¯r (r  ) n 1 n 2 ⎭ p¯r (r ) r 2 a

(compare with Eq. (3.75)). This matrix element can be expressed via classical Fourier components of electron’s coordinates, analogous to the solution of the previous problem. However, now these components should be calculated for the parabolic orbit with the transition frequency 1 1 ωn 1 n 2 = 2 − 2 , E¯ = 0. 2n 2 2n 1

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

91

Table 3.6 Coefficients C, Eq. (3.81) Transition

n 1 p − 1s

n 1 p − 2s

n1s − 2 p

n1 d − 2 p

WKB, Eq. (3.81) Exact value

2.00 2.17

6.82 6.63

2.33 1.91

7.15 7.66

These Fourier components can be calculated from Eqs. (3.75)–(3.77) of the previous problem by means of substitutions ωn ¯ → ωn 1 n 2 and E¯ = −ω¯ 3/2 /2 → 1/2n 1 n 2 . Then we find

n 2 l2 |r | n 1l1  =  ω n

1 n2

1    2 22/3 −Ai  t 2 + σt Ai t 2 . 5/3  (n 1 n 2 )3/2

(3.79)

Here the notations are introduced  1/3     ωn 1 n 2  ωn n 1 t ≡ l¯ + ; σ ≡  1 2  l = ±1.  2 2 ωn 1 n 2  Equation (3.79) is matched with the Eq. (3.77) from the previous problem, as it should be. The properties of the matrix element found in the end of the previous problem are valid as well as for Eq. (3.79). In the limit 1  n 2  n 1 Eq. (3.79) can be simplified; then it takes the form

n 2 l2 |r | n 1l1  =

C (n 2 , l2 , l1 ) 3/2

n1

,

(3.80)

where the notations are introduced  C (n 2 , l2 , l1 ) = u≡

4n 22

4 n2

l¯ + 1/2 (2n 2 )2/3

1/6 1    2 −Ai  u 2 + l · u · Ai u 2 ;

(3.81)

; l = l1 − l2 .

Since the localization region for the wave function of n 2 l2 −state is much less than the region for the wave function of n 1 l1 -state because of the inequality n 2  n 1 , then the dependence (3.80) for the radial matrix element on the principal quantum −3/2 number n 1 (∼ n 1 ) is the general property for Rydberg states which we discussed in Problem 3.5. It should be noted that the applicability of Eq. (3.80) is expanded to small values of the quantum number n 2 . It is seen from the Table 3.6 for coefficients C: Exact values of these radial dipole matrix elements for hydrogen atom are taken from Ref. [12]. In conclusion let us remark that the obtained asymptotic expressions (3.79) and (3.80) for matrix elements with n  1 decrease as some negative powers of

92

3 Rydberg States of Atomic Systems

the principal quantum number n 1 . Therefore, we can derive radial matrix elements using WKB wave functions (3.3) in the classical region. Such property of asymptotic expansions follows from the non-analytical form of Coulomb potential U (r ) = −1/r in the origin. Indeed, the region near the origin is the most important in the above calculations of the matrix elements. When the orbital quantum number l increases, the Coulomb singularity is damped by the centrifugal potential barrier. In the latter case, the matrix element is the exponentially decreasing function of the principal quantum numbers (compare to the Problem 3.5).

Problem 3.13 Calculate the radial matrix element n 2 l2 |r | n 1 l1  between high-excited states of an atom with the quantum defect, if n  1.

Solution The WKB-method developed for calculations of matrix elements with |n|  1 in the previous problem is valid also when the Coulomb potential is distorted by the field of atomic core on small distances r ∼ a B . Therefore, we can use the relation of the previous problem for radial dipole matrix element: 1

n 2 l2 |r | n 1l1  = ∗ π(n 1 n ∗2 )3/2

∞ a

⎧ r ⎫ ⎨

⎬ r dr dr  cos +  . ωn 1 n 2 ⎩ ⎭ p¯r (r ) p¯r (r  )

(3.82)

a

We have changed the principal quantum numbers by the  effective principal quantum  numbers n ∗1,2 and added the phase  = π δl1 − δl2 into the argument of cosine function. Here the quantities δl1,2 are the quantum defects of the considered states (see (3.3)). The transition frequency is defined now as ωn 1 n 2 =

1 1 − ∗2 . ∗2 2n 2 2n 1

Besides of this, we have neglected the centrifugal energy in the expression for p¯r . In fact, the quantum defect is nonzero only for states with small values √ of orbital quantum numbers (in practice, for l = 0, 1, 2 and 3). Hence, p¯r (r ) = 2/r . Substitution of the integration variable r = ξ 2 allows to simplify the above expression for radial matrix element √

2

n 2 l2 |r | n 1 l1  = π(n ∗1 n ∗2 )3/2

∞ 0

√  2 ωn 1 n 2 ξ 3 +  dξ. ξ 4 cos 3

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

93

The integration by parts simplifies the form of the integrand: 2

n 2 l2 |r | n 1l1  = ∗ ∗ π(n 1 n 2 )3/2 ωn 1 n 2

√

∞ ξ sin

 2 3 ωn 1 n 2 ξ +  dξ. 3

0

Sine function of the sum in the integrand can be presented as the sum of two terms. Substituting the integration variable ξ by √  2  ωn 1 n 2  ξ 3 x= 3 we obtain

n 2 l2 |r | n 1 l1  =

22/3

∞ 



5/3 31/3 π(n ∗1 n ∗2 )3/2 ωn 1 n 2  0

dx {cos  · sin x + σ sin  · cos x} , x 1/3

Here σ = ±1 and the sign of σ coincides the sign of the transition frequency ωn 1 n 2 . This integral is simply derived using the formula

∞

exp (i x) dx =  (2/3) exp (iπ/3) ;  (2/3) = 1.3541 . . . x 1/3

0

by means of separation of real and imaginary parts of this expression. Finally, we obtain

n 2 l2 |r | n 1 l1  =

22/3  (2/3) sin (π/3 + σ) .   31/3 π(n ∗1 n ∗2 )3/2 ωn n 5/3

(3.83)

1 2

It should be noted that the matrix element oscillates as a function of . In conclusion let us underline that the most important contribution in the integrand of (3.82) is given by the region  −2/3 r ∼ ξ 2 ∼ ω n 1 n 2  . These values are large compared to the radius of the atomic core, if |n|  1. Therefore, Eq. (3.83) is valid both under the condition 1  |n|  n 1,2 , and also when |n| ∼ n 1,2 . In the case of  = 0 Eq. (3.83) reduces to results of two previous problems (at the small values of the orbital quantum number).

94

3 Rydberg States of Atomic Systems

Problem 3.14 Calculate the rate for spontaneous radiation transition between high-excited atomic states with the principal quantum numbers n, n   1 and n = n −n   1. Average this rate over all values of orbital and magnetic quantum numbers in the assumption that the populations of all sublevels are the same.

Solution The states with orbital quantum numbers l   1 are the most important in this problem. Therefore, we can neglect quantum defects and use the radial matrix elements for hydrogen atom (see Eq. (3.77)). Substituting these matrix elements into Eq. (3.62), we obtain the required transition rates 1    22  27/3 2l¯ + 1  1  2 −Ai t + l · t Ai t2 . (3.84) w nl → n l  = 3 3c 2l + 1 ω 1/3 (nn  )3 Here ω = ωnn  =

1 1 − 2 2n 2 2n

is the frequency of the emitted spontaneous photon. Also, the notations are introduced    1 1  ω 1/3  ¯ ¯ l +l ; t = l + . l = l − l = ±1; l = 2 2 2 

Since the typical values of l¯  1 we can put l¯ ≈ l; then the total rate of dipole transition between the considered atomic levels is given by simple expression       w nl → n  = w nl → n l + 1 + w nl → n l − 1 1    2 210/3 Ai 2 t 2 + t 2 Ai 2 t 2 . = 3 3c3 ω 1/3 (nn  )

(3.85)

Let us average this expression over initial orbital and magnetic quantum numbers in the assumption that all n 2 sublevels are populated with the same probability: 

n     1 # w¯ n → n  = 2 (2l + 1) w nl → n  n

(3.86)

l=0

 (it should be noted that since lmax = n  − 1, the dipole transitions with l > n  into  states of n −level are forbidden). Substituting Eq. (3.85) into Eq. (3.86) and changing the sum by the integration, we obtain

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

  w¯ n → n  =

210/3 3 3c n 3 n 5 ω 1/3

n 

95

   2 1 2l Ai 2 t 2 + t 2 Ai 2 t 2 dl.

(3.87)

0

Here, t = l (ω/2)1/3 . Let us underline that the quantity t  1 at l  ∼ n   1 and n  1. Therefore the integrand in Eq. (3.87) is exponentially small near the upper limit; hence, the upper integration limit can be expanded to infinity. We rewrite Eq. (3.87) substituting x = t 2 :   w¯ n → n  =

16 3 3c n 3 n 5 ω

∞ 1

2 Ai 2 (x) + x Ai 2 (x) dx.

0

 √ −1 The integral in this expression is equal to 2π 3 . In order to calculate it we should use the connection between Airy function and McDonald functions: 1 Ai (z) = π



z K 1/3 3



 2 3/2 ; z 3

z Ai (z) = − √ K 2/3 π 3 



2 3/2 z 3



and also by Eq. (6.576) in Ref. [5]. Thus, we obtain the final expression (so-called Kramers formula):   8 w¯ n → n  = √ ; n  1, n   1. 3 3 3πc n 3 n 5 ωnn 

(3.88)

  Summing the rates w¯ n → n  over all values of n  , we obtain the total rate for spontaneous transition. It should be noted that the values of n  , for which Eq. (3.88) is inapplicable, do not contribute significantly any contribution into the total transition rate. Indeed, though the transitions with n ∼ 1 correspond to maximum values of 3 radial dipole matrix elements (∼ n 2 ) , the partial rates for such transitions w ∼ ωnn  are strongly diminished due to smallness of the corresponding transition frequencies ωnn  ∼ n −3 . On the other hand, transitions to low excited states are important only for values of the orbital quantum number l ∼ 1 . Spontaneous rate is especially large for the transition into the ground state if this transition is permitted by the dipole selection rule. Nevertheless, the contribution of transitions with n  ∼ 1 is negligibly small in the above sum due to relatively small number of the corresponding final states. In order to summarize Eq. (3.88) over n  we change the sum by the integration over dn  (we have dn  /n 3 = −dωnn  ) . The integrations limits are [∼1, n]. Thus, we find the averaging rate of spontaneous transition with logarithmic accuracy from Rydberg state with the principal quantum number n:

96

3 Rydberg States of Atomic Systems

w¯ n ≈

8 ln n . √ π 3c3 n 5

(3.89)

Further discussion of obtained results continues in the next problem.

Problem 3.15 Find the total rates of dipole transitions w (nl → l ± 1) for atomic Rydberg states with l  1 summed over principal quantum numbers n  of final states. Compare rates with transitions with l  = l + 1 and l  = l − 1. Calculate also the total rate for spontaneous transitions from nl-states and average it over orbital quantum numbers of the initial level. Solution Equation (3.84) gives starting expression for rates of spontaneous emission. Now we summarize the rates over values of n  changing it by the integration and using the relation dn  = −dωnn  . n 3 Then we obtain   #   w nl → l  = l ± 1 = w nl → n l ± 1 ≈

(3.90)

n

4 ≈ 32 3 ¯ c l n

∞

$

−Ai  (x) + l ·

√

x · Ai (x)

%2

dx, l = l − l 

0

Here we have made the substitution of the integration variable x = l¯2 (ωnn  /2)2/3 . Calculation of the integral in (3.90) is analogous to the calculations in the previous problem. Finally, we obtain   w nl → (all n  ) l  =

2



√ π 3c3 l¯2 n 3

It should be noted that the ratio 1+ 1−

√ √

3π/6 3π/6

≈ 20.5

√  π 3 1 + l . 6

(3.91)

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

97

is numerically very large. Therefore, the rate for the transition l → l + 1 is small compared to the rate for the transition l → l − 1. This is so-called Bethe’s rule [3]. According to this relation the spontaneous radiative transition occurs with decrease of electron’s angular momentum, as a rule (see also Problem 3.11). It follows from (3.91) that after addition of transition rates with l  = l ± 1 we obtain very simple expression wnl =

4 . √ π 3c3 l¯2 n 3

(3.92)

This expression determines the rate for spontaneous emission of Rydberg nl-state. Averaging (3.92) over all n 2 degenerate states with a given value of n in the approximation that they have equal population probabilities, we obtain the final result n−1 1 # 8 ln n . (3.93) w¯ n = 2 (2l + 1) wnl ≈ √ n π 3c3 n 5 l=0

(since ln n  1, inaccuracy in values of transition rates with l ∼ 1 does not matter in this result). Of course, the Eq. (3.93) coincides the result (3.89) from the previous problem found using Kramers formula.

Problem 3.16 Using WKB approximation, calculate the cross-section for photoionization of high excited atomic level with the principal quantum number n by the electromagnetic field with the frequency ω.

Solution The transition rate i → f produced by the monochromatic electric field with the field strength F (t) = F0 cos ωt =

1 F0 (exp (iωt) + exp (−iωt)) 2

is given by Eq. (3.51). It corresponds to absorption of one photon; the quantity Vint should be substituted by eF0 r/2; the sign +ω should be chosen in the argument of Dirac delta-function (see Ref. [1], Sect. 42). In order to obtain the differential cross-section, we must divide dw by the density j of the photon flux. The time-averaged density of the energy flux in the electromagnetic wave is given by Poynting vector

98

3 Rydberg States of Atomic Systems

  S¯  = c F 2 cos2 ωt = c F 2 , 4π 0 8π 0 while the density of photon flux is obtained from this expression dividing by ω,  i.e. j = S¯  /ω. The atomic system of units e =  = m e = 1 will be used in this problem. Thus, we find the cross-section of photoionization dσion,nl =

  4π 2 ω | f |rF0 | i|2 δ E i + ω − E f dν. 2 cF0

(3.94)

WKB wave functions of the initial and final atomic states are of the form ⎧ r ⎫  ⎨

2 π⎬ sin pi dr + i ≡ nlm = Ylm (n) ; ⎩ πn 3 pi (r ) 4⎭ a ⎧ r ⎫ 

⎨ 2k π⎬ p f dr +  f ≡ kl  m  = sin Yl  m  (n) . ⎩ π p f (r ) 4⎭ a

(see Eq. (3.3) and Problem 3.7). Here, the notation is introduced for WKB radial momentum  2  li( f ) + 1/2 2 . pi( f ) (r ) = 2E i( f ) + − r r2 The radial wave function  f of continuum state is normalized by the Dirac delta function δ k − k  ; the notations are also introduced Ei = −

1 ; 2n 2

Ef =

1 2 1 k f = ω − 2 > 0; k = p f (∞) . 2 2n

In order to calculate the cross-section, we summarize over final continuum states in Eq. (3.94) using the rule:

dν (. . .) =

#

dk (. . .) .

lm

After averaging over magnetic quantum numbers m of the initial atomic state we obtain # 2π 2 ω l + l  + 1    k f l  |r | nl2 . σ¯ ion,nl (ω) = · (3.95) 3ck f 2l + 1  l =l±1

Here we have used the rules for summing over m (see Eq. (3.54)). We took also into account that the substitution of the radius-vector r by its z-component in the matrix element produces the factor 1/3 ( z = rF0 /F0 ).

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

99

The difference between the matrix elements k f l  |r | nl in Eq. (3.95) and in Eq. (3.79) is only in the normalization factor of the radial wave function of the final  continuum state. We should substitute the factor 2/πn 32 by the factor 2k f /π. Then the cross-section for photoionization, Eq. (3.95) can be rewritten in the form σ¯ ion,nl (ω) =

# l  =l±1

   22 27/3 π 2 1  2 −Ai t ± t Ai t2 . 3cω 7/3 n 3

(3.96)

Here the notation is introduced    ω 1/3 1  ω 1/3 t ≡ l¯ + ≈l 2 2 2   (we substitute the quantity l¯ = l + l  /2 by l due to l ∼ n  1). Summing the right side of Eq. (3.96) over l  we also average this sum over all values of initial orbital quantum number l. Analogously to the solution of Problem 3.14 we obtain the final result 1 8π ; ω ≥ 2. σ¯ ion,n (ω) = √ 5 3 2n 3 3cn ω

(3.97)

Let us make some concluding remarks. WKB approximation is valid when both the initial binding state with n  1 and also the final continuum state is quasi-classical states. Thus, the transition frequency ω should not be too large. The expression (3.97) is correct under the condition ω  1 (in atomic units), or ω  me4 /2 in usual units (see Eq. (3.4)). It follows from Eq. (3.97) that the photoionization cross-section is nonzero in the threshold, i.e., at ω → 1/2n 2 :   1 64πn σ¯ ion,n ω = 2 = √ . 2n 3 3c

(3.98)

This property of the cross-section is explained by the long-range type of the Coulomb potential: this attractive potential U (r ) = −1/r decreases slowly with rise of r (see comments to the Problem 3.7). It should be noted that in the case of short-range binding potential the ionization cross-section vanishes in the threshold. It is explained by the decrease of the density of final states in the vicinity of the continuum edge. The applicability of Eqs. (3.97) and (3.98) can be extended also to the values of n ∼ 1. For example, according to Eq. (3.98) the threshold cross-section for photoionization of the ground state of hydrogen atom is   1 64π = √ , σion,1s ω = 2 3 3c

100

3 Rydberg States of Atomic Systems

while the exact threshold value is   1 29 π 2 exact ω= = 4 . σion,1s 2 3e c Here e = 2.718 . . .. The difference between these values is only 25 %. We have described the incoming photon flux in the frames of classical electrodynamics. However, the obtained results have more extensive region of applicability. Using Eqs. (3.48)–(3.51) we can check that in the dipole approximation the photoionization cross-section does not depend on the detailed form of distribution of population numbers for photon quantum states in the incoming photon flux (at the fixed value of the radiation frequency ω). It is seen that the photoionization cross-section, Eq. (3.97) decreases quickly with rise of the principal quantum number n. It is explained by weaker electron bonding; in the case of a free electron the absorption of a photon is forbidden by the energy and momentum conservation laws. Equations (3.97) and (3.98) are written in the atomic system of units. In the usual system Eq. (3.97) takes the form 64π σ¯ ion,n (ω) = α √ 3 3n 5



I0 ω

3 a 2B .

