Задачи по квантовой механике. Часть 2: Пособие по курсу ''Квантовая механика''

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Author: Корнев А.С.

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+ A

9 0 7 0*? n EF !8- 2 ",*(EF "OcF) !#6 SE 8(%!8-Q21 N% "% %!# ^ * ) !#` ; EF!#`-Q &("X Ψ(x)3 ^ 6"O 3`;8 - 2 c! 2 86" % O 3 2 7RI !#%!# ^ 457698;:=A@BDC −

}2 d 2 Ψ(x) + V (x)Ψ(x) = EΨ(x). 2m dx2

E B=F E=GIHJGLK 4MB=NO> E BQP HSR < E DB : DE T >.8U5V6 G 4WBUP E > GYX[ZDGIK QB \ G 4 TX B R <MF^] E 8Y6_>[4 T \`B 7

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G F2> EDc B9<[69] E <[Ped E > R&H BUP*BQ\ >[>fFhg GYE >f: EDTijR [< k 9R T 4A4jF G :lgU> B95onp;nqp?<Jr;n"s^5[5V6_> K[G 4I<WF^]t4 G 6 EDG 4M8;uv@8 E g c BDu4wdMF GMi F G :lgU>[n x=a mR x(y{z|y{}~yI ( ; EF%!8.L-Q`&("X` " N LN*!#;#

T !#->&( S 66-.%!#H!#6 S6E 8 !8-Q x7 = a

0

V(x)

(à)

a

x

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a

x

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b "

}2 [Ψ0 (a + 0) − Ψ0 (a − 0)] = 0. − 2m

N6) ^ R B E <[69B=:lB=Btg GME f> : DE GIHJG H BDB 4 G 6 EDG 4I<[P @8 E g c BQP 4jF G :lg > R <[k R9T 4?< X 8 4 \w>[5fF2>.5 G 5[4 G > i > R 4 GMi RG BQk;4 GWKoEDGYi A Ψ(a − 0) = Ψ(a + 0);

R o< k 9R T I4 < G F|> EDc B < 69] EDGYi F R > RT K >fF G 5f2

Ψ0 (a − 0) = Ψ0 (a + 0).

d E >R C 4 EDGYi

9 0 7 ?

M Ψ(a ± 0) )# *!#. !8-# ,; . ) &(%"X !#;%9 !8-% ? ( " x = a ;!## !# Q\N-) %!#\ 7 @ ,6T 2 9 0 7 ? )E % c! 2 6"O 3T;6-% ! 6T% 2 -Q &(%"X` "%3) EF% ; X/- 7 @ e 6T 2 9 0 7 ? / L! &(V*-6.3*A ¯ ¯ Ψ0 (x) ¯¯ Ψ0 (x) ¯¯ = , Ψ(x) ¯a−0 Ψ(x) ¯a+0

O-

¯ ¯ ¯ ¯ d d = . Ψ(x)¯¯ Ψ(x)¯¯ dx dx a−0 a+0

9 0 7 Z?

/7 7 4 F G :lgU> g GME > : DE GIHJGR [< k 9R T 4?< G 2 F > DE c 9B <[6_< 6 GWH < R BD@-\.BD:=>[5 gU< P RG Blk?4 GIK[E < P 4 G 6 EDG 4 GYi @(8 E g c BDB X 8 K >fF E > R > RT 4 E < 7* !8- 2 9 0 7 ? H9 0 7 Z? ] "%6- E,7 ¤

R!

- W\"% x = a ! 2 X >[5 g GME > : EDG 4 T 5 G Ug <[P G F2> EDc B< 69] E < P 5 F|> E C Ug < 9<!7 0 7 0 N ?8^ H4!# !# !T * 0 # + 9 0 7 0/^ ? 5 R E g;B4 G 6 E=G 4I<[P @(8 E g c BQPF G N K > 5fF24?> E=EDGjGMX R < 2fF25 P 4 E 8M69]^ *7 7 ] LN-!8.\!#X E > RGYE BDgUfF 7I !8 !# ! !8-* ; EF %!#H; x = a - 2 &(%"X 2 - L3N.%! 2 *-.A 9 0 7 1? Ψ(a − 0) = Ψ(a) = 0; Ψ(x > a) ≡ 0. I ] % !8-\L;) !c &(%"X`W"% x = a "O"Q8 -NL+ 6 _ QH,"O- * EF%6 ! 2 7 I )*"\- c ; !8-HN )V "O)*6-.%!8%F!#; X&(*!#"R!#Q!#% % +$&(+ T 2 ^ "%EF !#66-% c! 2 ' &(%"X6-.+ /-)*A 0/? 4?5 >d E > R&H >fF^B=:=> 5fg?B> 8 RG 4 E B X 8 K 8 F E >[4 TRG N K > E=EDT \.B ? 4 G 6 E=G 4 T >L@(8 E g c BDB 5fF2< c B GME < REDT Z 5 G 5fF G P E B i \ G N EDG 4 TX R <MF^] 4?> C 2>[5fF24I> EDEDT \`B "! #%$&(')+*-,&'.(-0/2131546 7--*8(1:9

: Z? >[5769B G 5 DE G 4 EDG > 5 G f5 F G P E 9B > E 8M\ > RG 4I<MF ^ ] B E K f> gU5 G \ 9> R 4 G > 4 G k X 8?N C

p B F n K n F G 4 GYX 6_ DE EDG > HSR < E B c=T 4 G 6 EDG I4 <[P @8 E g c BQP n C HJG 4 G k X 8?N GYX R -< 2fF25 P 4 E M8 69] R DB :=>V\ 4I5[> E 8M69B X 8 K

69Blk?< c BDB:= > K > EDEDGIH G 5 G 5fF G P E BUP RG 4 EDG n R <[k 8 F E >[4 TRG N K > EDE=T \.B 7

V"#v&2a*2"!D # #)G M X6-. 2 ] + 2 δ N)1c %!#N%!8.3"% "%`"/%# _'

x=a

A

V (x) = U (x) + Ωδ(x − a),

+ U (x) "%!8 ; EF 2 &("X 2 ! )/ LE H) EF%a" O Q -E 3"% x = a ^ Ω = const 7 4 NNe5T. !#,6T 2 9 0 7 ? - 2 ; X/-T + ;7 9 F24?>fF A +

m

2m ΩΨ(a), }2

Ψ0 (a + 0) − Ψ0 (a − 0) =

Ψ(a + 0) = Ψ(a − 0);

! ! T!#XE,7 ?

# 'V #h2

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V(x)

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(*2*-, - 1 5*8(' ) (, 1 0/

[ % 0 # + 49 0 7 0/? ! ;X/-% V (x) =

(

0,

0 < x < a;

+∞,

x 6 0, x > a

9 0 7@? ?

a*- E a+E H!8- 2 E < X >[5fg GYE >f: EDG 5fF^B 7 M( X6- 9 0 7@? ?:2 - 2 ! 2 "%!8 ; EFE ! N*!8" PE !#"O `!8 "OB7M(] % !# !8H!L9 0 7 1? 2 E - 2 &(%"X 2 N! 2 4*

*!# EFQ *-.4 ; `*-EFL) * 2 a+ 6O X 8 2 E ^ ;-%!#,. c !#+ -!# ! 2 !H+ E %! - 2 7 b ; . % 9 0 =7 ? ? ;, 5 6 ! 2 )*;!8EF%! 2 4;% "%> 5 a 9 0 7: ? }2 00 − Ψ (x) = EΨ(x) a*- E a!8- 2 E < HSR 2m < E B c < Z RG \w> N 8?F g < A 9 0 7 [? Ψ(0) = Ψ(a) = 0. I !#!# !# !#Q!#% * % + `&(+ a T 2 ,N 5T 7 6T >- Q+ * + &,&( X6-.+ %6 2 N ;# !#6.7 5T!# Qa&(A 9 0 @7 m ? Ψ(x) = A sin kx + B cos kx, + √ 9 0 7 0*5? k = 2mE/}, ; -

5T(;# #- \c "%!#6E,7M(%!8 2 2 k 6" T,;* A B - LW;# #- \c ^ ;%!#"%-." EF6 6 ! 2 )\ )*!#EF ) * 2 ] + * 7 h(!8- ) *!# E 8 # X ; jE 63 ! - + E 8 !8-Q 9 0 7 [? 7I ]% "O"%+ ; 2 ^ ;%!#"%-." 7 6\ % 2 9 0 7 : ? ;# - 2 ! 2 ! %!#,. c E > E 8Y6_>[4 GWHJG G 5fF G P EDE=GIHJG \ E=G N BDF2>V6QP 9<Q "%!#66E ? 7 a - 2 EF;- 2 ; + P+ + ,!8- 2 9 0 7 [? ,"% x = 0 N 8* L;-* L. B = 0 ^ 4+ A !8-8 W!#6. Qa"%O !#6QB7 2 k ^ ! -# %6#-. ^ T] + E ^ ;# - 2 c ! 2 E + E '!8-*_9 0 7 [? "% x = a ^ /7 7)P 2

a

%6 - 2 ! 2 ; !8-# 1c 5TQ8 K 9B 5fg R f> F DE T Z ) O 2 8 kA 9 0 7 00/? k = πn/a, n = 0, ±1, . . . sin ka = 0.

n

m N6) ^ !# !#_!$9 0 7@m ? k 9 0 7 00/? E ;-* !8-# !c T 5 EF6 T 2 - 2 ] + QaH-E 8 &("XQBA

b "

π 2 }2 n2 , 2ma2

En =

n = 0, ±1, . . . ;

Ψn (x) = A sin

πnx . a

9 0 7 0/? 9 0 7 0*Z?

I O &(%"X`9 0 7 0Z? ; 3 SP;-.!#.(EFN76T 2 4?> |> C 5fF24I> EDEDGYi &( 9 0 7@m ? 7 !# ; . ;%!# ) * 2 n 7 M(

H%!# + )* ^ ; n = 0 &(%"X 2 9 0 7 0Z? N 53! 2 /

*!# EFQ *-. ^ /7 7 ;-% R;# S!#66- 2 c 5T &()*!#"%+ *! 6-. 76\ % j 2 0 # + 7M(] k E <M:=> E B> n = 0 E > GMX ZDGIK BQ\ G B5 gM69u :lBDF^] Blk R E BUP 7 a 6- ^ ;3)* R *- + n → −n ) * 2 ] +H9 0 7 0/? %!86 c6 ! 2 ; ^ >-EF&("X 9 0 7 0*Z? %!8-# !# %!8 !#! 2 c ) " ^ /7 7O* c ! 2 "%!#6 −1 7 l ) &() *!#"1 c ] "%6- %!#. Ψ (x) Ψ (x) 7 M(] % G F R B c <WF2>V69] EDT >|k E <W:=> E BQP n F2<Mg?NO> E > GMX[ZDGWK BQ\ G nB5fgM69u :lB=−n F^] Blk R E BQP^ 6"L "O6 L1 5T *! 2 E / 3 ] + *!#"Q8W% Q ;T) 6" n 3!% - !#!# /7 ( 2 "%!#66 - 2 !8+ !8- 2 EF! - A n=1 ; 0 7 0 h,!# 0 7M(# -+#* !%!# 2 -.TNeN 5T.T ) *-.66T !8-Q n > 1 7M(# * "%#-.EFQ> A En =

π 2 }2 n2 , 2ma2

Ψn (x) =

r

n = 1, 2, . . . ; a πnx sin . 2 a

9 0 7 0 1%? 9 0 70 9 ?

C(6"%;%6-) * 76\ 7 4 % 2 N*!#"%(+ -N"O 2 ; X/-. 2T2 ,!# PN*!8" T! - Q`] + 7 l + !86XE 8`!#%!# 2 Q !# 8 + E _* )$;# #- 2 c! 2 "%E !8- n 7 U !8 !# 2 4 3 '!#%!8# ' 2 -Q ) !86 !W%- * "%+3!8-7 4 !#%P!8%!# 2 \c !# !# n = 1 7 4 E > E 8M6_> 4W8;u*d E > R C E 6T ; X/-.Q 2 VE,7 a O H BDu E1 = π2 }2 /(2ma2 ) ^ /7 7- LWj EFQ &," !#+ - !# ! 2 !,;X;% ;8 - %!8 A) N%!P) Q

05 "% E + W"% E W6)* 2 E ^ 6"W3;*-.%! ; !# GMX Pk;< E .P *- Q\6) N%! ) Q3%! -# !#R"% %!#3;6 Q>!8a !8% 9 0 70 ? ?

}2 h(∆x) ih(∆p) i > 4 2

2

9Y! 76" T\!8-# !c 5\Qa; ? 7 a - 2 ; "%!#X-- 2 XQ *E 76T 2 9 0 7 01? ^ 9 0 7 0 9 ? NA8 ; %.`6" ^ NE %!#% !#%!# 2 \c !#!#/- 7 a - 2 ] + ! #- * )* n → n + 1 A n=0 9 0 70 : ?

π 2 }2 (n + 1)2 , n = 0, 1, . . . ; En = 2 2ma r a πx Ψn (x) = sin (n + 1). 2 a

M(# -+#*$ ;.>! %!8 2 -. ; .W!#XO-- 2 X1c * ^ 36"O T\!#Q!#%\+ 6-.!8>a;-E &(%"XQ 9 0 7 0 9 ? 7 ¤ f+(*,+ #)G V G 8U5V6 G 4WBUP Z R > KoT(K 8 2> i k?< K <W:lB 4 5fF2< c B GYE < REDT Z 5 G 5fF G P9C

E UB P Z E < i F^B*5 R > KoE B9>`k E <W:=> E BQP g GMGR&K B E <MF T B BQ\ D8M69] 5 < B B Z 5 R > K[E >fgU4I< K C R <MF^BD: EDT > G F^gM6 GYE > E BQP&n a RG 4I> R B=F^] 5 GYG F EDG 2> E B> E > G R > K >V6_> E=EDG 5fF2> ieK 6lP g GYG R&K B E <WF T B BQ\ D8M69] 5 <|4 5 F|< c B GYE < REDT Z 5 G 5 F G P E BQP Z n x(y{z|y{}~yI / * * ;/-+,! ; Z 7 Z ,!# 0^ -."%F "/