(3.99)

Here α = e2 /c ≈ 1/137 is the fine structure constant, I0 ≡ me4 /22 is the ionization potential of the ground hydrogen state, a B = 2 /me2 is the Bohr radius. Equations (3.97) and (3.98) are valid for an electron moving in the Coulomb field of the charge e. In order to generalize these results to the case of the atomic core with the charge Z e we should have in mind that the charge e is inserted into the above expressions by different ways in different places. First of all, the factor e describes the interaction of an electron with the radiation field. This dependence produces the factor e2 in the photoionization cross- section. In all other cases the charge is appeared in the combination Z e2 which describes the Coulomb interaction of an electron with an atomic core. Hence, in order to obtain the cross-section for photoionization of an atomic ion, we should incorporate the factor Z 2 into the expression for ionization potential I0 , and the factor Z −1 into the expression for Bohr radius a B . Therefore, the dependence of the cross-section on the charge Z is of the form: σ¯ ion,n ∼ Z 4 . In the threshold we find m Z 2 e4 ωthres = . 22 n 2 The corresponding threshold cross-section is σ¯ ion,n (ωthres ) ∼ Z −2 .

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

101

Problem 3.17 Calculate the cross-section for photorecombination of a slow electron with a proton into the high-excited hydrogen state with a principal quantum number n.

Solution In the recombination process a free electron radiates a photon with the frequency ω and goes into a binding atomic state. This process is inverse with respect to the photoionization considered in the previous problem. The cross-sections of mutually inverse two-particle processes are connected with each other by the detailed balancing principle (see Ref. [1], Sect. 144). According to this principle we have g1 pe2 σ¯ rec,n (v) = g2 pγ2 σ¯ ion,n (ω) . Here the notations are introduced v ≡ pe ,

pγ = ω/c, ω = v 2 /2 + 1/2n 2 .

Statistical weights of the considered states are g1 = 1, g2 = 2n 2 . The factor 2 in g2 corresponds to two independent polarization states of a photon. Using Eq. (3.97) from the previous problem for the photoionization cross-section, we obtain the photorecombination cross-section: 16π ; v  1, n  1. σ¯ rec,n (v) = √ 3 3c3 n 3 ωv 2

(3.100)

It is seen that the photorecombination cross-section increases with the decrease of n at the fixed value of the velocity v. Thus, the recombination occurs mainly into the ground state of an atom (it should be noted that the applicability of WKB result (3.100) at v  1 can be extended to the binding states with n ∼ 1: this is typical for results of WKB approximation). The physical reason for this statement is analogous to that in the previous problem. Indeed, the atomic electron with the principal quantum number n  1 is almost free, but the radiation of a photon by a free electron is forbidden by energy and momentum conservation laws. It should be noted that the dependence of the photorecombination cross-section on the electron velocity is σ¯ rec,n ∼ v −2 at v → 0. In the case of a short-range potential the cross-section of inelastic processes is of another form: σ ∼ v −1 . More sharper dependence of the photorecombination cross-section on the electron velocity is explained by strong influence of long-range Coulomb attractive potential upon a slow electron (compare to the Problem 3.7 and 3.16).

102

3 Rydberg States of Atomic Systems

Finally, let us rewrite the photorecombination cross-section (3.100) in the usual system of units; besides this, we generalize it upon the case of recombination of a slow electron with an atomic ion with charge Z e: I0 I0 I0 128π Z 4 · · · σ¯ rec,n (v) = α √ 3 3n 3 mc2 ω mv 2



2 me2

2 .

Here I0 = me4 /22 and ω = mv 2 /2+ Z 2 I0 /n 2 (compare to the previous problem).

Problem 3.18 Using WKB approximation, calculate the spectral distribution dσbs /dω for the bremsstrahlung of photons in the collision of a slow electron with a proton. The electron’s velocity is v  e2 /. Compare the obtained result with the spectral distribution calculated in the frames of classical electrodynamics.

Solution The bremsstrahlung process of a photon is similar to the photorecombination process. The only difference is that now the final electron state belongs to the continuum. However, at E → 0 the properties of the continuum states (E > 0) are similar to the properties of the quasi-continuum spectrum of Rydberg states (E n < 0), since Rydberg levels are condensed at E n → 0. Here we have an analogy with the case of a free particle: its continuum states are similar to the states of a particle inside the large box. Therefore, we can add each with others photorecombination cross-sections to the final states with the principal quantum numbers in the interval dn, in order to obtain the differential cross-section for bremsstrahlung dσbs = σ¯ rec,n (v) dn.

(3.101)

Here, the photorecombination cross-section is given by Eq. (3.100) of the previous problem. Using the energy conservation law for photorecombination process v 2 /2 = ω + E n = ω − 1/2n 2 , with the fixed initial electron’s velocity v, we express the differential dn via the differential of frequencies of the emitted photon: dn = n 3 dω. Then according to Eq. (3.101) we obtain the spectral distribution of bremsstrahlung, taking into account Eq. (3.100):

3.2 Interaction Between a Rydberg Electron and an Electromagnetic Radiation

dσbs =

e4 16π Z 2 e2 dω · . · √ · 2 c ω (mcv) 3 3

103

(3.102)

Here, we have used the usual system of units and we have generalized the result to the case of collision of an electron with the atomic ion having the charge Z e (compare to the results of two previous problems). The continuum edge ω = mv 2 /2 is a regular point of Eq. (3.102), as it should be. Thus, this expression does not contain any singularities. If we multiply Eq. (3.102) by the photon energy ω, we obtain so-called “effective radiation” [9]: 16π Z 2 e6 dω. (3.103) dκ = ω · dσbs = √ 3 3m 2 c3 v 2 This expression does not contain the Planck constant, and therefore it is of a classical nature. Namely, it coincides the corresponding expression of classical theory of bremsstrahlung in Coulomb potential under the condition of sufficiently large photon frequency ω  mv 3 /Z e2 (see Ref. [9], Sect. 70). However, if the classical approach is valid under the condition ω  mv 2 only, such condition is not needed in the quantum-mechanical calculations: Eqs. (3.102) and (3.103) are correct as well as in the case of ω ≥ mv 2 , when the final electron velocity differs strongly from its initial velocity. On the other side, in the limit of extremely small frequencies, i.e., under the condition ω ≤ mv 3 /Z e2 , classical effective radiation is described by an expression which differs from Eq. (3.103) (see Ref. [9]). Hence, Eqs. (3.102) and (3.103) are incorrect in this limit. Let us discuss the reasons of this contradiction. The derivation of Eqs. (3.102) and (3.103) in this problem is based on Eq. (3.100). We have used WKB-approximation for radial dipole matrix element n 2 l2 |r | n 1 l1  under the condition n = n 1 − n 2  1. On the other hand, we have  2  2   m Z e2 m Z e2 1 1 n ω= − 2 ∼ · . 2 3 23  n2 n1 n 1 n 22 It follows from this relation that in the case of n  n 1 ω≈

mv 3 n . Z e2

Therefore, in the limit of low frequencies, ω → 0 WKB matrix elements of the dipole moment with n ∼ 1 should be used (their relation with classical Fourier components of the dipole moment was obtained in Problems 3.10 and 3.11).

104

3 Rydberg States of Atomic Systems

References 1. L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd edn. (Pergamon, Oxford, 1977) 2. A.I. Baz, Ya.B. Zeldovich, A.M. Perelomov, Scattering, Reactions and Decays in NonRelativistic Quantum Mechanics (Plenum, New York, 1975) 3. H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, 1957) 4. A.A. Radzig, B.M. Smirnov, Reference Data on Atoms, Molecules and Ions (Springer, Berlin, 1986) 5. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, 4th edn. (Academic, New York, 1965) 6. N.F. Mott, H.S.W. Massey, The Theory of Atomic Collisions (Clarendon Press, Oxford, 1965) 7. F.J. Burke, Potential Scattering in Atomic Physics (Wiley, New York, 1975) 8. V.M. Galitsky, B.M. Karnakov, V.I. Kogan, Problems in Quantum Mechanics, 2nd edn. (Nauka, Moscow 1992, in Russian) 9. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, 3rd edn. (Pergamon, Oxford, 1971) 10. L.D. Landau, E.M. Lifshitz, Mechanics (Pergamon, Oxford, 1977) 11. A.B. Migdal, Qualitative Methods in Quantum Theory (Benjamin, New York, 1977) 12. V.B. Berestetsky, E.M. Lifshitz, L.P. Pitaevsky, Quantum Electrodynamics (Pergamon, Oxford, 1986) 13. M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions (National Bureau of Standards, Washington 1964)

Chapter 4

Penetrability of Potential Barriers and Quasi-Stationary States

One of the important applications of WKB-approximation is related with derivation of penetrability coefficients of potential barriers. The penetrability coefficient of one-dimensional barrier shown in Fig. 4.1 is given by expression ⎫ ⎧ ⎬ ⎨ 2 b | p (x, E)| dx , D0 (E) = exp − ⎭ ⎩ 

(4.1)

a

where a, b are the classical turning points and E is the energy of a particle. Let us make some remarks with respect to conditions of applicability of Eq. (4.1) and its accuracy. 1. Its derivation is based on the assumption that the condition of applicability of WKB-approximation (1.6) is valid everywhere, except of narrow regions near the turning points. In order to match WKB-solutions one with other on both sides from the turning points a, b, linear expansion of the potential [see Eq. (1.8)] should be used. In addition to matching relations (1.9–1.12) used in the investigation of binding states, we give once more important condition for matching of WKB-solutions (see Fig. 4.1): ⎡ x ⎤     π 1 C ψ (x) = √ p x dx + ⎦ , cos ⎣  4 p (x) ⎡

b

1 C exp ⎣ ψ (x) = √  | p (x)|

b

⎤       p x  dx ⎦ ,

x > b,

a < x < b.

(4.2)

x

The exponent of Eq. (4.1) (let us denote it as −ξ) should be the large negative number in the condition of applicability of this expression. Hence, D0 = exp(−ξ),

ξ  1,

B. M. Karnakov and V. P. Krainov, WKB Approximation in Atomic Physics, DOI: 10.1007/978-3-642-31558-9_4, © Springer-Verlag Berlin Heidelberg 2013

105

106

4 Penetrability of Potential Barriers and Quasi-Stationary States

Fig. 4.1 The typical potential barrier

U(x) E a

b

x

i.e., the penetrability is exponentially small quantity. 2. The accuracy of Eq. (4.1) for the barrier penetrability is determined by higher on  quasi-classical corrections. Taking into account the first correction on  we find that the barrier penetrability in WKB-approximation is given by expression ⎫ ⎧ b ⎬ ⎨ F 2 (x)  ∂2 (4.3) dx · D0 (E). Dq (E) = 1 − ⎩ | p(x, E)| ⎭ 12 ∂ E 2 a

Here the notation is introduced F = −U  (x). Quasi-classical correction is estimated as 1/ξ on the order of magnitude (here ξ is the WKB-parameter, see above); the accuracy of calculations increases essentially at moderate values of the parameter ξ if to take into account this correction. It should be noted that the accuracy of derivation of barrier penetrability related with neglection of exponentially decreased part of solution under the barrier is δ D ∼ D02 on the order of magnitude. 3. WKB-expression for the barrier penetrability should be modified if the matching conditions based on the linear expansion of the potential are inapplicable. The preexponential factor should be introduced so that Dq (E) = c(E) · D0 (E). Here the quantity D0 (E) is given by Eq. (4.1); it is responsible for exponential smallness of the penetrability coefficient. In the general case the pre-exponential factor c(E) is on the order of magnitude of 1 [(however, it vanishes in the case of short-range potentials at E → 0 (slow particles)]. 4. Equation (4.1) is inapplicable when the energy of a particle E approach to the top of the potential barrier U0 . However, in this case we can obtain as well as the analytic WKB-expression for the penetrability coefficient D(E). The potential U (x) is expanded near its maximum in Taylor series: 1 U (x) ≈ U (x0 ) − k (x − x0 )2 , 2

k = −U  (x0 ) > 0.

Then we use the exact solution of Schrödinger equation for parabolic potential and match it with WKB-solutions on large distances from x0 . The result is of the form (see [1], Sect. 50 and also Problem 4.6):  1 m 1 , ε= (4.4) D(E) = (E − U (x0 )) . 1 + exp (−2πε)  k

4 Penetrability of Potential Barriers and Quasi-Stationary States

107

U(x)

E0n - iΓn/2

x a1

b1

a2

Fig. 4.2 One-dimensional potential in Problem 4.1

This expression reduces gradually to WKB-formula (4.1) when the energy of a particle decreases. It is applicable as well as in the case of above-barrier reflection until the energy of a particle exceeds strongly the top of the potential barrier. Problems in this Chapter are devoted mainly to consideration of quasi-stationary states of a particle which are related with penetrability of potential barriers. More information can be found in Ref. [3].

4.1 Quasi-Stationary States of One-Dimensional Systems Problems and Solutions Problem 4.1 Using WKB approximation, derive the energies and widths of quasi-stationary states of a particle in one-dimensional potential described in Fig. 4.2. Generalize results to the case when the barrier has non-zero penetrability on the left from the potential well (Fig. 4.3) also.

Solution Let us remember that energies and widths of quasi-stationary states are determined from the condition that the solution of Schrödinger equation with a complex energy E = E 0n − in /2 (see Ref. [1, 2]) on large distances should be a diverging wave. Such boundary condition is called also the condition of emission; it corresponds to exponentially decreasing solutions of Schrödinger equation in classically forbidden region.

108

4 Penetrability of Potential Barriers and Quasi-Stationary States

The wave function of the quasi-stationary state is of the WKB-form (except of the vicinity of the classical turning points) in the regions [−∞, a1 ] and [a1 , b1 ]: ⎡ a ⎤ √ 1  i iC  + (x) = √ exp ⎣ x < a1 ; p x  dx  ⎦ ; (4.5)  2 p (x) x

and ⎡ x ⎤   1 π C  + (x) = √ p x  dx  + ⎦ ; sin ⎣  4 p (x)

a1 < x < b1.

(4.6)

a1

This wave function can be written analogously in the other WKB-regions [b1 , a2 ] and [a2 , ∞]: in the WKB-region [b1 , a2 ] ⎤ ⎤ ⎡ ⎡ a a2 2   i C i C 1 1  + (x) = √ p x  dx  ⎦ + i √ p x  dx  ⎦ ; exp ⎣− exp ⎣   p (x) 2 p (x) x

x

(4.7) in the WKB-region [a2 , ∞] ⎡ x ⎤   i C 1  + (x) = √ p x  dx  ⎦ . exp ⎣  p (x)

(4.8)

a2

Here Eq. (4.5) decreases exponentially at x → −∞; Eq. (4.6) is written taking into account the matching conditions (1.9–1.10). Equation (4.8) describes the wave propagating to the right (this corresponds to the outcoming particle), while Eq. (4.7) is related with Eq. (4.8) by matching condition (1.11–1.12) and (4.2). It should be noted that the classical turning points are complex quantities due to complexity of the energy of the considered level. However, this fact does not influence upon the matching conditions because of exponential smallness of imaginary part n of the energy. Let us underline also that if we neglect this imaginary part, then the momentum p (x) is pure imaginary one in under-barrier region, and i p (x) < 0. We should match now Eqs. (4.6) and (4.7) each with other. Two approaches can be suggested to solve this problem. The first approach is based on the neglection of the second term in Eq. (4.7) in the matching procedure. Indeed, nearly of the turning point x = b1 this term is exponentially small compared to the first term in Eq. (4.7). Then we can use relations (1.11–1.12) for matching of solutions. Thus, we find Bohr-Sommerfeld quantization rule

4.1 Quasi-Stationary States of One-Dimensional Systems

1 

109

  b1  1 , 2m (E 0n − U (x))dx = π n + 2

(4.9)

a1

and obtain the connection between the coefficients C1 and C : ⎤ ⎡ a2  1√ ⎢ 1      ⎥ p x dx ⎦ . i (−1)n C exp ⎣− C1 = 2 

(4.10)

b1

However, this approach does not allow to calculate the value of the energy width n . The width is exponentially small, and we cannot determine it directly in this approach due to neglection of the second term in Eq. (4.7). Nevertheless, we can calculate this width using the some additional considerations and above results. The energy width determines the rate for decay of the system (this is tunneling of a particle through the potential barrier in the considered case). Let us calculate the density of a flux on the right from the turning point x = a2 . Using Eq. (4.8), we find i j =− 2m



∂ ∂   −  ∗ ∂x ∂x ∗

 =

|C1 |2 . m

(4.11)

Let us normalize the wave function  + (x) using the condition b1

 +   (x)2 dx = 1.

a1

Then we should choose 4m |C| = ; T (E 0n ) 2

b1 T (E 0n ) = 2m a1

dx p (x)

(4.12)

(see Eqs. (1.14–1.15); here T (E 0n ) is the period of motion of a classical particle in the considered potential well. Such normalization of the wave function allow to calculate directly the rate for decay w = j. Using relations (4.10–4.12), we obtain the width of the level for the considered quasi-stationary state (so called Gamow formula): ⎡ n = wn =

ω (E 0n ) D (E 0n ) ⎢ 2 exp ⎣− = T (E 0n ) 2π 

a2 b1

⎤ ⎥ | p (x)| dx ⎦ .