%

76

h

*!#-E 8L&(%"XQW!#;-.) * 9 0 7 0 9 ? ^ , + % ;* a;V# L"% 5 a 7 G( + %.a)# !#.H Na;4N) ) Q ; Q ξ = x/a 7%I !# EF"O-/ " "%%O ! -.T! !# 2 #-. ^ 3)# *!#. ;# * -\6T. " EFQ> )/-.66A a hxi = ; 2 hpi = 0;

· ¸ 2 1 a 1 h(∆x)2 i = − ; 2 6 π 2 n2 ¶2 µ π}n 2 , n = 1, 2, . . . h(∆p) i = a

8

,6

, !#63;-%E T) *-./6P; !# T ;#

-%!# Q 9 0 7 0 ? ? !#6X6E 8>!#%!# 2 2 8 9 0 7 0 9 ? A }2 h(∆x) ih(∆p) i = 4 2

2

µ

¶ π 2 n2 −2 . 3

+ "% O . ^ * L#-.$ !#"N"O 8 ' ;Q !# N N-.,6T # XE ;a%!# 8L ;%!#E 8>) * 2 8 n 7 ¤

<

00

f+(*,+`#)G G ! EDc BDC < 69] EDGYi Pl\w>, g G F GR < P GWHSR < E BD:=> E <j5V6_>[4I< X >[5fg GME >f: E=G 4 T 5 G g GYi G F2> E=c B< 69];C < 5 R E ];g GMi g GYE >f: EDGYi 4 T 5 G F T V m R B5on p?n ;<{r?n EDGYi 5fF2> E g GMi ) < i F^B d E > R&H BDBe5[4 Pk?< E=EDT Z 5fF2< c B GME < R9EDT Z 5 G 5fF G P E B i :=
0

x(y{z|y{}~yITI "%

; X6-. 2 ] + 2 ; ; % "% EFQ\) EF ^ ; ] %$% 0 # + 9 0 7 0/? L 76 .' 8 !#%!8# Q8 N-!# 2 8SA I (0 6 7 x 6 a) II (x > a) I N-!# I )*/ * ;-%!#,. c ] "%6 - 'Q ^ ) N6 $; 0 7 O^ ;] %a/ W!8) EF;! . 7 6T ! N*E '!8- *-A 9 0 7 0*[? ΨI (x) = A sin kx, + A ;) -. 2 *- % 2 "%!#6O 6 ^ 4- !8- k !# 2 )*6 ! ] + Q !#, 6T *_9 0 7 05? 7 o\ T )/ L -."% ;a%! - x = a

V(x)

(à)

E I

II V0 x

a

0 w(x)

(á)

x

a

0

&UoDoI

9 0 70 m ? N;8 EF%.F;, 6T P . 2 0 # + FN-!# A II 9 0 7 5? }2 00 ΨII (x) + V0 ΨII (x) = EΨII (x). − 2m ( * N8 .! 2 ^ % \c 9 0 7 5? - 2 !8-# ! c 5 2 &(%"X 2 ^ N 53 c 5 2 ! 2 /-.; x → +∞ A 9 0 7 O0/? ΨII (x) = B e−κx , + E < V0 ,

U7;6T ) 2 -9 0 .7 0*[ 2?

p q 2m(V0 − E) κ= = K02 − k 2 , } p K0 = 2mV0 /},

9 0 7 ? 9 0 7 Z?

*- 2 "%!8667 9 0 7 O0*? NA8 ! 6T. "%(" + T) E %H; X6- x = a !8 !#$! 9 0 7 Z?8^ ; - +# 2 Ψ(a − 0) = Ψ (a) ^

B

I

0* _p_ 2a

k0

_p a

_3p _ 2a

k1

_2p _ a

_5p _ K0 2a k

0

&UY=Y ;

'*7 7 I + ; -* !8-8 1c T 5 !8X ^ - 2 E A

Ψ(a + 0) = ΨII (a) % - 2 k ^ T)

tg ka = − p

k K02 − k 2

.

9 0 7 61?

M W;)-. ) * ;6 K \*- + \6 ( -*!#"%+ 6T 2 7 a - 2 ] + *!#"%+ 0 !#; " ; . * &(O *-SA 90 7 9 ? }2 kn2 , n = 0, 1, . . . , En = + k ;-* L-.EFW9<;2m * ? "%H%6 2 9 0 7 /1? "7 l + n !#"W% ! c! 2 )8 *!#.4 !# !8'!L!#XO-- 2 XQ Q 7 a - 2 6-)*` ) *-.666 %!8;-.) ! 2 + &(!#" * % 7 6\ 2 2 9 0 7 61? 9 ! 7!67 0 7 1? ,7 n &(" + (- Q !# 2 - 2 ! 2 6 + !# QB7K N%!#X!!#E "` \;*!8 2 !,+ &("%%$;Q4!# 2 - 2 c! 2 "% 2 H9 0 7 /1? 7 n &("\;6Q\!#3 "O6-.1 c !#;O " ; k = K ^ ;] :lB95V6 G 5[4[Pk?< EDEDT Z 5 G 5fF G P E B i 4 Pl\w> g GYE >f: EDGYi H 698 X B EDT GWHSR < E BD:=> ED0 G 7 ,P!7 0 7 1 ;# !#6- W!8-%6Q ! 2 !# 2 )E !#%!8 2 2 %!#E L; E ) N% 3 E 7 H/ L,;"6)*. ^ - 2 ; 2 - 2 2 ;!8-8 6-.%!8 % Q E ^ 7 7 7 ^ E ; 0 n T

a

. 8 ! Uc 2 K 0

K0 > (2n + 1)

M(

π . 2a

90 7 ? ?

5[4[Pk?< EDEDT ` > 5 G f5 F G P E BQP :=.4 GMGYX 2> G F25f8;F25fF24W8?C

u F 7M(>%! -W T

9 /7 7; V → ∞ ? ;6% 2 !#. 2 9 0 7 /1? !#*! 2 " *-\c ^ ] + *!#"Q0 !#;" !#6! 2 66" T ^ 2K0 a < π

K0 a À 1

0Z "O" !8- N*!#"% 4+ -%N"%Q ; X6-.Q 2 E ^ /7 7 ; 8 9 0 7 0 : ? 7 b " N6) ^ !# !8 ! 9 0 7 Z? ^ 9 0 7 9 ? ] + *!#"Q !#; " !#XE ;-%!#.,c ;# #- 2 ! 2 ! !#Q ^ H6"O 3 ;) # *'; a2 mV 7 ,9 0 7 61? ^ 0)*/ ^ F!8-\6T"%%4 N - 2 ! - + 76\ 2 ^ 6">"O"> N 3;!#"O\*!8"-."Q8 + 3"% Q 7 + ^ "% ^ * 6 ]-*6Q+ W/ L,;N) 6.("] "6- N- N% ^ EF% c 5T* + ) %!8.TN E 8L+ *!#"U8H&("XQ A ka = π(n + 1) − arcsin √

! 2 T 5 \W;

}k . 2mV0

90 7 : ?

%6a9 0 7 : ? E > XMG 6_>[> GIK C EDGWHJG G 6 G N BDF2>V69] E=GIHJG g G R9E P K 6lP gU<MN KoGIH G n >- + "%76 ! 2 !8- 7 4 N> 6- 2 75T * !#N!#. O T 2 "O %7SI Q4)*/ * N-!8. x > a 2 - 2 ! 2 gM6_ 5 g;B E > KoG 5fF^8 DC EDGYi ^ 6"a"O"a Q E < V (x) 7 b *$ , !#!#4!9 0 7 O0*? ] Q N- !8 4 G 6 EDG 4I<[P @8 E g c BQP E > GMXR < 2fF25 PO4 E 8M69] F G N K >[5fF24I> EDE=G l ) RGYE BDg EDG 4?> E B>j:=[5 g;B E > KoG 5fF^8 E 8?u GMX 6_
+4!#%!# 2 2 ! 2 >!7 0 7 Z N7 M -%!#.H 2 %!# N% T 2 !#XE ] "O!#; X6-.')*1 8% '; %+ -N- "-! !8*!#"_# !#%;! c N- !8. 9 ∼ e−2κx ? -7 l &,&( " 2 + -N ;" 2 d ∼ (2κ)−1 ;8 - 2 ! 2 E !#Q !#; ." V ] + Q !#XE E 7M( 2 ] "O!8 ; 66-. ;*

S0+ &,6"64 )*/ `; ; "!#"%;%!8,)* -. E 8P;N87 ) <[69BD:QB> !#XE'* L )*6 &("O!#%6. ^ ;# 2 , !#!# 2 4! ;# - E ';*-.%!#% K7 o " X 2 9 0 7 O0/? H!#!# 6"%%'!#%!8 2 UcT7 C(!#E T>; O

3 ;" 2 "!#XE "-! !8*!#" 8 %!8%;! c N -!#. 2 - 2 ! 2 -.EFQ>]*&,&( "H9 ! 7O 3 ? 7I !8- N*!8" TE !#"%Q !8%; ." 9 V → ∞ ? !8X EF66-"%6 ! 2 ) "-! !#!#" # !#%;Q>N-!# 09 D → 0 ? 4E ;Q8 '"a)*/ *(; 0 7 7 ¤

n = 0, 1, . . .

V"#v&2a*2"!D

"%`"/%# _'

BGh,!#X

#

! !8E m O 3! 2 >* Q ; 2 + -.Q`N*!#"% + -%N"%Q ; X6-.Q 2 6TE a 7 ,QN- 2 EFF;-

T 2 !#XE ^ ;# Qa ; () N

,!8%!# 2 7 9 F24?>fF A% %T;-/ T 2 3! !# 2 a L!# "B7 ? 4

01

V Gh,!#X

! !8E m O 3! 2 >* Q ; 2 + -.Q`N*!#"% + -%N"%QW; X6-.Q 2 6TE a 7 ,Qa!## () " *!#"%Q>] + a!8XE ;) -.'!#6X%'!#%!# 2 7 9 F24?>fF A E = π2 }2 n2 ^ + n = 1, 2, . . .? n 2ma2 ∗ G h,!8X !!#E m O T ! 2 % Q'; 2 + -.Q$!8 6 Qa; X6-.Q 2 ,"% Q>+ -NE,A V (x) =

8

,Q ] + !#6XE

   V1 ,

0,   V , 2

x < 0; 0 6 x 6 a; x > 0.

!#%!# 2 Q !8XE,7 ( M "O"%%_!8- !#%!8 2 Q ,C,"% "O!# 6- . !8-% Q

2 PPN !# 2 )*6E 8 ;a)*/ E 8 V ^ V a 9 F24?>fF A] + 14EF2 !8- 2 c ! 2 ; &(*- 9 0 7 9 ? "% . % 2

kn

}kn }kn − arcsin √ , kn a = π(n + 1) − arcsin √ 2mV1 2mV2

;-/ -.EFQ n = 0, 1, . . . ,

;8 - 2 *EFQa! - ? 7 3 ∗ G h,!#X ! !#E m ;* S ) !(!#"% * g N%!#-\c ;%+ !c + ) /-.1c ; 8O%!8.7 ,Q ] + !#6XE 8_!#%!# 2 Q 7 1 F24I>fF A ! 7 9 a 7 0/? 7 3 E = (mg 2 }2 /2)1/3 α n = 0, 1, . . . (α ) = 0 n

n+1

k

2") %% 2/2}&+O2 (v

I` >)# - E !!8-# *>"EF !#; "E B E @(B E BDF EDGWHJGK 4MBDC NO> E BQP^ 3 ; * - (!#X Qa;X/-.+ 3N%.O7 M(%!8.\;- V (x) 6"% ^ 3; X6-. 2 "% 2 ^ ) N6 T 2 !7 0 =7 9 ^ + 6Q N-!# +N ^ ' !#;" 9 x → ? E 8* ;- ^ /7 T !#X\!#6! 2 !#N E 7 b "%Q ±∞ + N ; 2 L)EF%. G F2> EDc B9<[69] E=T \ X < R ]U> RG \ 7 M(%!#6 )** Sa!!8-8 2 ;X*! ! ; * - 2 N%. L!# X Q ! !#Q m ^ O L! 5\ Q! 2 !P;-QH] + Q E 7 M(%!8-# 6#-.P! ! N * -% 2 !#XE -E `;6"% L7 6T 2 !# !81 c 5T Q>*!#6X6Q>)** S 9Y! 7; 1 7 [ !8 0/? 7 >R- ;!#Q ^ "% ^ ;# !#6- 2 ! 2 5fF2< c B GYE < R9E <[P )** S 3! !# 2 >N%6. 8U5 F|< EDG 4WB4 %2> HJG 5P+ G F G gU<2:=
0

9

V(x) jîòð

jïðîø

Vmax jïàä E Vmin

x

0

&UY=Y W

N 8 ") %6.a!8-# 1c 5T N-cR SVE -BA g G d @(C @B c B> E F G \ G F R E BQP^ ;# S!86- 2 c 5T + '!#NQ ,6\ `;-!# ;"O3/ 3 E 8 !#Xa"H;O-%!#H;"O3;/ c 5TQ8W!8X +

¯ j ¯¯ R= , j ¯x→−∞

j = |j |;

j = |j |

,6 ;O-%!# ;"O ;,6T# 6TU8H!#X " ;O-%!# ;"O ;/ c 5TQ8H/ !#X j | + j 90 7 m ? D= , = |j |. j | M -%!8 ;"OH;# ;-+c ! 2 ) 2 E `!#;*!#" * 6- E 8 !!# 2 2 8WN-!#>E 8 * T; X6- 3;O-7 a - 2 EF! - 2 ;O-%!# Q\;"OF@ NA8 ) . -1$c &("O X\c !!# %Q,N. F!#XE,7 + -!#P%! -\c )** S ^ N- 6 !# x → +∞ 7c ! 2 F G 69];g G R9G 2 % > K ^% B9> ) N. a!#XE 9Y!P;-

-.E ;*-.%!#% ? 7 M(] % !8;!#"Q -Q &("O Xa; x → +∞ ;# !#6- 2 L!#NQ ;-%!#"!c -SA RG[ZDG N K > E BQP^ ;# !#6- 2 T + !#NQ T

g G d @"@(B c B > E F G \

x→+∞

x→−∞

0

+

c5

9 0 7 [?