(4.13)

110

4 Penetrability of Potential Barriers and Quasi-Stationary States

This result for wn has the simple physical interpretation: the rate for ejection of a particle from the potential well is equal to the product of the penetration probability D through the barrier at one collision by number of collisions 1/T for 1 sec. Therefore it is obvious how to generalize this result to the case when the potential barrier has nonzero penetrability in both directions from the well: n =

ωn [D1 (E 0n ) + D2 (E 0n )] . 2π

(4.14)

Applications of these results are given in the next Problems. Also the generalizations will be considered for the situations where WKB approximation is inapplicable in some regions of particle’s motion. Let us consider now the second approach for matching of Eqs. (4.6) and (4.7). Though the second term in Eq. (4.7) (which was deleted above) is exponentially small compared to the first term, it can be taken into account by usual WKB-method. These terms can be considered as two independent solutions of Schrödinger equation due to factor i in the second term. Respectively, we can match these solutions independently in the vicinity of the turning point x = b1 . The first term in Eq. (4.7) is matched with the WKB solution in the region x < b1 ⎤ ⎡ ⎡ b ⎤ √ a2 1    π 1 2 −iC1 ⎥ ⎢ i (1)  (x) = √ p (x) dx ⎦ sin ⎣ p x dx + ⎦ , (4.15) exp ⎣−   4 p (x) x

b1

(see Eqs.1.11–1.12). The second term in Eq. (4.7) takes the other form in the region x < b1 at the analogous matching [compare to Eq. (4.2)]: √

⎡

i −iC1 ⎢i  (2) (x) = √ exp ⎣  2 p (x)

a2 b1

⎤

⎡ b ⎤ 1    π 1 ⎥ p (x) dx ⎦ cos ⎣ p x dx + ⎦ .  4

(4.16)

x

Sum of Eqs. (4.15) and (4.16) determines the wave function of the quasi-stationary state in the region of finite motion of a classical particle. Taking into account smallness of the exponential factor in Eq. (4.16), we write the wave function in the form ⎡ b ⎤ √ 1  1 i π 2 −iC1  + (x) =  (1) (x) +  (2) (x) = √ p x  dx  + + A2 ⎦ . sin ⎣  4 4 A p (x) x

(4.17) Here the notation is introduced ⎡ ⎢i A (E) = exp ⎣ 

a2 b1

⎤ ⎥ p (x, E) dx ⎦

(4.18)

4.1 Quasi-Stationary States of One-Dimensional Systems

111

(the quantity A2 (E) determines the penetrability of the barrier D at the real values of the energy E). The Eqs. (4.6) and (4.17) present the same solution of the Schrödinger equation. The condition for their identity is that the sum of phases of sine functions should be integer of π. Hence, we obtain 1 

b1

  i 1 − A2 (E n ) . p (x, E n ) dx = π n + 2 4

(4.19)

a1

This relation gives the generalization of the Bohr-Sommerfeld quantization rule to the case of quasi-stationary states; here E n = E 0n −in /2. We can expand p (x, E n ) over small quantity n in the left side of Eq. (4.19) and omit n in the right side of this equation, in expression for A (E n ) . Then the real part of Eq. (4.19) reduces to the quantization rule (4.9) for energies of quasi-stationary levels, while its imaginary part determines the widths n of these levels in agreement with Gamow formula, Eq. (4.13). When the energy E 0n increases, the width of the level and the barrier penetrability rise also. If the energy E 0n is nearly of the top of the potential barrier, Eq. (4.19) is inapplicable. The generalization of the quantization rule to this case is given in Problem 4.4.

Problem 4.2 Derive shifts and widths of the levels of linear oscillator (the potential energy is U0 (x) = mω 2 x 2 /2) produced by “small” anharmonicity potential V (x) = −λx 3 . Find the conditions for applicability of obtained results. Solution Qualitative form of potential is shown in Fig. 4.3. We derive energies and widths of quasi-stationary states of anharmonic oscillator using Eqs. (4.9) and (4.13) from the previous Problem. The main difficulty of derivations is a calculation of integrals in these expressions. We use perturbation theory to calculate the shifts of oscillator’s levels. We underline that the perturbation V (x) is small compared to the U0 (x) in the classical region only. In the non-classical region of large distances (under the barrier) no restrictions for V (x) exist. Moreover, the anharmonicity term dominates in the latter region. The tunneling of a particle is possible due to V (x) only and it spreads the levels. First we calculate the barrier penetrability. We should derive the integral 1 

x2 x1

1 | p (x)| dx = 

x2 

 2m

x1

 1 mω 2 x 2 − λx 3 − E 0n dx, 2

(4.20)

112

4 Penetrability of Potential Barriers and Quasi-Stationary States

U(x) E0n - iΓn/2

Fig. 4.3 One-dimensional potential in Problem 4.2

where x1,2 are classical turning points. In order to derive this integral, we divide the integration interval on two regions: [x1 , d] and [d, x2 ]. The value of d satisfies the conditions  E 0n mω 2 . (4.21)

d

2 mω λ In the first integral the term λx 3 is a small perturbation, while in the second integral the term with E 0n is a small perturbation. We expand on these small parameters and obtain 1 

d x1

1 | p (x)| dx ≈  ≈

d 

λmx 3

x10



dx (mωx) − 2m E 0n −  (mωx)2 − 2m E 0n 2

E 0n E 0n 2mω 2 d 2 λd 3 mωd 2 − − ln , − 2 2ω 2ω E 0n 3ω

(4.22)

and also 1 

x2 d

1 | p (x)| dx ≈ 

x20 (mωx) d

2

− 2mλx 3

m E 0n



− dx (mωx)2 − 2mλx 3

λd 3 E 0n 2mω 2 mωd 2 m 3 ω5 + − ln . − ≈ 15λ2 2 3ω ω λd

(4.23)

Sum of these integrals gives 1 

x2 x1

  m 3 ω5 8m 3 ω 6 E 0n | p (x)| dx = + ... 1 + ln 2 − 15λ2 2ω λ E 0n

(4.24)

(of course, the arbitrary parameter d is excluded from the final result). Equation (4.24) determines the value of the integral (4.20) in the form of expansion over the small parameter λ2 E 0n /m 3 ω 6 [(compare with Eq. (4.21)]. The next 2 /(ωm 3 ω 6 ). Since this term of expansion is on the order of magnitude of λ2 E 0n

4.1 Quasi-Stationary States of One-Dimensional Systems

113

integral (4.20) is contained in the exponent of barrier penetrability, then Eq. (4.24) is applicable under the condition 2 λ2 E 0n

1. 3 m ω 7

(4.25)

If this inequality is fulfilled, shifts of the oscillator levels produced by the anharmonicity term V = −λx 3 are equal (see Ref. [1], Sect. 38) 15 λ2 E n = − 4 ω



 mω

3   11 2 n +n+ . 30

(4.26)

They are small compared to the interval between the unperturbed oscillator levels, E n ω under condition (4.25). Then we can substitute E 0n by E n(0) = ω (n + 1/2) in Eqs. (4.20) and (4.24). Thus, the final expression for the width of oscillator’s levels is ω n = 2π



8m 3 ω 5 2 λ  (n + 1/2)

n+1/2



 1 2m 3 ω 5 +n+ exp − . 15λ2  2

(4.27)

Problem 4.3 Generalize the result of the Problem 4.2 to the case of low-lying quasi-stationary states (with n ∼ 1). Use parabolic approximation of the potential U (x) near of its minimum. Solution Results of the previous Problem are needed to modify, since WKB approximation is inapplicable at n ∼ 1 in the region of classical motion of a particle. However, we can use here the exact solution of Schrödinger equation based on the parabolic expansion of the potential energy in the vicinity of its minimum 1 U (x) ≈ U (xm ) + U  (xm ) (x − xm )2 . 2

(4.28)

We assume that WKB approximation is valid already on the boundary of the region where such expansion is correct. The energy spectrum of low-lying levels is given by the expression  1 ; ≈ U (xm ) + ω n + 2





E 0n

ω=

U  (xm ) ; m

(4.29)

114

4 Penetrability of Potential Barriers and Quasi-Stationary States

(we can take into account anharmonic corrections using the perturbation theory). Normalized wave functions are of the form     x − xm 1 (x − xm )2 exp − n (x) ≈  , (4.30) √ Hn a 2a 2 2n n!a π √ where Hn (z) are Hermite polynomials, and a = /mω (see Ref. [1]). Asymptotic expressions for these functions at |x − xm |  a are given by the expression 

  2n (x − xm )n (x − xm )2 exp − √ an 2a 2 πn!a

n (x) ≈

(4.31)

(we have used the relation Hn (z) ≈ (2z)n at z → ∞). WKB momentum of a particle in the considered region is of the form p (x) ≈



  2m ω (n + 1/2) − mω 2 (x − xm )2 /2 .

Let us derive now the integral: 1 

x x1

      p x  dx = 1 

x  m 2 ω 2 (x  − xm )2 − (2n + 1) mωdx  x1

  1 1 1 mω (x − xm )2 − n+ 2 2 2    1 2mω (x − xm )2 ln − n+ 2 (n + 1/2) 

≈

√ (here x1 is the right classical turning point and (x − xm )  2n + 1a). Thus, the wave function (4.31) takes the WKB form and can be written in the region of right barrier: ⎡ ⎤  x      1 2mω  p x  dx ⎦ exp ⎣− n (x) ≈ γn (4.32) π | p (x)|  x1

where the notation is introduced  n/2+1/4 1 1/4 2n + 1 γn = √ π e 2n/2 n!

(4.33)

(e = 2.718...). Analogous expression can be written also for the left under-barrier region.

4.1 Quasi-Stationary States of One-Dimensional Systems

115

Difference of Eq. (4.32) from the usual WKB function in under-barrier region is given by additional factor γn . Respectively, we should insert the square of this factor into Eqs. (4.13– 4.14) for widths of levels. In particular, widths of quasi-stationary states in the case of potential depicted in Fig. 4.2, are described by n =

1 ωγn2 D (E n ) . 2π

(4.34)

Here the quantity D (E n ) is the barrier penetrability derived in the WKB approximation, as before. It should be noted that γ → 1 at n → ∞, as it should be. Moreover, the quantity γn differs a little from its WKB value γ∞ = 1 as well as in the case of quantum numbers n ∼ 1. Indeed, according to (4.33) we find γ0 = 1.037; γ1 = 1.014; γ2 = 1.0083 and γn ≈ 1 +

1 at n → ∞. 48 (n + 1/2)

Finally, let us apply Eq. (4.34) for the anharmonic oscillator potential considered in the previous Problem. It follows from Eq. (4.27) that   ω 3n+1 2n+1 2 2 n = √ γ˜ exp − γ˜ , 2 15 πn!

(4.35)

where the notation is introduced  γ˜ =

m 3 ω5  1. λ2

This expression determines widths of all quasi-stationary oscillator states under the condition that anharmonic corrections are small (it was used in the solution of Problem 3.2).

4.2 Quasi-Stationary States and Above-Barrier Reflection Problems and Solutions Problem 4.4 Obtain the quantization rule for quasi-stationary states of a particle with the energy in the vicinity of the top of the potential barrier (see Figs. 4.2 and 4.4). Compare with result of the Problem 4.1. Find that the width of quasi-stationary state is small

116

4 Penetrability of Potential Barriers and Quasi-Stationary States

(a)

E U0

(b)

E U(x)

x0

x1

xm

x2

x

Fig. 4.4 Under-barrier a and above-barrier b quasi-stationary states of a particle

compared to the energy interval between this state and the neighboring level even at E → U0 (U0 is the top of the potential barrier).

Solution We should generalize the quantization rule (4.19) to the case when the energy of the considered level E 0n is nearly of the top of potential barrier U0 . WKB approximation is inapplicable in the vicinity of this point x = xm , U0 = U (xm ) . In other regions the WKB wave function is described by Eqs. (4.5, 4.6, 4.8) as before [but not by Eq. (4.7)]. We use parabolic approximation for the potential 1 U (x) ≈ U0 − λ2 (x − xm )2 ; 2

λ=

 −U  (xm ) > 0

(4.36)

in order to find the wave function in the vicinity of x = xm , and then we match this function with WKB-solutions. After substitution ρ=

√

2λ (xm − x) ;

a=

U0 − E λ

(4.37)

the Schrödinger equation takes the form ( = m = 1) : d2 ψ+ dρ2



 ρ2 − a ψ = 0. 4

(4.38)

This is equation for the functions of parabolic cylinder. Its solution should satisfy the condition of running to the right wave at x → ∞ [then ρ → −∞ according to

4.2 Quasi-Stationary States and Above-Barrier Reflection

117

(4.37)]. Let us consider the values of x − xm where the parabolic expansion (4.36) is still valid, and WKB approximation, Eq. (4.8) is already valid. We obtain ψ(ρ) = const × D−ia−1/2 (ρ exp(3iπ/4)).

(4.39)

Indeed, using the asymptotic representation 9.246 from Ref. [4], we find ⎤ ⎡ x     1 ψ (ρ) ∼ ρ−ia−1/2 exp iρ2 /4 ∼ √ p x  dx  ⎦ at ρ → −∞. exp ⎣i p(x) xm

(4.40) (in this case we have arg (ρ exp (3iπ/4)) = −π/4). Here WKB-momentum is determined by relation     1 2 1 2 p (x) ≈ 2 E − U0 + λ (x − xm ) = −2λa + λρ2 . 2 2 Analogously √we find the asymptotic representation of Eq. (4.39) to the left from xm (then ρ  a and arg (ρ exp (3iπ/4)) = 3π/4):   2   ρ const 3iπ 3πa exp i ψ (ρ) = √ exp − − ia ln ρ + ρ 8 4 4 √   2 ρ πa π 2π exp −i + ia ln ρ + +i . +  (ia + 1/2) 4 4 2

(4.41)

Let us introduce the notation √ ! π 2π exp exp (2iξ) ≡ (i − a) .  (ia + 1/2) 2 Then we can rewrite Eq. (4.41) in the form  2  ρ C ψ (ρ) = √ cos − + a ln ρ + ξ ρ 4

(4.42)

which can be matched with WKB-solution (4.6) taking into account the boundary condition at x → −∞ : ⎡ x ⎤   C π ψ (x) = √ cos ⎣ p x  dx  − ⎦ . (4.43) 4 p (x) x0

118

4 Penetrability of Potential Barriers and Quasi-Stationary States

Here x0 ≡ a is the left classical turning point. The integral in Eq. (4.43) can be written in the form x

 p x  dx  =

x0

x1

 p x  dx  −

x0

x1

 p x  dx  .

(4.44)

x

Here x1 ≡ b1 is the second classical turning point (see Figs. 4.2 and 4.4). Its definition in the case of E > U0 will be given below. The second of the integrals in the right side of√Eq. (4.44) can be derived using parabolic expansion; we obtain in the region ρ  a : x1

ρ

 p x  dx  =



√ 2 a

x

ρ2 a ρ ρ2 − a dρ ≈ − − a ln √ . 4 4 2 a

(4.45)

Now we obtain the generalization of Bohr-Sommerfeld quantization rule to the case of resonance states nearly of the barrier top using coincidence of solutions (4.42) and (4.43–4.45) each with other: x1



1 p (x) dx = π n + 2



1 − ϕ (a) , 2

(4.46)

x0

where n = 0, 1, 2, ... and ϕ (a) = a (1 − ln a) +

1  (ia + 1/2)  . ln 2i  (−ia + 1/2) 1 + exp (−2πa)

(4.47)

Here we have used the relation  (ia + 1/2)  (−ia + 1/2) =

π . cosh (πa)

Let us consider some conclusions from the obtained results. (1) If the barrier penetrability is small, i.e. a  1, then we find from Eq. (4.47) ϕ (a) =

i 1 + ... + exp (−2πa) 24a 2

(4.48)

(we omitted here high-order real terms on the parameter 1/a). The quantization rule (4.46) with ϕ (a) from (4.48) corresponds to result (4.19) of the Problem 4.1. Indeed, in the parabolic approximation (4.36) for the potential we obtain

4.2 Quasi-Stationary States and Above-Barrier Reflection

⎡ A (E) = exp ⎣i

x2

⎡

⎤

p (x) dx ⎦ = exp ⎣−

x1

x2 

119

⎤ 2λa − λ2 (x − xm )2 dx ⎦ = exp (−πa) .

x1

(4.49) Therefore exponent in Eq. (4.48) corresponds to the barrier penetrability. The real term 1/24a and the next real terms in Eq. (4.48) present high-order WKB corrections on (or on 1/n) in the usual quantization rule which does not take into account the barrier penetrability. Their contribution into the energy shift of the level should be taken into account simultaneously with the similar corrections from the main region of localization of the wave function. (2) The most interesting are applications of Eqs. (4.46–4.47) to resonance states nearly top of the potential barrier, so that E n ≈ U0 and |a| 1. We consider this case in detail. In order to derive the integral in the left side of Eq. (4.46) we rewrite it in the form x1 I (E) ≡

d pdx =

x0

x1 pdx +

x0

pdx,

(4.50)

d

where the value of d is chosen so that the parabolic approximation of the potential is still valid at x = d, but already λ2 (d − xm )2  |E − U0 | .