Ψ(x)|x→+∞ = eik x ,

9 0 7 Z5?

9 0 7 Z0/? o "X 2 9 0 7 Z0/? %6 !#;* ;-%!8T;"O ;,6\#) 6TQ8 !#X4!("O-! !#!#"QH!#"%%!#.,c 9 ! 7; 1 7 Z !# 0/? A k0 =

j

p

2m(E − V ) . }

|x→+∞ =

}k 0 . m

9 0 7 Z?

0

?

?k < K <W:=> G R P E BDB := EDc B < 69] EDG \ X < R ]U> R >w4 Ug 4M:=>[5fF24I> HSR <MC E B=: EDGIHJG 8 5V6 G 4MBQP g 8 R E BDu R > K B EH > R < 9 0 7 0/? E > GMX[ZDGWK BQ\ G B5 G 69]Wk G 4?<MF^] [5 g;B i 4MB K 4 G 6 EDG 4 GMi ( @ 8 E g c BDB 9 0 7 Z5? 7

I N-!# x → −∞ c! 2 "O">;/ c 5T ^ 6">>6 T EF(!# XE,7 @ + -!#,; 1 7 1 !# 0^ * %!#N T N "6TEF/ 3 \;T) "; "X ;*-.%! 7M(] % !#;O *!#"Q 7 6T 2 % e 2 0 # + ]Q`N- !#`;# !#6- 2 ! 2 !8-# ! c 5\'N)% A 9 0 7 ZZ? Ψ(x)|x→−∞ = A eikx + B e−ikx , + - !8- k ;# -Ta9 0 7 0*5? 7OM( !8-+ a9 0 7 ZZ? !# !# ;"(;* S c 5TU8(!8X ^ ;"(6 T E 8P!#X 7 M -%!8 ;"% EF! - 2 c! 2 ; 2 . 34 !#!# !4&(*-Q ; 1 7 [ !# 0 A 9 0 7 Z1? S 2 "%!#6E A B ^ - +"3EF!8-. "%]*&,&(X E 6 T 2 ;;8*

2 ; &(*- 9 0 7 [? ^ 9 0 7 m ? ^ 9 0 7 Z? 9 0 7 Z1? A j |x→−∞ =

f+(*,+T#)G 3G

}k |A|2 ; m

|B|2 ; R= |A|2

j |x→−∞ =

}k |B|2 . m

9 0 7Z 9 ?

k0 D= |A|−2 . k

%

G gU<[k?<WF^] :lF G 4 5 BDF^8U< c BDB 5.8U5fF2< EDG 4MB4 ^BQ\w5 P+ G F G g G \ R + D = 1.

8

9 0 7Z ? ?

8

x(y{z|y{}~yII !#6XE H!8%!# 2 2 >;O-%!#. 2 !#H\)*6!8

3* 7M(] % ^ L!# !#H! ;EF%!# ^ ;-6 %!#.;"OT 2 !#>\ 2 ! 2 3 %!8* ; 2 3H%!# Ox A 9 0 7Z : ? jx (x, t) = j = const . M(8 -+ 3%!#%6 9 0 7 ZZ? ! %!# 2 -. N8 .! 2 % ^ TN-!# x → −∞ ;/ c 5\QWW6 T EFQW;" E >B E F2> R @ > R B R 8;u F3^ /7 7 9 0 7 Z[? j | = (j − j )| . +j )| = (j

1- /"! 9 7-: ,* 1./21-2-' :*-) +- 1 ./2*-*2) + *2521 ,-! 461 2) +3 4- (1 21 /-23)2- , *-- -$ ( 9!"# 9!$ - --*2,?+2*-1 .7-*--.7 *-18(*2,*-2/&% 2, 7-R -*2,D 89 x x→−∞

,x

,x

x→−∞

x→−∞

0 I N-!8

:

;" 2 - 2 ! 2 X-"%%';e6T# 6T A

9 0 7Z m ? G() N,5TQ8L&(*- 9 0 7 [? ^ 9 0 7 m ? ,%!# 9 0 7 Z : ? k 9 0 7 Z m ? ;-* &(% /-. !#,6T `9 0 7 Z ? ? ^ EF/ c 5T )"% !#A86 2 5T !#%3;H! !# 2 >!#Xa3N. 7 ¤ U ! !# 2 ("!#X ; X6-.% N%. !#! 5T!# T*-O ! 2 L/-++ 6"%!#"%;*!#"%+3;X*! ! 7 U ! ! ^ ; E 8 ^ !8-Q V < E < V 7 I 6"% 4?5[> X >Jk2B5fgM69u :=> E BUP !#XE_;+ 3/ 6-!#.3NE 3N%. ^ /7 7 R = 1 7I "% !#XE + (! *- Q\ 2 %!#,. c ;"O. ("O- ! !#O *!#"W# %!#;1 c_; N. ! c N-!#. 9Y! 7%)* \"X;# E ! 5T +F;%"6 ? 7 !8- N%.PR! -U6T"%% 6T" 9<; 2 "OR # > Q- %!#"%Q -E ? ^ 4!#XE N HN6 L%6.%! 2 ! *-Q 2 %!#.,c ;;-/ Q !#TN%. ^ /7 7 D 6= 0 7 h,!8XE "O" NE ; ! % c ! 2 !#") .PN%. ^ O- %-! c*7 b "% ;A 8*

!#X !#") . ; X6-.EFQ N%. ^ );7 5T "O- !!#*!#"%Q 8%"%Q ^ / ) EF ! 2 F^8 EDE >769] EDT \ d @ @ >fg;F G \ 7 a EFQ ]*&,&( "H `!#`" 1 c ; 7 @ RGYc >[5[5 ;; 8*

2 !#") .PN%. ;X;6-., * 3 3NEF. ) >]"!8; 66-. ^ /7 " 7 N- !8WN%. ;*-.%! a;# - + ) * 2 )* 2 ! 2 "O- !!#*!#" ;N/ -7 l "O!#; 66-.H&("O!# ! 2 -U6T ! @ <Mg;F ;; 8*

2 ]-" '!#") . N%.B7 b %-.EFQ ]*&,&(" !8;-.) ! 2 ']-""% 9<-.EF * E ?8^ ; EFEF%6 ]-"`! ;1 8!# ;* "O 7 6T']- "*!#"4;-* 9 8-* 2 ]! !8 2? L JnO% 2 %!#. α !#;/ FEF!8- 2 ! 2 ;# ;-* 3 \%-.%4 8/ )*R E 8* α !#XE ) 2 6!#O-. ;!# 2 O-T) "%!# ]- "*!#"%P;-P; " F^8 E=E >V69] EDGYi B GME Blk?< c B=B 6%7 U ! ! ; .4!8-%Q E > V 7BI "% 4I5[> X >JkAB95fgM69u :=>fC E BUP !#XE ;* - - NE_!# T 3;-/ -. ;- %!8 Ox ^ )* ) L !#Q ;/-.%! ^ *7 7 D = 1 7I "% *!#A 86 ;*-.%! H4 6T !#O-%_;- ;* 4" % ^ LN- !8aN%6.T; 2 - 2 c! 2 !#%!# 2 2 !#%!# )** E a) * 2 ;*-.%! ^ % !8-P 5 KVR 8 H BQ\ > HJG k E <Wg G \ 7M("H! 6"E a!# ;- E W;*-.%! &( ! c\ +P! %+ % ^ 3"% O % + T; W" ; 2 - \c 6N- cR S6* E 8H/ 3 E 84!#X ^ /7 7 7 b "%Q "O- !!#*!#" );7 5TEFQ ]&,&( "4) EF%6 ! 2 E < KoX < R ] > R C R > 0 EDT \ G F R <MN> E B>V\ 7 a - 2 ; 2 - 2 / N%. + >]*&,&("6a!#1 5T*!# T!#-."%LN. ^ !#"%-."%6-\!#O-+ ;- 2 L+ 6 QaN- 6 !# ;%!#!#%7 M(] G F R F E >jF G 69];g G X < R ]U> R EDG B x → +∞

jx |x→+∞ = j

|x→+∞

0[ Pl\w< 7

b "

N) ^ !!# 2 "!#X ; X/-.% N%6. !#"% ;*!8"1c 6-+ Uc ^ *>"O- ! !#*!8"1c$ 8%*!#"!cT7M( X6-.EFQ N%. / >;* N.a!## !T;* E ;"6)*- ;-% - 2 7

f+(*,+ #)G G

a G gU<[k?<WF^] O:lF G [> 5V69B :=fF2<Mu F bF G R B E >fBlk[\w> E=EDT Z G 5fF2< 69] EDT +∞ mg R Bd @ 5on @p;B n{B9r > BQk F MG X 6_ →BUPe B RG ZDG N K > E BQP G 5fF2<Mu F|5P R > N E G c E T G R E x(y{z|y{}~yI/ I N T*4!8-R;) -.+ !#6X+ !#%!# 2

,5

E < X < R ]U> R Z 8U5V6 G 4WBUP Z BQ\`Bn

2 !#;O *!#"Q -Q &("X ;# !#6- 2 !#NQ !8; ;) XUc 18W;O-%!8"Q8 - ^ !8;%!#6 2 c 5\U8%! 2 N,!8E_%!# Ox A ; x → −∞; Ψ(x) = A− eikx + B− e−ikx 9 0 7 1%5? ; x → +∞. Ψ(x) = A+ eik x + B+ e−ik x M(%!8"-." ] EF/ 3 2 ;# S!86- 2 c !#NQ !#;*!8" &(O E * + + 37 6\ 2 - Q+ &,&( X6-.+ % 2 ^ 3 "%]*&,&(X 6 A ^ B A ^ B !#1 5T*!# H- Q 2 !# 2 ) .7 M(!#. A = αA +βB ^ + −α ^ β− ;%!#+ 2 +EF(9 eN 5T + 2 ^ "%%;O- "O!8 EF ?8^ )*+! 2 5TV− "%−" + ;- 2 V (x) 7K 6-+ !#, 6T - 2 * L3+ \;! .T %!#%6 !#N6 T Q ^ !# 2 )* E 8W! 5T B+ !# !#e. c !#6X+ A 2 0 # + 49 0 7 0/? A*!8- Ψ(x) *!8. 7 6T 3 S+ a * 2 0 # + ^ 4 Ψ∗ (x) *!#.47 6T +

T,% 2 7K!#;*!#" E ; x → −∞; ∗ ikx Ψ∗ (x) = A∗− e−ikx + B− e ; x → +∞ ∗ ik x e Ψ∗ (x) = A∗+ e−ik x + B+ - c! 2 9 0 7 15? -\6\. N) H;%!# 2 E 8,"%]*&,&(X ; ] %W * B ∗ = αB ∗ + βA∗ ^ O- B = α∗ B + β ∗ A 7 b "'N) ^ "%]*&,&(X E +`9 0 7 1%5−? !# 2 )*−E + !+ %+ %$−!#, 6T− 2 > 9 0 7 10/? A+ = αA− + βB− ; B + = β ∗ A − + α ∗ B− . a - 2 !#X ^ /- 6 c 5TU8 'N. ;-* L#-.% ;- 9<!7 0 =7 9 ?8^ &("X 2 8D9 0 7 1%5? ! -# ;-/ . B = 0 ; ] % + ^ + ∗ ∗ 7 I ;;-* L% ! -%T;-+6* B− /A− = −β /α A− = 0 7 a - 2 "]*&,&(X >6 T 2 4!#!# !H9 0 7 Z 9 ? A+ /B+ = β/α∗ * A 0

0

0

¯ ¯ ¯ ¯ ¯ B− ¯2 ¯ β ∗ ¯2 ¯ =¯ ¯ , R1 = ¯¯ ¯ α∗ ¯ A− ¯

0

¯ ¯ ¯ ¯ ¯ A+ ¯2 ¯ β ¯2 ¯ =¯ ¯ , R2 = ¯¯ ¯ α∗ ¯ B+ ¯

0

m

"* 2 !# ^

9 0 7 1? a - 2 "%]&,&(X ;A8/ 3 2 $!#- 9 0 7 Z ? ? ;-* 6-+ !#,6T 7 ¤ @ e 6T 2 9 0 7 Z ? ? ^ 9 0 7 1? EF6

c * -. )!#EF !#Q!#% ! !# 2 2 3; X6-.% N%. 7 U ! ! ; . 6 T '; X6-. N%. )*/ Q &(E V (x) EF!8- "%]*&,& X E / 3 2 ;; 8*

2 "6" &(%"Xa] +a!#XE ^ 7 c 5\ Q` !!# m 7 R1 = R 2 .

f+(*,+ #KG X G

R > K >769BDF^]g G d @ @B c B9> E F G F R E BQP|:= EDc B< 69] EDGYi 5 F 8 > E ];g?B4 T 5 G F T V0 B G Fvd E > R&H B=B|:=<MC m5fR F^BB 5on_p;n r 4 :=k;[4I< E <MC c=T E m R
V(x) V0