(4.51)

In the first of the integrals in right side of Eq. (4.50) we can expand over (E −U0 ), and in the second of these integrals we use parabolic approximation for the potential (compare to the analogous derivation of the integral (4.20) in Problem 4.2). Thus, we find d d d dx , (4.52) pdx ≈ p¯ (x) dx + (E − U0 ) p¯ (x) x0

x¯0

x¯0

√ where p¯ (x) = 2 (U0 − U (x)). We can put p¯ (x) = λ (xm − x) in these integrals at upper limit. Then we can extend the integration up to point xm in the first of the integrals and find its explicit dependence on the parameter d : d

xm p¯ (x) dx =

x¯0

xm p¯ (x) dx −

x¯0

xm p¯ (x) dx ≈

d

p¯ (x) dx − x¯0

λ (xm − d)2 . 2

(4.53)

Analogously we can transform the second integral in Eq. (4.52), defining diverging part at d → xm with the help of relation

120

4 Penetrability of Potential Barriers and Quasi-Stationary States

d x¯0

dx ≈ p¯ (x)

xm  x¯0

 1 1 1 xm − d dx − ln . − p(x) ¯ λ (xm − x) λ xm − x¯0

(4.54)

The value of the second integral in the right √ side of Eq. (4.50) is determined by Eq. (4.45) if we change x → d and ρ → 2λ (xm − d). Finally, we find the expansion of the quantization rule integral (4.50) at |a| 1 with the accuracy up to terms on the order of a 2 : 1 I (E) = I0 + a ln a − I1 a + ..., 2 where

(4.55)

xm I0 =

p¯ (x) dx x¯0

and  ! 1 1 + ln 2λ (xm − x¯0 )2 + I1 = 2

xm  x¯0

 λ 1 − dx. p(x) ¯ (xm − x)

(4.56)

According to Eq. (4.47) the analogous expansion of the function ϕ (a) is of the form    i 1 ϕ (a) = (ln 2 − πa) − a ln a + 1 + ψ a + ..., (4.57) a 2 where ψ (1/2) = −C − 2 ln 2 = −1.9635... (this is the logarithmic derivative of gamma-function). Substituting expansions (4.55) and (4.57) into the quantization rule (4.46), we find the spectrum of the considered resonance states E n = U0 + λ

2π (n + 1/2) − 2I0 − (i/2) ln 2 L + iπ/2

(4.58)

where L = 2I1 − 1 − ψ (1/2) .

(4.59)

Finally, separating real and imaginary parts in Eq. (4.58) (E n = E 0n − i/2), we obtain the energies E 0n and widths n of resonance states E 0n = U0 +

2πλL (n − n ∗ ) ; + π 2 /4

L2

Here the notation is introduced

n =

2π 2 λ λ ln 2 + 2 (n − n ∗ ) . (4.60) L L + π 2 /4

4.2 Quasi-Stationary States and Above-Barrier Reflection

n∗ =



1 π

I0 −

π π ln 2 + 2 8L

121

 ≈

1 I0 π

(4.61)

for the number of the resonance when the energy is equal to the top of the potential barrier. The width of resonance state increases with the rise of n, as it should be. If n = n ∗ , the width of the state is n = (λ ln 2) /L . The interval between the neighboring resonances 2πλ ∂ E 0n ≈ , E n = ∂n L so that the ratio

n ln 2 ≈ 0.110, = E n 2π

n = n∗.

(4.62)

We can conclude from these relations that resonance states do not overlap each with other, analogously to the states deep under the top of the potential barrier. This conclusion is valid also for some first above-barrier resonances (n > n ∗ ), though their widths increase rapidly with rise of n (and they overlap each with others, unlike the under-barrier resonance states). Equation (4.60) is applicable in the case of |a| 1, i.e. under the condition 2π |n|

1; L

n = n − n ∗ .

(4.63)

The region of its applicability depends on the value of the parameter L, Eq. (4.59). This parameter increases as well as with rise of n ∗ , but more slowly (logarithmic law). We consider, as an example, the case of parabolic potential U (x) = −λ2 x 2 /2, restricted from the left by infinite wall, so that U (x) = ∞ for x < −l [here we have ψ (−l) = 0 and we should substitute (n + 1/2) → (n + 3/4) in the quantization rule (4.46)]. According to above expressions we find (U0 = 0, xm = 0, x¯0 = −l): I0 =

1 2 λl ; 2

I1 =

9 L = ln n ∗ + ; 2

  1 1 + ln 2λl 2 ; 2

n ∗  1.

n∗ ≈

3 λl 2 − ; 2π 4 (4.64)

If the condition (4.63) is violated, i.e., |a| ∼ 1, the numerical solution of Eq. (4.46) is required; neighboring resonance states begin to overlap in this case. In the next Problem we consider the case of the above-barrier resonances at |a|  1. In conclusion, we like to give the reference for the paper Popov, Mur, Sergeev, Sov. Phys. - JETP, 100, 20 (1991). In this paper the reader can find also further development and applications of quantization rule taking into account the barrier penetrability.

122

4 Penetrability of Potential Barriers and Quasi-Stationary States

Fig. 4.5 Turning points and integration contours in the complex plane

(a)

C

x0

x

x1

x2

(b)

C′

x

x 0 x2

x1

Problem 4.5 Using the analytical continuation of the Bohr-Sommerfeld quantization rule (without barrier penetrability) for the potential depicted in Fig. 4.4, into above-barrier region, find the energies and widths of above-barrier resonance states. Consider, as an example, the above-barrier resonance states for anharmonic oscillator U (x) = ω 2 x 2 /2 − gx 3 in the limit g  ω 5/2 .

Solution This Problem is a continuation of the previous Problem. First of all, we like to make some remarks about the classical turning points. All three turning points x0 , x1 , x2 are real at the real values of the energy below the top of the potential barrier U0 . The turning points x2 → x1 at E → U0 . If E > U0 , then these turning points are complex quantities (and the energies E are real). The Bohr-Sommerfeld quantization rule is of a simple form for deep under-barrier levels, E n < U0 (neglecting their width) 1 2

x1

" pdx ≡ C

  1 ; pdx = π n + 2

n = 0, 1, 2...

(4.65)

x0

Here the turning points x0 and x1 are surrounded by the contour C in the plane of complex variable x. These points are root branching points of the function p (x) =



2 (E − U (x))

(4.66)

(we should cut this plane for single-valued definition of the momentum p(x)). The classical turning point x2 is also the branching point of p(x), but it is found outside of the contour C (see Fig. 4.5a). The quantization rule (4.46) takes into account the barrier penetrability, and it is more accurate than Eq. (4.65). The energies E n = E 0n − in /2 contain imaginary parts n > 0 so that the classical turning points are the complex quantities.

4.2 Quasi-Stationary States and Above-Barrier Reflection

123

Their shifts are shown in Fig. 4.5a by arrows. It should be noted that we have some restrictions for the parameter a = |a| exp (iα) (see Eq. 4.37): 0<α<

π 2

π <α<π 2

for under-barrier resonance states, E 0n < 0; for above-barrier resonance states, E 0n > 0.

(4.67)

√ Since we have x1,2 = xm ∓ 2a/λ nearly of the top of the potential barrier, then the classical turning points x1 and x2 are found in the complex plane, respectively, below and above the real axis, both for under-barrier and for above-barrier states. Let us investigate the above-barrier resonance states and consider the limit |a|  1, Re a < 0. In order to simplify the function ϕ (a), Eq. (4.47), we use the well-known asymptotic expansion of the gamma-function  (z) =

     √ 1 1 + ... exp z − ln z − z ; 2π 1 + 12z 2

|z|  1,

|arg z| < π.

(4.68) We should be careful in choice of phases of arguments of gamma-function  (±ia + 1/2) , since the restriction |arg z| < π in Eq. (4.68) fixes the choice of the branch of logarithmic function. Taking into account the second from the conditions (4.67), and also the inequality |a|  1, we should choose   3π 1 ≈α− ; arg ia + 2 2

  1 π arg −ia + ≈α− . 2 2

(4.69)

Then we obtain the asymptotic expression of the function ϕ (a) for above-barrier resonance states   1 |a|  1, ϕ (a) = −2iπa + O , Re a < 0. (4.70) a Thus, we obtain the quantization rule for above-barrier resonance states x1

  1 , p (x, E n ) dx − iπa (E n ) = π n + 2

(4.71)

x0

where a (E) = (U0 − E) /λ. This equation is correct for above-barrier resonance states which are found not too nearly of the top of the potential barrier, since the condition |a|  1 should be required. However, these states should be not too far from the top of the potential barrier, because we have used the parabolic approximation for the potential. The generalization to the case of higher resonance states will be given below.

124

4 Penetrability of Potential Barriers and Quasi-Stationary States

Let us illustrate the application of Eq. (4.71) on the example of the parabolic potential barrier restricted by infinite wall which was considered above. Using the result (4.45), we obtain from Eq. (4.71) (U0 = 0, ReE n > 0):     En 2π (n + 3/4) n − n∗ En − ln − + 1 + ln 2 + 2πi = −1≈ . (4.72) λ2 l 2 λ2 l 2 λl 2 n∗ Here the quantity n ∗ ≈ λl 2 /2π coincides approximately with the above √expression (4.64). Further, the result (4.45) is valid under the condition ρ  a. This condition restricts from above the energies: |E n | λ2 l 2 . From the other side, the condition |a|  1 means that |E n | /λ  1. Thus, Eq. (4.72) describes resonance states with parameters:   ln n ∗ (4.73)

n − n∗ n∗. 1; 2π Further, let us underline that the choice of phase for the argument of logarithm in Eq. (4.72) is determined by the second of inequalities (4.67). Respectively, we have ln (−E n ) = ln E n + iπ,

where now −

π < arg E n < 0. 2

We rewrite now Eq. (4.72) in the form  −1 En n − n∗ En − ln = − + 1 + ln 2 + iπ . λ2 l 2 n∗ λ2 l 2

(4.74)

This form is appropriate for approximate iterative solution (with logarithmic accuracy): n − n∗ E 0n ≈ [ + ln ]−1 . λ2 l 2 n∗

n n − n∗ ≈ 2π [ + ln ]−2 . λ2 l 2 n∗

(4.75)

Here the notation is introduced  ≡ ln

2n ∗ + 1  1. n − n∗

It is seen from Eq. (4.75) that n n − n∗ , ≈ 2π E n 

(4.76)

so that neighboring resonance state overlap strongly because of the inequality (4.73). Now we find the relation between the quantization rule (4.71) for above-barrier resonance states in the case |a|  1 and analytical continuation of the Bohr-Sommerfeld quantization rule (4.65). It should be noted that nearly of the top of the potential bar-

4.2 Quasi-Stationary States and Above-Barrier Reflection

125

rier (when parabolic approximation of the potential is applicable) we have x2 p (x, E) dx = −iπa (E) .

(4.77)

x1

Here the function p (x, E) is determined on the same sheet of Riemann surface as in the Eq. (4.71).Therefore the quantization rule (4.71) can be written in the form 1 2

x2

" pdx ≡ C

  1 , pdx = π n + 2

(4.78)

x0

where the integration contour C  is shown in Fig. 4.5b. The quantization rule (4.78) for above-barrier resonance states having the form of the Bohr-Sommerfeld quantization rule (4.65), gives the solution of the problem. It is seen that analytical continuation of Eq. (4.65) reduces to the change of the integration contour: the contour C is substituted by the contour C  after collision of the classical turning points x1 and x2 at E = U0 . It should be noted that this connection between the quantization rules for underbarrier and above-barrier resonance states can be checked also on the above considered example of parabolic barrier with the infinite wall. Indeed, let us take the real 2 negative values  2 of the energy E n = −κ /2 in the Eq. (4.74). Here argE n = π and lnE n = ln κ /2 + iπ. Then this equation contains the real quantities only, and it coincides with the Bohr-Sommerfeld quantization rule. We restricted above by the values of E n for above-barrier resonance states near the top of the barrier using the parabolic expansion for the potential U (x). However, substitution of Eq. (4.77) for a(E) generalizes this investigation to the case of the arbitrary, sufficient smooth potential which satisfies the conditions for applicability of the WKB approximation. Analogous substitution for under-barrier resonance states a (E) =

x2 

− p2

1/2

dx

(4.79)

x1

generalizes Eqs. (4.46–4.47) to “deep” levels under the barrier. Using this substitution in the expansion (4.48), we obtain the quantization rule (4.19) from the Problem 4.1, as it should be. Now we consider the example of the anharmonic oscillator with cubic nonlinearity. We can omit the term ω 2 x 2 /2 in the potential U (x) because of the inequality g 2  ω 5 . Then Eq. (4.78) takes the form

126

4 Penetrability of Potential Barriers and Quasi-Stationary States

Table 4.1 Comparison of WKB and exact results for above-barrier resonances g

n=0 q E 0n

n=0 δ

n=0 ηexact

n=1 q E 0n

n=1 δ

n=1 ηexact

100 10 5 1 0.5

3.6852 1.4670 1.1120 0.5841 0.4426

0.9463 0.9464 0.9464 0.9474 0.9490

0.72654 0.72646 0.72630 0.72320 0.71610

13.77 5.483 4.455 2.183 1.654

0.9952 0.9952 0.9952 0.9958 0.9967

0.72654 0.72651 0.72643 0.72500 0.72180

  x2   1 3 . 2 E n + gx dx = π n + 2

(4.80)

x0

The values of the classical turning points are (see Fig. 4.5b; we suggest g > 0):  x0 = −

En g

1/3

 ;

x1,2 =

En g

1/3

  iπ . exp ∓ 3

(4.81)

The integration contour in Eq. (4.80) is chosen to be from two straightforward segments in complex x−plane: (x0 , 0) and (0,x2 ). Using the value of the integral   √ 1  1 3 π(1/3) 1 3 , , 1 − x dx = B = 3 3 2 5 (5/6) 0

we obtain the WKB-expression for the energy of above-barrier resonance states of the considered cubic nonlinear oscillator: q En

6/5   3/2  iπ 5π (2n + 1) = exp − g 2/5 √ 1 1 5 6 3  6 !1/5  π π × 0.72195 (2n + 1)6 g 2 = cos − i sin , 5 5

n = 0, 1, ...

(4.82)

(under-barrier resonances are absent at g → ∞). Then q

n π q = 0.72654. q ≡ ηn = tan 5 2E 0n

(4.83)

We present comparison of WKB and exact results for considered above-barrier q resonances in the Table. 4.1. In this Table the quantity E 0n is WKB energy of the q resonance state according to Eq. (4.82); δ ≡ E 0n /E 0n where E 0n is the exact value of the energy; finally, the parameter η is given by the ratio (4.83). Results of exact derivation are taken from the paper: Alvarez, Phys. Rev. A 37, 4079 (1988) (they correspond to the case of ω = 1/2).

4.2 Quasi-Stationary States and Above-Barrier Reflection

127

It is seen that WKB approximation allows to determine with high accuracy even the parameters of lowest resonance states. In conclusion, let us remark that above-barrier resonance states considered in this Problem have large widths which exceed the intervals between the neighboring states. Therefore there is no any simple individual properties of these states.

Problem 4.6 Find the coefficients of penetration and reflection of a particle with the arbitrary energy E in the case of parabolic potential barrier U (x) = −mω 2 x 2 /2. Use the exact solution of the Schrödinger equation.

Solution We suggest that particles are moving from the left to the right (of course, coefficients of penetration and reflection do not depend on the direction of motion of particles). We should solve the Schrödinger equation   1 2  2 2 ψ (x) + E + mω x ψ (x) = 0, 2m 2

(4.84)

which is of the form of wave going to the right at x → ∞. This solution was obtained in Problem 4.4:  ψ (x) = const · D−ia−1/2 (ρ exp (3iπ/4)) , a = −E/ω. ρ = −x 2mω/, (4.85) Asymptotic representations of this function are given by Eqs. (4.40–4.41). The second term in Eq. (4.41) corresponds to falling particles, while the first term corresponds to reflecting particles. In order to normalize wave function (4.85) by the condition of unit density of the falling flux j (x) = 1, we should choose the const to be equal     iπ πa 2m 1/4  (ia + 1/2) − . (4.86) exp − const = √ ω 4 8 2π Densities of fluxes are derived according to asymptotic representations, Eqs. (4.40– 4.41). Then we find the amplitudes of reflected wave (A (E)) and of transmitted wave (B(E)), and also the coefficients of transmission and of reflection:    πa  1 1 B(E) = √  ia + exp − ; 2 2 2π

128

4 Penetrability of Potential Barriers and Quasi-Stationary States

1 . 1 + exp(−2π E/ω)    πa  −i 1 exp ; A(E) = √  ia + 2 2 2π 1 . R(E) = |A|2 = 1 + exp (2πω E/ω)

D(E) = |B|2 =

(4.87)

We have D + R = 1, as it should be. In conclusion, it should be noted that the solution of this problem can be made by another method (see Ref. [1], Sect. 50). WKB approximation is used in this method, and connection between amplitudes of transmitted and reflected waves is obtained by means of shift to the complex plane of the variable x.

Problem 4.7 Using WKB approximation and the result of the previous problem, find the coefficients of transmission and reflection of a particle with energy nearly of the top of potential barrier in the case of an arbitrary smooth potential. Generalize the obtained solutions to the case of above-barrier reflection of a particle with arbitrary energy. Illustrate WKB results on the example of the potential barrier U (x) = U0 / cosh2 αx and compare with exact solution.