I

II x

0

&UY=Y I #! !8 !!7 0 7@?7:M( X6-. 2 "% 2 2 - 2 ! 2 "%!# ; EFQ>!" E ) EF% 3*-7M( ] % % 0 # + > NA8 a76T.H -.> - 2 "O6

Q )3N-!# Q I II ^ W) ! 6T.>] 7 6T 2 "%3) EF% x = 0 !# !8H!L9 0 7 ? 7 I N-!8 I ;!#!#! c`"O" ;/ c 53 2 ^ 6" 6 T 2 -E,7 M(]% 7 6T Ψ (x) EFN* 9 0 7 ZZ? 7 U7 6T PN- !8I II !#1 5T*!# )*!#3 ) "O E − V 9<)*V ^ -, eN 5T + ! -% 2 T SQa)/ S V = V 0= V ? 7 M( E < V N- !#. II 2 - 2 ! 2 "O- ! !#*!#"'# %!#;Q ^ ;0] - 2 c 5T0 (!#6 S6E !8- 2 7 6T ,!%!#,. c_ ;%!8 2 +LV/ - 2 EFN* 9 0 7 1%Z? ΨII (x) = e−κx , + κ ;# - 2 ! 2 !#e 6T * 9 0 7 ? 7 < + "%L; . ^ W!#%!# 2 9 0 7 1%Z? ;-%!8.3;"O\\*- c\7M(] %L"%]*&,&(X T;; 8* 2 D = 0 7%U "%% * ! !# 2 #-. ;-.TEF6 T 2 - 2 A a3%!#%6 9 0 7 Z 9 ? )*!#V349 0 7 Z ? ? ;"O)*. ^ R = 1 7 B b " N) ^ *!8-'N-!#. II 8 %!8%;` "O- ! !#*!#"%% ! E !8- ^ !#XE ;-%!#,. c / c ! 2 !#; ." 7I *!#>!L* ^ N-) H" 2 %!86 ! 2 *- % 2 2 !#. N% T 2 !8X 9Y! 7 ;O 0 7 1%? 7 !#.* L ;*!#a6-+ \c !;-E 6 T *';"%7

5 M E > V N-!#. II 2 - 2 ! 2 "O- !!#*!#"H %!#;Q ^ ;]%> ( 6T ΨII (x) 0EFN O '9 0 7 Z5? ! 6T* ! ΨI (x) 7 U "%% * 6"O 3 ! %!# 2 -. ;-%. k + k0 A= , 2k

k − k0 B= . 2k

;A8/

2 EF!8- 2 * ; &(*-

C(]&,&(X E 6 T 2 9 0 7Z 9 ? A

√ µ√ ¶2 ξ− ξ−1 √ = √ R(E) = ; ξ+ ξ−1 p 4 ξ(ξ − 1) + 4kk 0 √ √ D(E) = = , (k + k 0 )2 ( ξ + ξ − 1)2 µ

4

k − k0 k + k0

¶2

ξ=

E . V0

^ R 6= 0 ^ /7 7 \*!#\/ N%. 6 T 7%I Q !#%X HN `66-+ a!#% / 3 Uc !# Q -E ; 6-. ;/ + 6X`)# - a!8# !36)-E ; "O)*6- 2 `;- - 2 7 M(# -+6* ! %!# 2 -.L%N# .%! 2 !#;/ # -!# 9 0 7 Z ? ? H;%!#.+ &(">)*!#%!# Q R(E) D(E) 7 I ;# #-.% !8-%6>"-! !8*!#"%Q 8%" 9 } → 0 ? "%]&,&(X 6 3 2 - T N.! 2 4*-.7 H

* ;- LEF6 T - 2 R(E) %!# !# % P;%!8 2 QLM - "O C -! !#!#"%%\; - !#!# a!8-%6Q ^ "+ >RQ- %!8"O 2 ->-E !#XE 6- \;L!# \c ! 86" !#*!#"a)* ! !8 λ = 2π}/p % c 5TQ8 * %!8Qa; X6- ^ *7 7; !# \c ! !!# 2 2 ^ a"% E 8 ) 2 ! 2 ;- V (x) 7 I ! ! %*% T ! 8* 6 *!#"%% ; P] ! !8 2 T*\- c 9< "%P) EF% ?8^ 6"> ;# -.EFQa; 8* H\* 3 NEF. ;) # 7 ¤

f+(*,+ #KG gG !
\wfB4I<Wu F25P E < R Pl\ G 8 HJG 69] EDG \ 5fBU\\w> F R B=: E=G \ G F2> EDc B< 69] EDG \ X < R ]U>fC R > B R B EDT a B 4 T 5 G F T V0 m R B5on p;n Vr;n R > K >V69BDF^] g G d{@ @B c B> E F T G F R <MNO> E BQP B R >V6 G \6_> E BUPtgU<Wg|@8 E g c BDBjd E > R&H B=B E n x(y{z|y{}~yI IR8 * !#!8* "%

%

V(x) V0

II

I

III a

0

#! !8_! !7 0 7 :7 @ 6X &UY=Y W % 0 # + N;8 #-. 763. "O6

Q$)H 8 N- !8 Q A I ^ II ^ III 7 I_N-!# I Ne5TQH 76\ 2 ! 2 EF6 T * 9 0 7 ZZ? 7 a - 2 N-!# III 7 6T P N ;8 S!#6.

ΨIII (x) = eik(x−a) ,

x

0 "% -6 ! 2 9 0 7 Z5? ;!# 2 E / -* e−ika ^ k0 = k = √ %!8-8 S!#R!# TN%.O7 U76T N- !8 II !#!5T!# 2mE/} )*!# )"O E − V 7 M( E < V ; N. 0 2 N- !#. 2 - 2 ! 2 "O- !!#*!#" # %!#;QB7 M(]% 7 6T 0,% A 2 0 # + 3EFN6 ! 2 9 0 7 11? ΨII (x) = F eκx + G e−κx , + κ ;# #- 2 ! 2 !#, 6T * 9 0 7 ? 7I'- ,;# E ! 5\ + ,;O ^ )# *!#.\E N 2 )*E !#A 86.PL!#! 5TF7 6\ !8-# !# g GME >f:lC EDG 5fF^B&%^B R B EDT X < R ]U> R < 7 M(%!8-F! 6T% 2 7 6T Q ("O 8 x = 0 x = a !8 !#!P9 0 7 ? ;-%6* !8-# 1 c 5T1 c !#!#*H Qa - 2 "%]*&,&(X A ^ B ^ F A G  A + B = F + G;      iκ   (G − F );  A−B = k   F eκa + G e−κa = 1;        F eκa − G e−κa = ik ; κ

9 0 71 9 ?

+ "%`)*. ^ !#!#* 9 0 7 1 9 ? / ) N.`H W NEF - 2 76T 2 ;* !#!#E,7 M(%!8- N;8 E 8 EF! - Q 9<; #- .`! !# 2 -. ? * A

<

i A = ch κa + 2

·

κ k − k κ

I !#!#H!L9 0 7 Z 9 ? ^

¸

sh κa;

i B=− 2

·

κ k + k κ

¯ ¯2 ¯B ¯ (k 2 + κ 2 )2 sh2 κa ¯ ¯ R=¯ ¯ = 2 2 ; A 4k κ + (k 2 + κ 2 )2 sh2 κa 4k 2 κ 2 −2 D = |A| = 2 2 . 4k κ + (k 2 + κ 2 )2 sh2 κa

¸

sh κa.

9 0 71 ? ? 9 0 71 : ?