Solution (1) We have in the vicinity of the top of the potential barrier x = xm 1 U (x) = U0 − mω 2 (x − xm )2 + ...; 2

mω 2 = −U  (xm ) ,

(4.88)

where U0 is the top of the barrier. If the energy E is nearly of U0 , then WKB approximation is inapplicable in the vicinity of xm . However, we can use the exact solution of Schrödinger equation for parabolic potential barrier. This solution allows to match WKB solutions far from the point xm which correspond to incoming, reflected and transmitted waves. Using Eq. (4.87), we find 1 ; 1 + exp [−2π (E − U0 ) /ω] 1 . R(E) ≈ 1 + exp [2π (E − U0 ) /ω]

D(E) ≈

(4.89)

4.2 Quasi-Stationary States and Above-Barrier Reflection Fig. 4.6 Turning points and integration contours

(a)

129

(b)

(c) x2

x1

C′ x2

x2

C

x1

The condition for applicability of Eq. (4.89) is that parabolic expansion (4.88) of the potential energy is valid up to the values of x (in matching region), where WKB-condition |dλ/dx| 1 is fulfilled for E = U0 , i.e.   |x − xm |  . (4.90) mω Here the energy E should be not too far from U0 , in order the classical turning points x1,2 defined by the expression  1/2     x1,2 − xm  =  2 (E − U0 )   mω 2  would be inside the region, where the expansion (4.88) is still valid. These two conditions allow to match WKB-solutions using the parabolic expansion of the potential. They are equivalent to the conditions |ρ|  |a| and |ρ|  1. Then asymptotic Eqs. (4.40–4.41) are valid for the function of the parabolic cylinder, Eq. (4.39). (2) Generalization of Eqs. (4.89) to the case when the energy E is not too close to the top U0 of the potential barrier, can be obtained by means of the next consideration. Let us investigate first the case of E < U0 (under-barrier tunneling). When U0 − E > ω, the penetrability of the barrier diminishes up to very small quantity. In the parabolic approximation the next relations are valid

1 π (U0 − E) = πa ≡ ω 

x2 x1

1 (−i p) dx = 

x2 | p| dx,

E < U0 ,

(4.91)

x1

where x1,2 are the classical turning points (see Fig. 4.6a). Then the transmission coefficient (4.89) takes usual WKB expression at a > 1: ⎤ ⎡ x2 2 | p| dx ⎦ . (4.92) D (E) ≈ exp ⎣−  x1

Obviously, applicability of Eq. (4.92) at E < U0 does not require parabolic expansion in all under-barrier region.

130

4 Penetrability of Potential Barriers and Quasi-Stationary States

The case of E > U0 can be considered similarly. The formal difference is in that now the classical turning points x1,2 (E) are in complex plane (see Fig. 4.6b). In the parabolic approximation we have  2 (E − U0 ) x1,2 = xm ∓ i (4.93) mω 2 (the interval between points x1 and x2 increases with the rise of the energy E). In this case we have 1 π (E − U0 ) = − πa ≡ ω 

x2

i (−i p) dx = − 2

x1

" pdx.

(4.94)

C

The integration contour is shown in Fig. 4.6b. The coefficient of above-barrier reflection is small at E − U0 > ω. According to (4.89) and (4.94) it is described by ⎡ ⎤ " i R (E) ≈ exp ⎣ p (E, x) dx ⎦ (4.95)  C

This WKB-expression is valid already at further rise of the energy E, when the parabolic approximation for U (x) is incorrect. We make two comments concerning Eq. (4.95). First, let us underline that the integral in the exponent of Eq. (4.95) is pure imaginary √quantity. This follows from the relation p(x) = p ∗ (x ∗ ) for the momentum p(x) = 2m (E − U (x)) (in particular, x2 = x1∗ ; compare to Eq. (4.93) which is valid in the parabolic approximation for U (x)). Second comment refers to more accurate derivation of Eq. (4.95). It is described in the book, Ref. [1], Sect. 52. The coefficient of above-barrier reflection is  1 σ! σ = Im pdx, (4.96) R = exp −4 ;  2 C

the integration contour C  is shown in Fig. 4.6c (its ends are found on the real axis of x , and this contour does not intersect the cut going from the branching point x1 ). This expression coincides Eq. (4.95), since the relation takes place (followed from the first comment)   i pdx. 2σ = Im pdx = − 2 C

C

(3) As an example of application of Eq. (4.95), we consider the case of the potential barrier U (x) = U0 / cosh2 (αx). The classical turning pointsare found on the imagi  nary axis x ≡ i y of the complex variable x. We have p(x) = 2m E − U0 / cos2 (αy) .

4.2 Quasi-Stationary States and Above-Barrier Reflection

131

Derivation of the integral in the exponent of Eq. (4.95) gives the result i 

 C

2 pdx = − 

y0  −y0

√    2π 2m √ U0 dy = − 2m E − E − U0 . cos2 αy α 

(4.97) turning points which are nearest to the real Here x1,2 = ∓i y0 are the classical √ axis of x, and y0 = α−1 arccos U0 /E. The WKB coefficient of above-barrier reflection is     2π 2m E 2mU0 k= R(E) ≈ exp − ; k0 = . (4.98) (k − k0 ) , 2 α  2 The exact value of the coefficient of reflection ([1], Sect. 25) is R(E) =

1 2 .    2 1 + sinh (πk/α) / cosh (πk0 /α) 1 − (α/2k0 )

(4.99)

It is seen that WKB result (4.98) and exact result (4.99) coincides each with other under the conditions k0  1, α

2 (k − k0 ) > 1. α

In this case R(E) 1.

Problem 4.8 Find energies of high excited levels in the hydrogen molecular ion.

Solution 1. Introduction The WKB approximation is widely applied for analysis of high excited (Rydberg) states in various one-dimensional problems of quantum mechanics. Also it is applicable for three-dimensional problems with the central potential when variables are separated in spherical system of coordinates. The hydrogen molecular ion presents the simplest molecular system. It contains two protons with the separation distance R between them and also one electron. The goal of this short text is to consider analytically this quantum mechanical problem in the WKB approximation and to

132

4 Penetrability of Potential Barriers and Quasi-Stationary States

find the energies of high excited (Rydberg) states when the separation distance R is large, or small compared to the range of the considered electron state.

2. The Separation of Variables The stationary Schrödinger equation for an electron in the field of two protons with the distance R between protons is of the form (in atomic units)   1 1 1  = E. − − − 2 r1 r2

(4.100)

Here r1 = |r − R/2| , r2 = |r + R/2| are distances from an electron to two protons, E < 0 is the energy of the considered highly excited state; R is the distance between protons. Variables in Eq. (4.100) are separated after introduction of three elliptical coordinates: r1 + r2 ; R 1 < ξ < ∞; ξ=

r1 − r2 ; R − 1 < η < 1.

η=

ϕ;

Substituting  = X (ξ)Y (η) exp(imϕ), one obtains two ordinary differential equations    dX   E R 2   m2 d  2 2 2 ξ −1 + ξ − 1 + 2Rξ − s − 2 X = 0; dξ dξ 2 ξ −1      dY  d E R2  m2 X = 0. 1 − η2 + 1 − η2 + s 2 − dη dη 2 1 − η2 Here s 2 is the separation constant. The next substitution K (ξ) ; X (ξ) =  ξ2 − 1

Y (η) = 

L(η) 1 − η2

results in equations d2 K (ξ) + R 2 (ξ)K (ξ) = 0; dξ 2 d2 L(η) + Q 2 (η)L(η) = 0. dη 2

;

4.2 Quasi-Stationary States and Above-Barrier Reflection

133

Here notations are introduced for electron momenta: 

1/2 E R2 2Rξ − s 2 m2 − 1 + 2 − ; R(ξ) = 2 2 ξ −1 ξ2 − 1  1/2 E R2 s2 m2 − 1 + Q(η) = − . 2 2 1 − η2 1 − η2

(4.101)

3. The Langer Transformation The WKB Langer transformation is to substitute in Eq. (4.101) m 2 − 1 −→ m 2 for correct consideration of high excited states. Let us discuss this Langer correction on the example of the equation for the wave function L(η): d2 L(η) + Q 2 (η)L(η) = 0 dη 2 where  Q(η) =

E R2 s2 m2 − 1 + − 2 2 2 1−η 1 − η2

1/2 .

The reason for such correction is that the condition for applicability of the WKB approximation dλ/dη 1 (is the electron De Broglie wavelength) is not fulfilled in the vicinity of the values of η = ±1. Indeed, for example, near the value η = +1 we can simplify this equation, introducing the new variable r = 1 − η → 0. Then this equation can be rewritten in the form   s2 m2 − 1 E R2 d2 L(r ) + − L(r ) = 0. + dr 2 2 2r 4r 2 It follows from this equation that in the vicinity of r → 0 the derivative dλ/dr ∼ 1/|m|. Thus, the WKB approximation is inapplicable when the magnetic quantum number |m| ∼ 1. Following Langer, we now change the independent variable r = exp(x). The values of r → 0 correspond to x → −∞. Then the above equation takes the form   s2 m2 − 1 E R2 d2 L(x) dL(x) + exp(2x) + exp(x) − L(x) = 0. − dx 2 dx 2 2 4 In order to remove the first derivative of the wave function, we change further the wave function L = exp(x/2)ψ. Then the considered equation takes the form:

134

4 Penetrability of Potential Barriers and Quasi-Stationary States

  d2 ψ(x) E R2 s2 m2 ψ(x) = 0. + exp(2x) + exp(x) − dx 2 2 2 4 It is seen that at x → −∞ the electron momentum is constant so that quantity dλ/dη → 0 (excluding the case m = 0 which should be considered specially). The Bohr quantization rule for this equation is of the form    x2 s2 m2 E R2 1 exp(2x) + exp(x) − dx = n 2 + π. 2 2 4 2 x1

Returning to the variable r , one obtains this rule in the form:    r2 s2 m2 E R2 1 + − 2 dr = n 2 + π. 2 2r 4r 2 r1

Returning to the variable η, this equation can rewritten in the final form #   η2 $ $ E R2 s2 m2 1 % + π. − 2 dη = n 2 + 2 1 − η2 2 1 − η2 η1

Hence, one obtains simple expressions for electron momenta:  R(ξ) =  Q(η) =

E R2 2Rξ − s 2 m2 + 2 − 2 2 ξ −1 ξ2 − 1 E R2 2

+

s2 1 − η2

m2

1/2 ; (4.102)

1/2

− 2 1 − η2

.

The effective potentials are of the form: V (ξ) =

m2 s 2 − 2Rξ +  2 ; ξ2 − 1 ξ2 − 1

V (η) = −

s2 m2 +  2 . (4.103) 1 − η2 1 − η2

The potential V (η) is shown in Fig. 4.7 for parameters n 1 = 0, n 2 = 1, m = 1, n = 3, R = 4; s 2 = 3.2, E = −0.2, E R 2 /2 = −1.6 when the distance R between protons is less than the radius of the electron orbit n 2 . The potential V (ξ) is shown in Fig. 4.8 for the same values of parameters.

4.2 Quasi-Stationary States and Above-Barrier Reflection

135

−1 < η < 1 -1

1

-b

b

ER 2 / 2

Q 2 (η )

Fig. 4.7 The effective potential V (η) at the small separation distance R between protons

ξ >1 ER 2 / 2

R 2 (ξ )

Fig. 4.8 The effective potential V (ξ) at the small separation distance R between protons

According to Fig. 4.8 the Bohr quantization rule for the potential V (ξ) is of the standard form: ξ2 R(ξ)dξ = π (n 1 + 1/2) . ξ1

(4.104)

136

4 Penetrability of Potential Barriers and Quasi-Stationary States −1 < η < 1 -1

1

-b

-a

a

b

ER 2 / 2

Q 2 (η )

Fig. 4.9 The effective potential V (η) at the large separation distance R between protons

ξ >1 ER 2 / 2

R 2 (ξ )

Fig. 4.10 The effective potential V (ξ) at the large separation distance R between protons

Here n 1 = 0, 1, 2, 3... is an integer, and ξ1 , ξ2 are left and right classical turning points. Analogously we present the potential V (η) in Fig. 4.9 for parameters n 1 = 0, n 2 = 0, m = 1, n = 2, R = 10; s 2 = 7, E = −0.22, E R 2 /2 = −11 when the distance R between protons is larger than the radius of the electron orbit n 2 . The potential V (ξ) is shown in Fig. 4.10 for the same values of parameters.

4.2 Quasi-Stationary States and Above-Barrier Reflection

137

4. Splitting of Levels for Large Separation Distance Between Protons Let us find now the Bohr quantization rule for the potential V (η) described in Fig. 4.9. It is found by matching of the WKB wave functions in different regions. In the case which is shown in Fig.1.1a, matching conditions for the WKB wave functions can be found, using Airy functions in the vicinity of the turning point a (Q is the kinetic electron momentum): ⎞ ⎛ a ⎛ x ⎞   2 π 1 exp ⎝− |Q|dη ⎠ → √ sin ⎝ Qdη + ⎠ ; √ 4 |Q| Q ⎛

x

a

⎞

⎛ x ⎞ a  1 π 1 exp ⎝ |Q|dη ⎠ → √ cos ⎝ Qdη + ⎠ . √ 4 |Q| Q x

(4.105)

a

In the case presented in Fig. 1.1b, matching conditions for WKB wave functions are of the analogous form: ⎞ ⎛ x ⎛ a ⎞   2 π 1 exp ⎝− |Q|dη ⎠ ; √ sin ⎝ Qdη + ⎠ → √ 4 |Q| Q x

a

⎞ ⎛ x ⎛ a ⎞   1 π 1 exp ⎝ |Q|dη ⎠ . √ cos ⎝ Qdη + ⎠ → √ 4 |Q| Q x

(4.106)

a

Let us consider the WKB wave function in the classically forbidden region (Fig. 4.9) (−1 < η < −b). It is of the form decreasing to the left (we do not normalize this wave function): ⎛ −b ⎞  1 L(η) = √ exp ⎝− |Q|dη ⎠ . |Q|

(4.107)

x

Now we form the WKB wave function in the classically permitted region (−b < η < −a) based on the matching rules, Eq. (4.105): ⎞ ⎛ x ⎛ −a ⎞   −a 2 π π 2 L(η) = √ sin ⎝ Qdη + ⎠ = √ sin ⎝ Qdη + − Qdη ⎠ (4.108) 4 4 Q Q x −b −b ⎧ ⎛ −a ⎞ ⎛ −a ⎞⎫   ⎨ π π ⎬ 2 =√ cos A · cos ⎝ Qdη + ⎠ + sin A · sin ⎝ Qdη + ⎠ . 4 4 ⎭ Q⎩ x

x

138

4 Penetrability of Potential Barriers and Quasi-Stationary States

Here notation is introduced: −a A=

b Qdη =

−b

Qdη.

(4.109)

a

Further we find the WKB wave function in the classically forbidden region (−a < η < a) using matching with the wave function, Eq. (4.106), based on the matching conditions, Eq. (4.107): ⎧ ⎛ x ⎛ x ⎞ ⎞⎫   ⎬ 1 2 ⎨ cos A · exp ⎝ |Q|dη ⎠ + sin A · exp ⎝− |Q|dη ⎠ L(η) = √ ⎭ 2 |Q| ⎩ −a −a ⎧ ⎞ ⎛ a  2 ⎨ =√ cos A · exp(B) · exp ⎝− |Q|dη ⎠ |Q| ⎩ x ⎛ a ⎞⎫  ⎬ 1 + sin A · exp(−B) · exp ⎝ |Q|dη ⎠ . (4.110) ⎭ 2 x

Here notation is introduced:

a B=

|Q|dη.

(4.111)

−a

Then we find the WKB wave function in the classically permitted region (a < η < b) using matching with the wave function, Eq. (4.110), based on the matching conditions, Eq. (4.105): ⎛ x ⎞  2 π L(η) = √ 2 cos A · exp(B) · sin ⎝ Qdη + ⎠ 4 Q a ⎛ x ⎞  2 1 π sin A · exp(−B) · cos ⎝ Qdη + ⎠ +√ 4 Q2 a ⎡ ⎛ b ⎞  π 2 = √ 2 cos A · exp(B) · ⎣sin A · sin ⎝ Qdη + ⎠ 4 Q x ⎛ b ⎞⎤  π + cos A · cos ⎝ Qdη + ⎠⎦ 4 x

4.2 Quasi-Stationary States and Above-Barrier Reflection

139

⎡ ⎛ b ⎞  2 1 π +√ sin A · exp(−B) · ⎣cos A · sin ⎝ Qdη + ⎠ 4 Q2 x ⎛ b ⎞⎤  π − sin A · cos ⎝ Qdη + ⎠⎦ . (4.112) 4 x

Finally we find the WKB wave function in the classically forbidden region (b < η < 1) using matching with the wave function, Eq. (4.112), based on the matching conditions, Eq. (4.106): ⎛ x ⎞  π 2 L(η) = √ 2 cos A · exp(B) · sin ⎝ Qdη + ⎠ 4 Q a ⎛ x ⎞  2 1 π sin A · exp(−B) · cos ⎝ Qdη + ⎠ +√ 4 Q2 a ⎛ ⎞ b π 2 = √ 2 cos A · exp(B) · sin ⎝ A − Qdη + ⎠ 4 Q x ⎛ ⎞ b π 1 2 + √ · sin A · exp(−B) · cos ⎝ A − Qdη + ⎠ 4 Q 2

(4.113)

x

or 2 L(η) = √ 2 cos A · exp(B) Q ⎡ ⎛ b ⎞ ⎛ b ⎞⎤   π π × ⎣sin A · sin ⎝ Qdη + ⎠ + cos A · cos ⎝ Qdη + ⎠⎦ 4 4 x

x

2 1 sin A · exp(−B) +√ Q2 ⎡ ⎛ b ⎞ ⎛ b ⎞⎤   π π × ⎣cos A · sin ⎝ Qdη + ⎠ − sin A · cos ⎝ Qdη + ⎠⎦ . 4 4 x

x

The WKB function in this region should contain only the decreasing expo *wave x nent exp − b |Q|dη . Therefore the coefficient in Eq. (4.113) at the increasing * x exponent exp b |Q|dη should be equal to zero. Thus, one obtains the equation:

140

4 Penetrability of Potential Barriers and Quasi-Stationary States

2 cos2 A · exp(B) −

1 2 sin A · exp(−B) = 0. 2

This equation can be rewritten in the form: 1 cot A = ± exp(−B). 2 Thus, one obtains the quantization rule in the form (taking into account that B  1): ⎛ a ⎞ b  1 (4.114) Q(η)dη = π (n 2 + 1/2) ± exp ⎝− |Q(η)| dη ⎠ . 2 −a

a

Signs ± in this expression correspond to even and odd wave functions with respect to η = 0, respectively. When R → ∞ it is possible to neglect by exponentially small term in the right side of Eq. (4.114). Then this quantization rule takes the simple form: b Q(η)dη = π (n 2 + 1/2) .

(4.115)

a

The energy E < 0 and the separation constant s 2 are determined from two equations (4.104) and (4.115). Oppositely, when the quantity a |Q(η)| dη ∼ 1 −a

Equation (4.114) is inapplicable, and we should solve exactly the wave equation with parabolic potential near the top of the potential barrier.

5. Splitting of Levels for Small Separation Distance Between Protons When the separation R between protons is diminished, the corresponding picture for the potential Q(η) is shown in Fig. 4.7. In this case one finds two real classical turning points −b and b, as well as two imaginary turning points, −ia and ia. There is now a vertical cut between points −ia and ia in the complex plane of the variable η. This cut should be gone around the turning point ia. Instead of Eq. (4.114), one finds ⎛ ia ⎞ b  1 Q(η)dη = π (n 2 + 1/2) ± exp ⎝i Q(ηdη ⎠ (4.116) 2 a

−ia

4.2 Quasi-Stationary States and Above-Barrier Reflection

141

It is of the same form as Eq. (4.114). It is applicable when the real quantity   ia      1 i Q(ηdη     −ia

is very large. Signs ± in the Eq. (4.116) again correspond to even and odd wave functions with respect to η = 0, respectively. If we neglect exponentially small splitting of even and odd levels, then one obtains from Eq. (4.116): b Q(η)dη = π (n 2 + 1/2) .

(4.117)

a

6. Energies of Excited Levels for Small Separation Distance Between Protons and m=0 Now we solve Eqs. (4.104) and (4.117) at R → 0. First we consider the simple case of zero magnetic quantum numbers: m = 0. Eq. (4.117) takes the form (λ = s 2 ): 1  2 0

|E|R 2 s2 − + 2 1 − η2

1/2 dη = π (n 2 + 1/2) .

or, changing variable in the integrand η = sin ϕ : π/2 2

1 s 2 − |E|R 2 cos2 ϕdϕ = π (n 2 + 1/2) . 2

0

Expanding this expression over R → 0, one obtains:  π/2 1 2 s − |E|R 2 cos2 ϕ dϕ = π (n 2 + 1/2) . 4s 0

Deriving the integrals, one finds the equation: s−

1 |E|R 2 = n 2 + 1/2. 8s

(4.118)

In the case of Eq. (4.104) we take into account that ξ  1 in the classical region. Then one obtains

142

4 Penetrability of Potential Barriers and Quasi-Stationary States

ξ2  ξ1

1 2R s2 − |E|R 2 + − 2 dξ = π (n 1 + 1/2) . 2 ξ ξ

Here ξ1  1 and ξ2  1 are left and right classical turning points, respectively. Direct calculation of the integral gives the equation:  2 (4.119) = s + n 1 + 1/2. |E| Substituting s from Eq. (4.118) into Eq. (4.119), one finds:  1 2 = |E|R 2 + n 2 + n 1 + 1. |E| 8s Substituting into this equation according to Eq. (4.119) the zero approximation for the separation constant s0 = n 2 + 1/2, one obtains:  |E|R 2 2 = + n 2 + n 1 + 1. |E| 8 (n 2 + 1/2) In zero approximation when R = 0 it follows the obvious relation |E| = 2/n 2 where n = n 1 + n 2 + 1 is the principal quantum number. In the next approximation we obtain finally: 2 R2 E∼ . =− 2 + 5 n n (n 2 + 1/2)

(4.120)

Thus, the binding energy decreases when the distance R between protons increases as it should be. Here the quantity R is separation between protons in the units of the Bohr radius 2 /me2 . The condition of applicability of Eq. (4.120) is R n 2 , i.e. the separation between protons should be small compared to the radius of the Rydberg orbital n 2 .

7. Energies of Excited Levels for Small Separation Distance Between Protons and m = 0 Now we consider the case R → 0 when m = 0. If R = 0 Eq. (4.119) takes the form: # a $ 2 $ s m2 − 2 %  2 dη = π (n 2 + 1/2) . 1 − η2 1 − η2 0

4.2 Quasi-Stationary States and Above-Barrier Reflection

143

Here a is the right classical turning point. Let us rewrite this equation in the form a  2 a − η2 2s dη = π (n 2 + 1/2) . 1 − η2

(4.121)

0

Here a 2 = 1 − m 2 /s 2 < 1. We have: a 0

dη π = √ .   2 1 − a2 1 − η2 a 2 − η2

We introduce notation: a  2 a − η2 f (a) = dη 1 − η2 0

Differentiation of this expression over the parameter a allows us to connect the derivative with the previous integral: 

a

f (a) = a 0

dη πa = √ .   2 2 2 2 1 − a2 a −η 1−η

Now we integrate this expression taking into account that f (0) = 0 :   π 1 − 1 − a2 . f (a) = 2 Thus, Eq. (4.121) takes the form:    s 1 − 1 − a 2 = n 2 + 1/2. Hence, in zero approximation the separation constant s = s0 is of the form: s0 = n 2 + |m| + 1/2. Now we find the first correction. The Bohr quantization rule is # a $ 2 $ s m2 |E|R 2 dη = π (n 2 + 1/2) . 2 % − 2 − 2 1−η 2 1 − η2 0

144

4 Penetrability of Potential Barriers and Quasi-Stationary States

We expand this expression over the small parameter R → 0. The shift of the classical turning point can be neglected since the corresponding contribution contains the electron momentum in the classical turning point, i.e. zero. Thus, one obtains # a $ $ 2 % 0

s2 m2 |E|R 2 − 2 dη − 2 1−η 2 1 − η2

a 0

 1 − η 2 dη   = π (n 2 + 1/2) . s 2 1 − η2 − m 2

or |E|R 2 π (s − |m|) − 2

a 0

 1 − η 2 dη   = π (n 2 + 1/2) . s 2 1 − η2 − m 2

Deriving the integral by changing the integrand variable η = a sin ϕ, we find first correction for the separation constant s: s = n 2 + |m| + 1/2 +

 |E|R 2  2 2 s . + m 0 8s03

(4.122)

Now we consider the second Bohr quantization rule with m = 0 at ξ  1: ξ2  s2 1 2R m2 − 2 + 4 dξ = π (n 1 + 1/2) . − |E|R 2 + 2 ξ ξ ξ

ξ1

The term m 2 /ξ 4 is small compared to other terms in the integrand. Therefore we can expand this integrand: ξ2  ξ1

s2 1 2R − 2 dξ + − |E|R 2 + 2 ξ ξ

ξ2 

ξ1

m2

2ξ 4 − 21 |E|R 2 + 2R/ξ − s 2 /ξ 2

dξ = π (n 1 + 1/2) ,

or using the previous derivations and changing the integrand variable ξ = 1/x: + π

, x2 2 m2 x 2  −s + dξ = π (n 1 + 1/2) . |E| 1 2 2 2 x1 2 − 2 |E|R + 2Rx − s x

Deriving the integral, one obtains   2 |E|R 2  2 − s − n 1 − 1/2 + 6 − |E|s = 0. 0 |E| 8s05

(4.123)

4.2 Quasi-Stationary States and Above-Barrier Reflection

145

Substituting Eq. (4.23) into Eq. (4.24), we exclude the separation constant s:   |E|R 2   2 |E|R 2  2 2 2 6 − |E|s − s = 0. + m −n+ 0 0 |E| 8s03 8s05 In the small terms of this equation which are proportional to R 2 we can substitute |E| → 2/n 2 . Then one obtains  + , R2 2 2m 2 3m 2 n 2 =n+ 2 + 2 . 1− |E| 4n s0 s04 s0 Thus, the energy of the high excited state is + , R2 2m 2 2 3m 2 n 2 + 2 . E =− 2 + 5 1− n n s0 s04 s0

(4.124)

Here s0 = n 2 + |m| + 1/2. When m = 0 Eq. (4.124) reduces to Eq. (4.120), as it should do.

8. Energies of Excited Levels for Large Separation Distance Between Protons and m=0 Now we consider the opposite limit of large separation R  n 2 between protons of the molecular hydrogen ion. In this limit the values of ξ and η near 1 are significant (if we consider an electron near the right proton) at the integration. We restrict ourselves the case of m = 0. The Bohr quantization rule is 1  η0

s2 |E|R 2 dη = π (n 2 + 1/2) . − 2 1−η 2

Changing the integrand variable η = cos ϕ, one obtains ( p 2 = |E|R 2 /2) a 

s 2 − p 2 sin2 ϕdϕ = π (n 2 + 1/2)

0

Here ϕ 1. Taking into account terms of zero and first order and expanding sin ϕ, we rewrite this equation in the form a  2  s 2 − p 2 ϕ − ϕ3 /6 dϕ = π (n 2 + 1/2) 0

146

4 Penetrability of Potential Barriers and Quasi-Stationary States

or after expansion over small quantity ϕ3 : a 

a s 2 − p 2 ϕ2 dϕ +

0

0

ϕ4  dϕ = π (n 2 + 1/2) . 6 s 2 − p 2 ϕ2

Here a = s/ p 1. After derivation of simple integrals one obtains the first relation between the energy E and the separation constant s:

s 2 = 4 p (n 2 + 1/2) −

s4 = 4 p (n 2 + 1/2) − 2 (n 2 + 1/2)2 . 8 p2

(4.125)

Now we consider the second Bohr quantization rule at R  n 2 and m = 0 : ξ0  − p2 + 1

2Rξ − s 2 dξ = π (n 1 + 1/2) . ξ2 − 1

Changing the integrand variable ξ = cosh ϕ, we rewrite this equation: ϕ0 

2R cosh ϕ − s 2 − p 2 sinh2 ϕdϕ = π (n 1 + 1/2) .

0

Since ϕ 1 then after expansion of cosh ϕ and sinh2 ϕ in Taylor series we can rewrite the quantization rule in the form ϕ0   2R − s 2 − p 2 ϕ2 + Rϕ2 − p 2 ϕ4 /3 dϕ = π (n 1 + 1/2) . 0

 Expanding the integrand over small quantity Rϕ2 − p 2 ϕ4 /3 we rewrite the equation in the form ϕ0  0

ϕ0  2 2 ϕ4 /3 Rϕ − p  2R − s 2 − p 2 ϕ2 dϕ + dϕ = π (n 1 + 1/2) . 2 2R − s 2 − p 2 ϕ2 0

Let us introduce the notation t2 =

2R − s 2 . p2

Then the quantization rule can be rewritten as

4.2 Quasi-Stationary States and Above-Barrier Reflection

147

ϕ0  ϕ0  2 Rϕ − p 2 ϕ4 /3 2 2  p t − ϕ dϕ + dϕ = π (n 1 + 1/2) , 2 p t 2 − ϕ2 0

0

or after substitution ϕ = t x: pt

2

1 

1  1−

x 2 dx

0

+t

2 0

Rx 2 − p 2 x 4 t 2 /3 dx = π (n 1 + 1/2) . √ 2 p 1 − x2

Evaluation of simple integrals results in the second equation connecting the electron energy E and the separation constant s: pt 2

π t2 R π π + − t4 p = π (n 1 + 1/2) , 4 2p 4 32

or 2R − s 2 = 4 p (n 1 + 1/2) +

s 4 − 4R 2 . 8 p2

(4.126)

Substituting s 2 from Eq. (4.125) into Eq. (4.126), one obtains 2R − 4 p (n 2 + 1/2) +

s4 s 4 − 4R 2 = 4 p + 1/2) + , (n 1 8 p2 8 p2

Let us again introduce the principal quantum number n = n 1 + n 2 + 1. Thus, we find a simple equation  1 − n 2|E| = −

1 . 2|E|R

(4.127)

Zero approximation gives in Eq. (4.127) the obvious result: E = −1/2n 2 . The first approximation is E =−

1 1 − . 2 2n R

(4.128)

The second term in Eq. (4.128) is the Coulomb energy of the left proton. We can take now into account the exponentially small splitting of levels changing in Eq. (4.128) according to Eq. (4.114) ⎞ ⎛ a  1 πn 2 → πn 2 ± exp ⎝− |Q(η)| dη ⎠ . 2 −a

148

4 Penetrability of Potential Barriers and Quasi-Stationary States

After derivation of the simple integral we find a |Q(η)| dη = −a

R e2 R + n − 2 (n 2 + 1/2) ln n 2n (n 2 + 1/2) ⎛

and p=

R n R + ∓ exp ⎝− 2n 2 4πn 2

a

⎞ |Q(η)| dη ⎠ .

−a

The final expression for the electron energy of excited states with m = 0 is: 1 1 1 E =− 2 − ∓ 2n R 2πn 3



e2 R n (2n 2 + 1)

2n 2 +1 exp (−R/n − n)

It should be noted that in the case of the ground state n 1 = n 2 = 0, n = 1 the exponent coincides with the known expression ∓2R · exp(−R − 1) with accuracy to numerical factor. 9. Terms of the Order of 1/R 2 in the Energies of Excited Levels for Large Separation Distance Between Protons and m = 0 Now we generalize the derivation of the previous section taking into account the terms of the order of 1/R 2 in the energies of excited states. In the Bohr quantization rule (see previous section) I =

a 

s 2 − p 2 sin2 ϕdϕ = π (n 2 + 1/2)

0

we now take into account terms of zero, first and the second order in expansion sin ϕ at ϕ 1. Thus, a   2 s 2 − p 2 ϕ − ϕ3 /6 + ϕ5 /120 dϕ I = 0

a   s 2 − p 2 ϕ2 − ϕ4 /3 + 2ϕ6 /45 dϕ. = 0

Changing the integrand variable ϕ = aψ, one rewrites this integral in the form (a = s/ p) :

4.2 Quasi-Stationary States and Above-Barrier Reflection

I = pa

2

ψ0 

149

1 − ψ 2 + a 2 ψ 4 /3 − 2a 4 ψ 6 /45dψ.

0

Changing of the integrand variable ψ=



1 + a 2 /3 + 8a 4 /45x

transform this integral to the form  I = pa 2 1 + a 2 /3 + 8a 4 /45 ×

1 

  1 − x 2 1 − a 2 x 2 /3 − 8a 4 x 2 /45 + 2a 4 x 4 /45 dx.

0

Expanding square roots in Taylor series, one obtains   a2 4 1 4 I = pa 1 + + ( − )a 6 45 72     1   1 1 2 2 4 4 2 1 4 4 2 − a x dx. 1−x 1− a x − a x + × 6 45 45 72 2

0

Deriving the simple integrals, one obtains after cancellation of many terms I =

! π 2 pa 1 + a 2 /8 + 3a 4 /64 = π (n 2 + 1/2) . 4

Thus, the separation constant is of the form s 2 = 4 p (n 2 + 1/2) − 2 (n 2 + 1/2)2 − (n 2 + 1/2)3 / p.

(4.129)

The first two terms in the right side of this expression were obtained already previously in Eq. (4.125). Let us consider now the second quantization rule: J=

ϕ0 

2R cosh ϕ − s 2 − p 2 sinh2 ϕdϕ = π (n 1 + 1/2) .

0

We now take into account terms of zero, first and the second order in expansion sinh ϕ and coshϕ at ϕ 1. Thus,

150

4 Penetrability of Potential Barriers and Quasi-Stationary States

ϕ0    J= 2R 1 + ϕ2 /2 + ϕ4 /24 − s 2 − p 2 ϕ2 + ϕ4 /3 + 2ϕ6 /45 dϕ 0

Let us introduce the quantity: t2 =

2R − s 2 . p2

We change the integrand variable ϕ = tψ. Then one finds ψ0     J = pt 1 − ψ 2 + Rψ 2 / p 2 − t 2 ψ 4 /3 + Rt 2 ψ 4 /12 p 2 − 2t 4 ψ 6 /45 dψ. 2

0

The next changing of the integrand variable ψ = ψ0 x allows us to rewrite this integral in the form 1    2 J = pt ψ0 1 − x 2 1 + αx 2 + βx 4 dx 0

where the notations are introduced   α = 2t 4 /45 + 1/3 − R/12 p 2 t 2 ψ04 ;

β = 2t 4 /45.

  ψ02 = 1 + R/ p 2 − t 2 /3

Since

then one obtains α = t 2 /3 − 8t 4 /45 + 7Rt 2 /12 p 2 . Thus, the integral after expansion of the integrand in Taylor series takes the form

J = pt ψ0 2

1  

1 − x2



! 1 + αx 2 /2 + βx 4 /2 − α2 x 4 /8 dx.

0

The derivation of simple integrals gives the quantization rule in the form: ! π 2 pt ψ0 1 + α/8 + β/16 − α2 /64 = π (n 1 + 1/2) . 4  Taking into account that ψ02 1 − R/ p 2 = (1 − α), we can rewrite this equation ! ! pt 2 1 − 3α/8 + β/16 − 13α2 /64 = 4 (n 1 + 1/2) 1 − R/2 p 2 − R 2 /8 p 4 .

4.2 Quasi-Stationary States and Above-Barrier Reflection

151

Substituting the expressions for the quantities α and β, we rewrite this equation in the form      R 5t 2 R2 7Rt 2 1 − = 4 p . 2R − s 2 1 − + 1/2) − − (n 1 64 32 p 2 2 p2 8 p4 Using the approximate expressions for the quantity t, one finds   ! 2R − s 2 = 4 p (n 1 + 1/2) 1 − R/2 p 2 − 2n 4 /R 2 +

 2n(n + 1/2)2 t2  1 2R − s 2 + [7n − 3 (n 1 + 1/2)] . 8 R

Substituting the separation constant s 2 from Eq. (4.129) into this expression, one obtains 2R − 4 p (n 2 + 1/2) +

! s4 3s 6 2 4 2 + = 4 p + 1/2) 1 − R/2 p − 2n /r (n 1 8 p2 64 p 4 2  2R − s 2 2n(n 1 + 1/2)2 + + 8 p2 R × [7n − 3 (n 1 + 1/2)] .

We simplify this equation 2R − 4 pn +

2Rn 6n (n 2 + 1/2)3 n 3 (n 1 + 1/2) =− −4 p R R 6n (n 1 + 1/2)3 R2 14n 2 (n 1 + 1/2)2 s4 R − + 2+ + . R R 2p 16 p 4

Taking into account that in the first approximation p = R/2n + n/2 we can rewrite this equation in the form  p = R/2n + n/2 + f /2R = R

|E| . 2

where f = 3 (n 2 + 1/2)3 + 2n 2 (n 1 + 1/2) − 7n (n 1 + 1/2)2 + 3 (n 1 + 1/2)3 − 2n (n 2 + 1/2)2 =

n2 (3n 2 − 3n 1 − n) . 2

152

4 Penetrability of Potential Barriers and Quasi-Stationary States

Hence, the electron energy can be written in the form ( p 2 = |E|R 2 /2) : |E| =

n2 1 1 f + + + . 2 2n R 2R 2 n R2

Substituting f we obtain after some algebraic transformations the final expression for the electron energy E =−

3 n (n 2 − n 1 ) 1 1 − − . 2n 2 R 2 R2

(4.130)

The last term in this expression is the interaction of a proton with dipole moment produced by the second proton and an electron which is placed near the second proton. It should be noted that this expression is valid also in the case when the magnetic quantum number m is nonzero. In this case the principal quantum number n = n 1 + n 2 + |m| + 1.

10. Energy Splitting at Small Separation Distance R Between Protons In conclusion we consider the energy splitting of high excited levels at small distances R between protons compared to the size of the electron cloud n 2 . We restrict ourselves the case m = 0 analogously as we made above for large separation distances R. According to Eq. (4.116) we have b Q(η)dη = π (n 2 + 1/2) ±

1 exp (−K ) 2

a

where ia 

ia K = −i

Q(η)dη = −2i −ia

− p2 + 0

s2 dη. 1 − η2

Here p = |E|R 2 /2 ∼ = R 2 /n 2 1 and the separation constant s ∼ = n 2 + 1/2. Substituting η = i sinh ϕ, one obtains  ϕ0  ϕ0  p2 2 1 − s 2 − p 2 cosh2 ϕdϕ ∼ 2s cosh ϕ dϕ K =2 = 2s 2 0

0

where ϕ0 ∼ = ln(2s/ p). After derivation of simple integral one obtains K = 2s ln

2sn  1. Re1/4

4.2 Quasi-Stationary States and Above-Barrier Reflection

153

Thus, the separation constant s can be written taking into account the splitting of levels [see Eq. (4.118)]: s−

1 1 |E|R 2 = n 2 + 1/2 ± exp(−K ). 8s 2

Substituting this expression for s into Eq. (4.119), we find the energies of high excited states with m = 0 at R n 2 taking into account the small energy splitting [see Eq. (4.120)]: 2 R2 2 ± 3 E∼ =− 2 + 5 n n (n 2 + 1/2) πn



Re1/4 n (2n 2 + 1)

2n 2 +1 .

(4.131)

We did not consider the case when the energy level of the hydrogen molecular ion is found near the top of the effective potential barrier for Q 2 (η) at η = 0. Then the potential can be approximated by parabola function. The corresponding wave functions are so called functions of parabolic cylinder. Using these functions instead of sine and cosine wave functions, it is possible to match wave functions with each other. Such procedure is sufficiently cumbersome.

References 1. L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd edn. (Pergamon, Oxford, 1977) 2. A.I. Baz, Ya. B. Zeldovich, A.M. Perelomov, Scattering, Reactions and Decays in NonRelativistic Quantum Mechanics (Plenum, New York, 1975) 3. B.M. Karnakov, V.P. Krainov, Quasi-Classical Approximation in Quantum Mechanics (WKBMethod) (MEPhI Publishing, Moscow, 1992, in Russian). 4. M. Abramowitz, I.A. Stegun (ed.), Handbook of Mathematical Functions (Nat. Bureau of Standards, Washington, 1964)

Chapter 5

Transitions and Ionization in Quantum Systems

Here, we consider transition and ionization processes in various atomic systems produced by constant external fields, or by fields which change with time slowly compared to the typical Kepler periods of electrons in atoms.

5.1 Adiabatic Transitions Problems and Solutions Problem 5.1 Find the transition probability in a quantum two-level system produced by the strong monochromatic low frequency field.

Solution Here, we consider strong but slow perturbations in quantum mechanics which produce transitions in the considered quantum system. Instead of a general consideration it is better to start with a simple example. We consider the two-level system perturbed by a strong monochromatic low-frequency electromagnetic field V (t) = V sin ωt. Here, V is an perturbation amplitude and ω is its frequency. We use further the system of units where the Planck constant  = 1. The energy of a lower level (1) can be chosen as −1 while the energy of an upper level (2) can be chosen as +1.

B. M. Karnakov and V. P. Krainov, WKB Approximation in Atomic Physics, DOI: 10.1007/978-3-642-31558-9_5, © Springer-Verlag Berlin Heidelberg 2013

155

156

5 Transitions and Ionization in Quantum Systems

The perturbation can be strong so that the values of V  1 can be as well. The lowfrequency approximation means that we assume the condition ω  V. The product ωV can be both larger and < 1. Under these conditions the strong perturbation of a two-level system occurs only in vicinity of times t = 0. Hence, we can simplify the perturbation by expanding of sine in Taylor series: V sin ωt  V ωt. Here, we choose only one period of the electromagnetic field. Our first goal is to solve quantum mechanical equations describing dynamics of the two-level system in such field. The temporal interval can be increased from −∞ up to +∞, since real transitions take place only in the small vicinity near time instance t = 0. Next periods of the external electromagnetic field are found far from this region. Therefore, we will not consider analogous transitions during the next field periods. The temporal Schrödinger equation for amplitudes a1 and a2 of population of levels 1 and 2, respectively, can be written in the form: da1 + a1 = V ωt · a2 ; dt da2 − a2 = V ωt · a1 . i dt i

(5.1)

Diagonal part of the Hamiltonian contains energies of the lower and upper levels, while the non-diagonal part mixes both states. Let us introduce the notation v = V ω. Let us introduce also the new amplitudes: a+ = a1 + a2 ; a− = a1 − a2 . Equations for these amplitudes are obtained from previous equations by means of their addition and subtraction, respectively: da+ − vta+ = −a− ; dt da− + vta− = −a+ . i dt

i

Unperturbed energy for the amplitude a+ is equal to vt, and unperturbed energy for the amplitude a− is equal to −vt. Differentiating the first of these equations over time and substituting the amplitude a− from the second equation, one obtains the differential equation of the second order for the amplitude a+ . Analogously, we find the equation for the amplitude a− :

5.1 Adiabatic Transitions

157

 d2 a+  2 + + 1 + iv a+ = 0; (vt) dt 2  d2 a−  2 + + 1 − iv a− = 0. (vt) dt 2

(5.2)

It is seen that from the mathematical point of view each of these equations describes one-dimensional quantum mechanical harmonic oscillator . The analytic solution of each of these equations is expressed via the function of parabolic cylinder E(a, x). In particular, one obtains   i 1 √ a+ (t) = A · E − − , t 2v . 2 2v (A is constant). The asymptotic expression of this function in the infinity is √ a+ (t) = A 2(t)−i/2v ; t → ±∞. At the transition from t = −∞ to t = +∞ the phase of time increases by π at the motion in the upper part of complex plane of time. Hence, one obtains      a+ (+∞)   = exp − π .   a (−∞)  2v +

(5.3)

Adiabatic solution of the system (5.1) is of the form     a1 (t) ∼ exp i E(t)dt ; a2 (t) ∼ exp −i E(t)dt

where E(t) =

1 + (vt)2 .

The energies ±E(t) are called adiabatic energies of the upper and lower terms, respectively. When the interaction turns off (v = 0), these energies reduce to the unperturbed values of energies ±1. Thus, diabatic solution a+ (−∞) corresponds to the lower adiabatic level (1), while the same solution a+ (+∞) corresponds to upper adiabatic level (2). Hence, square of the expression (5.3) determines the transition probability from the state 1 to the state 2:  π  . W (1 → 2) = exp − Vω

(5.4)

This is so-called Landau-Zener formula . It follows from Eq. (5.4) that this probability vanishes when the interaction turns off (v  1). Oppositely, this probability is equal to 1 at the super-intense interaction

158

5 Transitions and Ionization in Quantum Systems

(v  1). It follows from expression for a+ (t) that the typical interaction time is t ∼ v −1/2 . Hence, the condition ωt  1 corresponds to the condition ω  V of low frequency electromagnetic field.

Problem 5.2 Find the transition probability in a quantum two-level system produced by the strong perturbation which changes slowly with time.

Solution Now we consider again the transition from the state 1 to the state 2 of two-level system produced by perturbation which changes with time slowly. Unlike the previous case, we do not specify the form of this perturbation, but suggest that the transition probability is exponentially small. Equations for amplitudes of states 1 and 2 were obtained already previously: d2 a1,2 + E 2 (t)a1,2 = 0. dt 2

(5.5)

Here, the quantity −E(t) is an adiabatic energy of the lower level, while E(t) is adiabatic energy of the upper level. In WKB approximation these amplitude are of the form (we do not normalized these amplitudes): ⎡ t ⎤ 1 a1 (t) = √ exp ⎣i E(t )dt ⎦ ; E(t) ⎡ 0 t ⎤ 1 a2 (t) = √ exp ⎣−i E(t )dt ⎦ . E(t)

(5.6)

0

Idea of the method is to choose one of these amplitudes (for example, a2 ) in the infinity t → −∞ and to produce analytic continuation of this amplitude in the complex plane of time to t → +∞. During this procedure small addition of the amplitude a1 appears. Thus, we can calculate the transition probability 2 → 1. According to Eq. (5.6) the amplitude a1 appears at t = τ when E(τ ) = 0 (in the general case when adiabatic energies of the upper and lower levels differ from each other not only by sign, but also by a value, the corresponding condition is E 1 (τ ) = E 2 (τ )). Hence, the analytic continuation should be made in the vicinity of the point

5.1 Adiabatic Transitions

159

t = τ in the complex plane of time. The quantity E 2 (t) in Eq. (5.5) can be expanded in Taylor series. Thus, one obtains √ E(t) = C τ − t. The solution of Eq. (5.5) with this energy is √

a(t) = A τ

(2) − t H1/3



 2C 3/2 . (τ − t) 3

(5.7)

Here, H (2) (x) is the Hankel function of the second kind. The choice of this solution is related with its asymptotic expression at t → −∞: ⎡ t ⎤ 1 5π exp ⎣i E(t )dt − i ⎦ . a1 (t → −∞) ∼ √ (5.8) 12 E(t) τ

According to Eq. (5.7) it corresponds to the condition that a particle is in the state 1 at t → −∞. The motion to and from the point t = τ at the analytic continuation should be made upon the lines where t Im

E(t )dt = −

τ

 2C  Im (τ − t)3/2 = const. 3

Indeed, the WKB solutions, Eq. (5.6), are the asymptotic solutions of the differential equation. Hence, both exponents should have the same imaginary part (in the opposite case we should neglect exponentially small solution). In the vicinity of the point t = τ there are three such lines which diverge from each other by the angle 2π/3. Two of these three lines are going to t → −∞ and t → +∞,, respectively. Transition from the line t → −∞ to the line t → +∞ at the analytic continuation of the function, Eq. (5.8), corresponds to increasing the phase of complex time t by 2π/3. Hence, the argument of the Hankel function, Eq. (5.7), increases by π. The Hankel functions are satisfied to relation:  (2)  (2) (1) H1/3 z exp(iπ) = H1/3 [z] + exp(iπ/3)H1/3 [z] . We apply this relation to Eq. (5.7) and then consider the times t → +∞: ⎤ ⎤ ⎡ t ⎡ t 1 5π 5π exp(iπ/3) a1 (t → −∞) ∼ √ exp ⎣i E(t )dt − i ⎦ + √ exp ⎣−i E(t )dt + i ⎦ . 12 12 E(t) E(t) τ

τ

160

5 Transitions and Ionization in Quantum Systems

The first term in this equation corresponds to amplitude of population in state 1, while the second term corresponds to amplitude of population in state 2. Thus, exponentially small transition amplitude 1 → 2 is equal to the ratio of the second term to the first term, i.e., to ⎡ ⎤ ∞ −i exp ⎣−2i E(t )dt ⎦ . τ

Hence, the transition probability 1 → 2 is ⎡ W (1 → 2) = exp ⎣−4Im

τ

⎤ E(t)dt ⎦  1.

(5.9)

0

We took into account that the integral over the real axis of time does not contribute into this probability. The complex time τ is found from the equation E(τ ) = 0. This is so-called Landau-Dykhne formula . It is simple to generalize this relation to the case when adiabatic energies of the upper and lower levels differ from each other not only by sign, but also by a value. One obtains ⎡ ⎤ τ W (1 → 2) = exp ⎣−2Im (E 2 (t) − E 1 (t)) dt ⎦ . (5.10) 0

Here the time τ is derived from the equation E 1 (τ ) = E 2 (τ ). In particular, the Landau-Zener transition probability (see above) can be calculated from Eq. (5.9) if to substitute adiabatic energy in the form

E(t) = Then one obtains

1 + (vt)2 ,

v  1.

 π  1. W (1 → 2) = exp − v

The advantage of the Landau-Dykhne approximation is that it allows us to derive exponentially small transition probabilities for various types of the adiabatic energies, while the Landau-Zener approach is related with the definite type of the adiabatic energies, but allows to find as well as large transition probabilities. It should be noted that the pre-exponential factor in the Landau-Dykhne formula, Eq. (5.10), differs from one in the case when in the vicinity of the complex turning point τ some other singularities of the adiabatic energy occurs.

5.2 Ionization of Quantum Systems

161

5.2 Ionization of Quantum Systems Problems and Solutions Problem 5.3 Find the ionization rate of a hydrogen atom by a weak constant electric field.

Solution Let us first to find the ionization rate of a hydrogen atom by a weak constant electric field with exponential accuracy. The energy of a ground state of a hydrogen atom is E 1 = −1/2 (in atomic units). The energy of a final state (neglecting the Coulomb field of proton) is E 2 = Ft 2 /2 where F is the electric field strength. This energy corresponds to acceleration of an electron in a constant electric field. The classical turning point in upper part of the complex plane of time is found from an equation (see previous Problem) E 1 = E 2 (τ ), i.e. τ = i/F. The ionization rate of a ground hydrogen state is derived with exponential accuracy by the Landau-Dykhne formula (5.10) of the previous Problem: ⎡ W ∼ exp ⎣−2Im

τ

⎤

  2 ⎦ . (E 2 (t) − E 1 (t)) dt = exp − 3F

0

Our goal is to determine the pre-exponential factor in this dependence. The unperturbed wave function of the ground state of a hydrogen atom is of the form 1 (r ) = √ exp (−r ) . π

(5.11)

The WKB wave function of an electron under the potential barrier during its tunneling is determined by the expression (electric field is directed along the axis z): ⎛ z ⎞ C π (z, p⊥ ) = √ exp ⎝i pz dz + i ⎠ (5.12) pz 4 a

Here, the longitudinal projection of electron momentum pz is an imaginary quantity under the potential barrier, and it is given by relation (according to the energy conservation law, taking into account the electric field):



2 + 2F z = i 1 + p 2 − 2F z. pz = −1 − p⊥ ⊥

162

5 Transitions and Ionization in Quantum Systems

The quantity p⊥ is the transverse electron momentum in the plane (X, Y ). The quantity a is some coordinate under the potential barrier, where we should match the unperturbed wave function (5.11) and WKB wave function (5.12). The value of a  1/F in order we could neglect the electric field F at this point. On the other side, a  1 in order we could substitute r → z. Thus, we assume that F  1, i.e., the electric field strength F should be small compared to the atomic field strength. In order to match Eqs. (5.11) and (5.12) with each other, we should expand the wave function, Eq. (5.11) in Fourier series in the plane (X, Y ) on the transverse momenta p⊥ . First we rewrite Eq. (5.11) in the equivalent form at r  1: 1 (r ) = √ exp (−r + ln z) . r π

(5.13)

The Fourier component of the function (1/r ) exp(−r ) is equal to 2π(1/| pz |)

2 . Thus, we find the Fourier compoexp(−| pz |z). In this region | pz | = 1 + p⊥ nent of the wave function at z = a √ 2 π exp (−| pz |a + ln a) . (r ) = (5.14) | pz | The function (5.12) in this region is C . (z, p⊥ ) = √ | pz |

(5.15)

Matching the functions (5.14) and (5.15) with each other allows us to determine the constant C taking into account that p⊥  1 in the considered region: √ C = 2 π exp (−| pz |a + ln a) . Hence, after substitution of C, the WKB wave function, Eq. (5.12), under the potential barrier takes the form ⎞ ⎛ z √ 2 π π (z, p⊥ ) = √ exp ⎝i pz dz + i + ln a ⎠ pz 4

(5.16)

0

This wave function requires the correction related with the long-range Coulomb potential of a proton under the potential barrier (so-called Coulomb correction). In the exponent of Eq. (5.16) we should add the classical action related with the Coulomb potential energy U = −1/r : −iδS = −i

U dt = i

dt = r

b a

dz = z| pz |

b a

dz 2 = ln . √ Fa z 1 − 2F z

5.2 Ionization of Quantum Systems

163

Here b = 1/2F is the classical turning point which corresponds to the end of the potential barrier. Multiplying Eq. (5.16) by exp (−iδS) , we find that the arbitrary value of a cancels, and the final form of the Fourier component of the wave function under the potential barrier is ⎛ z ⎞ √ 2 π π 2 pz dz + i + ln ⎠ . (z, p⊥ ) = √ exp ⎝i pz 4 F 0

The current of probability of this function at z = b is ⎞ ⎛ b 16π j ( p⊥ ) = pz |(z, p⊥ )|2 = 2 exp ⎝−2 | pz | dz ⎠ . F

(5.17)

0

Here | pz | ≈

√ p2 1 − 2F z + √ ⊥ . 2 1 − 2F z

We took into account here that typically p⊥  pz (see below). Substituting this expansion into Eq. (5.17), we derive the simple integrals:   2 p⊥ 2 16π − j ( p⊥ ) = 2 exp − . F 3F F

(5.18)

This expression determines the distribution of ejected√electrons on the transverse momentum. It is seen that the typical value of p⊥ ∼ F  1 while the typical value of | pz | ∼ 1 under the potential barrier. Thus, p⊥  pz that was used above in the derivations. The ionization rate of a hydrogen atom can be obtained from Eq. (5.18) by simple integration over all transverse momenta:   2 4 d2 p⊥ exp − . (5.19) = w= j ( p⊥ ) F 3F (2π)2 The exponent was already found above.

Problem 5.4 Find the ionization rate from the ground state in three-dimensional zero range potential by a weak constant electric field.

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5 Transitions and Ionization in Quantum Systems

Solution Let us first find the ionization rate by a weak constant electric field with exponential accuracy. The energy of a ground s−state in three-dimensional zero range potential is E 1 = −κ2 /2 (in units m =  = 1). This is the only bound state in zero range potential. Further we can put also κ = 1. The energy of a final state is E 2 = Ft 2 /2 where F is the electric field strength. This energy corresponds to acceleration of an electron in a constant electric field. The classical turning point in the upper part of the complex plane of time is found from an equation (see Problem 5.2) E 1 = E 2 (τ ), i.e., τ = i/F. The ionization rate of a ground state is derived with exponential accuracy by the Landau-Dykhne formula (5.10) of the Problem 5.2: ⎡ W ∼ exp ⎣−2Im

τ

⎤

  2 ⎦ . (E 2 (t) − E 1 (t)) dt = exp − 3F

0

Now our goal is to determine the pre-exponential factor in this dependence. The unperturbed wave function of the ground state in zero range potential at r > 0 is of the form 1 (r ) = √ exp (−r ) . (5.20) r 2π The WKB wave function of an electron under the potential barrier during its tunneling is determined by the expression (electric field is directed along the axis z): ⎛ z ⎞ C π pz dz + i ⎠ (z, p⊥ ) = √ exp ⎝i pz 4

(5.21)

a

Here the longitudinal projection of electron momentum pz is an imaginary quantity under the potential barrier, and it is given by relation (according to the energy conservation law, taking into account the electric field):



2 2 − 2F z. pz = −1 − p⊥ + 2F z = i 1 + p⊥

(5.22)

The quantity p⊥ is the transverse electron momentum in the plane (X, Y ). The quantity a is some coordinate under the potential barrier where we should match the unperturbed wave function (5.20) and WKB wave function (5.21). The value of a  1/F in order we could neglect the electric field F at this point. On the other side, a  1 in order we could substitute r → z. Thus, we assume that F  1, i.e., the electric field strength F should be small compared to the atomic field strength. In order to match Eqs. (5.20) and (5.21) with each other, we should expand the wave function, Eq. (5.20), in Fourier series in the plane (X, Y ) introducing the trans-

5.2 Ionization of Quantum Systems

165

verse momentum p⊥ . The Fourier component of the function ϕ(r ) = (1/r ) exp(−r )

is equal to ϕ(z, p⊥ ) = (2π/| pz |) exp(−| pz |z). In this region | pz | = we find the Fourier component of the wave function at z = a

2 . Thus, 1 + p⊥

√ (z = a, p⊥ ) =

2π exp (−| pz |a) . | pz |

(5.23)

The function (5.21) in this region is C . (z, p⊥ ) = √ | pz |

(5.24)

Matching the functions (5.23) and (5.24) with each other allows us to determine the constant C taking into account that p⊥  1 in the considered region: C=

√

2π exp (−| pz |a) .

Hence, after substitution of C, the WKB wave function, Eq. (5.21), under the potential barrier takes the form ⎛ z ⎞ 2π π (z, p⊥ ) = √ exp ⎝i pz dz + i ⎠ pz 4 √

(5.25)

0

The current of probability of this function at z = b is ⎞ ⎛ b 2 j ( p⊥ ) = pz |(z, p⊥ )| = 2π exp ⎝−2 | pz | dz ⎠ .

(5.26)

0

Here | pz | ≈

√ p2 1 − 2F z + √ ⊥ . 2 1 − 2F z

We took into account here that typically p⊥  pz (see below). Substituting this expansion into Eq. (5.26), we derive the simple integrals:   2 p⊥ 2 − j ( p⊥ ) = 2π exp − . (5.27) 3F F This expression determines the distribution of ejected √ electrons on the transverse momentum. It is seen that the typical value of p⊥ ∼ F  1 while the typical

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5 Transitions and Ionization in Quantum Systems

value of | pz | ∼ 1 under the potential barrier. Thus, p⊥  pz that was used above in the derivations. The ionization rate can be obtained from Eq. (5.27) by simple integration over all transverse momenta:   2 F d2 p⊥ exp − w= j ( p⊥ ) = . (5.28) 2 3F (2π)2 The exponent was already found above.

Problem 5.5 Find the ionization rate of an atom by a weak low frequency linearly polarized electromagnetic field.

Solution In the problems 5.2 and 5.3 we considered the ionization of quantum systems by a constant electric field. Now we generalize solutions to the case of low frequency electromagnetic field. This means that the photon energy ω (ω is the frequency of an electromagnetic field) is assumed to be small compared to the ionization potential E i of the considered atom. The atomic system of units will be used: e = m =  = 1. Low frequency of the electromagnetic field means that the adiabatic perturbation takes place. We derive the ionization rate only with exponential accuracy, since calculation of the pre-exponential factor requires cumbersome derivations even for the ionization of the ground hydrogen state. We can neglect Stark shift of the ground state in the external electromagnetic field. This shift does not change the exponent in the ionization rate (see below). Thus, the initial energy is E 1 = −E i . In the final continuum state we neglect the Coulomb potential of the atomic core. Indeed, it follows from the solution of the Problem 5.2 that this potential changes only pre-exponential factor. The final electron energy in the field of the linearly polarized electromagnetic wave is F2 sin2 ωt. E 2 (t) = 2ω 2 Here F is the amplitude of the field strength F cos ωt. The turning point τ in the complex plane of time is determined from the condition E 2 (τ ) = −E i . Thus, one obtains √ ω 2E i i . τ = arcshγ; γ = ω F The dimensionless quantity γ is called the Keldysh parameter.

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167

We derive the ionization rate according to Landau-Dykhne formula (Eq. (5.10) of the Problem 5.2): ⎡ w = exp ⎣−2Im

τ 0

where

⎤

  2E i ⎦ g(γ) (E 2 (t) − E 1 ) dt = exp − ω

  1  1 1 + γ2. g(γ) = 1 + 2 arcsinhγ − 2γ 2γ

(5.29)

(5.30)

When γ  1 it follows from Eq. (5.29) the tunneling limit for ionization rate by a weak constant electric field:   2 (2E i )3/2 w = exp − (5.31) 3F (see problems 5.1 and 5.2). Oppositely, when γ  1, it follows from Eq. (5.29) the multiphoton limit which corresponds to absorption of (E i /ω) photons of the external electromagnetic field:  w=

√ 2Ei /ω F e . √ 2ω 2E i

(5.32)

Here e is the base of natural logarithms. Of course, Eq. (5.32) is valid when there are no resonances with excited bound atomic states at the absorption of one, or several photons. Thus, the Keldysh parameter determines the ionization type. At the increase of the electromagnetic field first multiphoton ionization occurs, but then tunneling ionization begins. These results are applicable when the field strength F  Fa = (2E i )3/2 . The quantity Fa is the atomic field strength. If in the final atomic state an electron has the transverse momentum p⊥ , then its final energy is 2 p⊥ F2 2 . sin ωt + E 2 (t) = 2ω 2 2 Hence, tunneling ionization rate, Eq. (5.31), can be simply generalized by this case:     2 3/2 2 2E i + p⊥ w = exp − . (5.33) 3F Analogously, if an electron has some longitudinal momentum (along the field strength vector) p then its final energy is

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5 Transitions and Ionization in Quantum Systems

1 E 2 (t) = 2



F sin ωt p + ω

2 (5.34)

Then instead of Eq. (5.31) according to derivations of the Landau-Dykhne expression in the tunneling limit one obtains 

 3/2 2 2 (2E i )3/2 2 ω (2E i ) w = exp − . − p 3F 3F 3

(5.35)

Problem 5.6 Find the ionization rate from the ground state in three-dimensional short range potential by a weak constant external field.

Solution First we consider single-electron system with the time-independent Hamiltonian (further  = m = 1) 1 Hˆ = Hˆ f + U (r) = −  + V f (r) + U (r) 2 where U (r) is a strong short-range potential (its radius is rs ) while V f (r) is a weak interaction potential between an electron and some external field. This interaction can produce ionization of an electron from its initial bound state. The integral form of exact Schrödinger equation Hˆ ψr(+) (r) =Er ψr(+) (r) describing electron bound free transition to the some final states f is: ψr(+) r =

 f

ψ f (r) Er − E f + i0



ψ ∗f (r )U (r )ψr(+) (r )dr .

(5.36)

Here the unperturbed Schrödinger equation is of the form: Hˆ f ψ f (r) =E f ψ f (r). Due to short-range type of the potential U (r) and due to small shift and spreading (+) of the initial energy level we substitute the exact wave function ψr (r ) in the (0) integrand by the function ψi (r ) describing the initial electron bound state in the

5.2 Ionization of Quantum Systems

169 (0)

potential U (r) with the energy of E i . Thus, one obtains from (5.36) ψr(+) (r) =

 f

ai f ψ f (r) (0) Ei

(5.37)

− E f + i0

where the notation is introduced for the matrix element: (0) ai f = ψ ∗f (r )U (r )ψi (r )dr . When r  rs then U (r) = 0 and we find the Schrödinger equation (0) Hˆ f ψr(+) (r) =E i ψr(+) (r)

reflecting the energy conservation law. Therefore, we can simplify the Eq. (5.37) for (+) ψr (r): ψr(+) (r) = −2πi



   ai f ψ f (r)δ E i(0) − E f = c f ψ f (r).

f

(5.38)

f

  (0) c f = −2πiai f δ E i − E f

The coefficient

determines the transition amplitude to the some final state f. The probability distribution of various final continuum electron states (dν f is the number of final states) is: dwi f = Substituting

 2 c f  dν f = 4π 2

 2    (0) ai f δ E i − E f  dν f .

   dt  (0) (0) δ2 Ei − E f → δ Ei − E f 2π

one obtains the transition rate in the form

i =

dwi f = 2π dt



  2  (0) ai f  δ E − E f dν f = 2π i

where

is the density of final states.

  (0) dρ f = δ E i − E f dν f



 2 ai f  dρ f

(5.39)

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5 Transitions and Ionization in Quantum Systems

We can simplify the expression (5.39) for the level width i considering the electron ejection from the spherical zero range potential U (r) = −2πr δ(r). In this case, the initial wave function and the initial energy of the only bound state are of the well-known form  κ exp(−κr ) κ2 ψi(0) (r ) = ; E i(0) = − . 2π r 2 Then

√ ai f = − 2πκψ ∗f (0)

and i = 4πκ

  ψ f (0)2 dρ f .

(5.40)

This expression allows to find the ionization rate for static electric and magnetic fields with various configurations.

Problem 5.7 Find the ionization rate from the ground state in three-dimensional zero range potential by a weak constant electric field and constant magnetic field. Both fields are parallel to each other.

Solution Let us first to find the ionization rate by a weak constant electric and magnetic fields with exponential accuracy. The energy of a ground s−state in three-dimensional zero range potential is E 1 = −κ2 /2 (in atomic units e = m =  = 1). This is the only bound state in zero range potential. The energy of a final state is E 2n (t) = Ft 2 /2 + (n + 1/2) H/c where F is the electric field strength and H is magnetic field strength. This energy corresponds to acceleration of an electron in a constant electric field and to various quantum electron energies on Landau levels produced for a free electron by magnetic field. n = 0, 1, 2... is the integer. The classical turning point in the upper part of the complex plane of time is found from an equation (see Problem 5.1) E 1 = E 2n (τ ), i.e.,

i κ2 + (2n + 1) H/c. τ= F

5.2 Ionization of Quantum Systems

171

The ionization rate of a ground state is derived with exponential accuracy by the Landau-Dykhne formula (5.10) of the Problem 5.2: ⎡ n ∼ exp ⎣−2Im

τ

⎤

    H 3/2 2 2 ⎦ κ + (2n + 1) . (E 2 (t) − E 1 ) dt = exp − 3F c

0

Due to smallness of the magnetic field strength we can simplify this expression by expanding exponent in Taylor series:   ω H (2n + 1)κ 2κ3 − n ∼ exp − 3F F

(5.41)

Now our goal is to determine the pre-exponential factor in this dependence. The unperturbed normalized wave function of the ground state in zero range potential at r > 0 is of the form  1 κ exp (−κr ) . (5.42) (r ) = r 2π The energy of the only binding level is E 0 = −κ2 /2. The Hamiltonian of the quantum system in the external static fields can be presented as Hˆ = Hˆ f + U (r) where

1 1 Hˆ f = −  − F z + ω 2H ρ2 . 2 8

Here ωH =

H c

is the Larmor frequency in the magnetic field, and ρ is the transverse coordinate. An electric field with the strength F is directed along the axis Z . The solution of the equation Hˆ f ψ f (r) =E f ψ f (r) is of the form ψ f (r) =ψn (ρ)ψε (z) where ψn (ρ) =

1 √ exp(−ρ2 /4a 2H ) · F(−n, 1, ρ2 /2a 2H ); a H = a H 2π



c ; H

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5 Transitions and Ionization in Quantum Systems

F(−n, 1, ρ2 /2a 2H ) is the confluent hypergeometric function, n = 0, 1, 2... is the integer. The Landau oscillator energy in the transverse direction is E n = ω H (n + 1/2). Further,  1/6   ε  4 ψε (z) = Ai − (2F)1/3 z + F F is the wave function for the motion of an electron along the axis Z in the constant electric field with the strength F. Ai(x) is so called Airy function. The quantity ε is the energy of an electron along the axis Z . The total final energy is E f = ω H (n + 1/2) + ε = −κ2 /2. Here we used the energy conservation law at the bound-free transition. The value of the final wave function in the origin r = 0 is  ψ f (0) =

H3 2Fc3 π 3

1/6

1/6    1/3   c3 2 ε ω H (2n + 1) + κ2 . Ai − 2/3 = Ai F 2F H 3 π 3 (2F)2/3

Using the asymptotic expression for the Airy function due to small value of the field strength F compared to the atomic field strength, one obtains √ ψ f (0) =

H/c

1/4  2π ω H (2n + 1) + κ2

  3/2  ω H (2n + 1) + κ2 exp − . 3F

Substituting this expression into Eq. (5.40), one obtains the ionization rate with the transition to some state with the quantum number n: in =

κH

1/2  c ω H (2n + 1) + κ2



3/2   2 ω H (2n + 1) + κ2 exp − . 3F

Expanding the exponent due to smallness of the magnetic field strength, one finds in =

  2κ3 ω H (2n + 1)κ H exp − − . c 3F F

The exponent in this expression was already obtained previously. The total width of the initial state is     ∞  2κ3 ω H (2n + 1)κ 2κ3 H H exp − − exp − i = in = = . c 3F F 2c sinh (H κ/Fc) 3F n=0

5.2 Ionization of Quantum Systems

173

In the case of a weak magnetic field when H κ/Fc  1, it follows from this relation that   2κ3 F exp − . i = 2κ 3F This result coincides with result (5.28) of the Problem 5.4 as it should be.

Index

A Accidental degeneration, 79 Adiabatic invariant, 12 Airy functions, 5, 7, 11, 137 Asymptotic coefficient in the origin, 54, 68

B Balmer formula, 25 Bethe’s rule, 89, 97 Bohr-Sommerfeld quantization rule, 4, 6, 15, 17, 108, 118, 124, 125 Bremsstrahlung, 102, 103

C Correspondence principle, 77

E Expansion of the effective radius, 73

Harmonic oscillator, 9, 11–13, 16, 22, 23, 32, 33, 48, 50, 52–54, 157 Hermite polynomials, 114 Hulthén potential, 41, 43, 46

K Keldysh parameter, 166, 167 Kramers relations, 3–4

L Lagerre polynomials, 58 Landau-Dykhne formula, 160, 161, 164, 167, 171 Landau-Zener formula, 157 Langer correction, 18, 20–22, 25, 28, 58, 63, 70, 133

M Method of secondary quantization, 75

G Gamow formula, 109, 111 Generalized functions, 81

P Parabolic approximation, 116, 119, 123, 125, 130 Photorecombination, 101–102

H Hadron atoms, 25, 72–74 Hankel functions, 159

Q Quantum defect, 57–59, 64, 66, 67, 69, 71, 75, 84, 92

B. M. Karnakov and V. P. Krainov, WKB Approximation in Atomic Physics, DOI: 10.1007/978-3-642-31558-9, Ó Springer-Verlag Berlin Heidelberg 2013

175

176 R Riemann surface, 14, 125 Rydberg correction, 57, 73

S Selection rules, 78, 83 Spin-orbit splitting, 66 Stark shift, 166

Index Y Yukawa potential, 43, 45

Z Zwaan method, 5