RI ^ D 6= 0 ^ /7 7VN ` . *!# F^8 EDE >V69] E=Ti d @ @ >fg;F 7 + * L N 2 !#.`* ^ + -%NH;" 2 !#XE ;* 'N%. !# !^6TQ N%6. 9 ! 73+ &(" ;-!# 2 !# !7 0 7 [ ? 7 M( E > V N-!#. II 6" T 2 - 2 ! 2 "-! !8*!#" %!#;Q ^ ;] W7 6\ P0

2 0 # + ;N 8 3EFN. 9 0 7 1%[? Ψ (x) = F eik x + G e−ik x , 00

II

00

w(x)

(à) E < V0

a

0

w(x)

x

(á)

E > V0 a

0

x

&UY=Y I

+ k 00 =

r

2m(E − V0 ) . }

U76T W9 0 7 11%? - ! 2 49 0 7 1%[? )* Q

b "!c

T,)* L ! #- .a`9 0 7 1 ? ? ^ 9 0 7 1 : ? A κ → ik 00 .

(k 2 + k 002 )2 sin2 k 00 a R = 2 002 ; 4k k + (k 2 + k 002 )2 sin2 k 00 a 4k 2 k 002 D = 2 002 . 4k k + (k 2 + k 002 )2 sin2 k 00 a

¤

9 0 71 m ? 9 0 7=9 5?

I N,5T !8- R 6= 0 ^ *7 7 *!# / N. 6 T 7 4 6" ; "%E 8 ) * 2 8 ] + N%. !#6! 2 N%!8-\c ;) E A R = 0 7 4 ; 2 .T 3(;!8-# L% ! 2 ;*!#"O 2 6-+ 2 !

Z ;;8*

* !# QL-E !8") .T!8-Q 75T*!8%,! E ;"6)*- ;-% - 2 A ; !#; X6-.'; N6Q -5T !8- 2 N-\cR S ! 2 ;%!# - 7M(# -+ ! !# 2 #-.\Q ] >) 2 ^ \6"O T ! !8-# %.L!8-# ! c 5\P!#EFP;# #-.EF !8- A 0/? E → ∞ 9Y&,"*!8" E À V ? ? !8-QaN%6.3 /-Q>;) 0*%!# (V − E)ma2 /}2 À 1 Z? E → 0 9Y&,"!#" E ¿ ma2 V 2 /}2 ^ E 0¿ V ? 0 1? ma2 V /}2 ¿ 1 ma2 E/}2 ¿ 1 0 9 ? E ≈ V00 7 M( ./P ;%!## S!8 ^ R + D = 1 7

V"#v&2a*2"!D # GU76\.3)*/ * ; 0 7 : - 2 !#X ^ !5TQ8! 2 T;;-/

%';- ^ *7 7 !#"OEF%c T 5 Q8%! 2 !#L!#%;."B7 XGU76T.)** S; 0 7 [ - 2 !#X ^ ;- 6c \5 U8 / 4!# O Qa; 2 + -.Q ; X/-.Q 2 Q T 6 E a a+ -NE V 7 9 F24?>fF A! 7:9 0 7 1 m ? ^ 9 0 7 9 5? ^ ;- 6+ 2

"%`"/%# _'

0

p 2m(E + V0 ) k 00 = . }

l + 2 !8EF ! 2 " 2 E,7 ? g G\,h !#XE ! !#E m "O6 3 S 2 ! !# % c! 2

δ N6) %`; X6-. N%6. V (x) = Ωδ(x) 7 ,Q "%]*&,&(X E / 3 2 ;A8*

2 "O"4&(%"Xa] + a!#X 7 9 F24?>fF A R(E) = (mΩ)2 /[(}2 k)2 + (mΩ)2 ] 7 ? !!#E m "O6

2 ! !# % c! 2 3!#!#*P) 1 8 6" ∗ G"h,!#XE E 8 δ N6) E 8a; X6-. E 8HN%. V (x) = Ω[δ(x) + δ(x − a)] 7 M( "O"U8 ) 2 8 ] + '!#X 2 !#!#* N ;) Q - 2 U8

22 2 lv'V /2 _$#

) EF%6 ! 2 !#X ^ !# O 6 c 5 2 T P; X6-.Q 2 V (x) = mω2x2/2 9<!7 0 7@m ?8^ + ! ! 4!#XE ^ ω !86`!#XO-- 2 O7 I gM6_[5 g G \ ! -% m !#XT!8 U6 /- (NE T ;\)*"% x(t) = x cos(ωt + ϕ ) 7K;- S x ) *L;# - 2 ! 2 ] + Q`%!#X-- 2 0 ^ "%0 2 a!8, c B E > iDE=T \ H < R \ GME BD:=>[5 g;BQ\ G 5 c BQ6 6lP9F GRG \

0

/1

hw

# .W/ T ; . E > R > RT 4 E=Ti 2 ) Q4%/- 5 ∞ 7 I "%4!#6X 2 ;!#6"O V(x) )*/ N 7 6\ 2 2 0 # (à) + }2 d2 ΨE (x) 1 + mω 2 x2 ΨE (x) = EΨE (x) − 2 2m dx 2 9 0 7 0/?

hw

=9

! + E !8- 2 Ψ (±∞) = 0 % !8-# !#,&(+ T O 3E 2 7 ID!8 !# ! N,5TQ Q ] O + *!8"Q !8; "$%!8XO-- 2 4N K B5 C g R >fF EDT \ A µ ¶ 1 En = }ω n + , 2

hw/2 -a

(á)

w0(x)

9 0 =7 9 ?

n = 0, 1, . . .

-a

x

a

0

a

0

x

!#;-/ T E ] " !#6 &UoDo I ! !# 2 }ω %+' +#7(I !# 6 !# !3N,5\_!8Q!#% * + H&(+ O T 2 E > 4 T C RG N K > EDT ^ /7 7"O6

%>!8 !# 3-."% !# !# 2 A

1 Ψn (x) = √ Ψn x0

+

µ

x x0

¶

9 0 7=9 Z?

,

9 0 7=9 1? !# !#,!#; %!# 2 . # %!#X H* -E *!8 "- 2 %!#EFX"O--/- 2 "L; ^ ; 8*) %- 2 "Lc N53 ) 2!8)*!5T O E $- x0 =

Ψn

(ξ) = p

r

1

} mω

−ξ √ Hn (ξ) e n 2 n! π

2

/2

;

9 0 7=9,9 ?

;-%ih( NEj6T ;kUl 6 9 ! 7 ;-/ T ? 7o\%"X_9 0 7=9 Z? %E 38 XWa+ 6-.E 3%!#Qa75T*!# Q>!#BA

Hn (ξ)

Z

+∞

Ψn0 (x)Ψn (x) dx = δn0 n . −∞

9 0 7=9e? ?

9

5 E=G 4 EDG >*5 G f5 F G P E B> G 5 c BQ6 6lP9F G R < BQ\w>[>fF E > E 8M6_>[4M8;u d E > R&H B=u E = < 9 # ! F E % 6 ! ; X6-.Q 2 E ? 7 W6"`) EF%6*0 2 2 }ω/2 d E > R&H BQP E 8M6_> 4 T Z g G 6_> X < E B i 7 ,6- *- E "%- N%6Q'a; R B EDc B D8 E > G R > K >V6_> EDE=G 5fF2> i ^ \;)- 2 T* !#XT;!#.%! 2 7 !# !#%!8 2 !# !# 3- 2 &(%"X 2

8

Uc

4

l

c5

µ

x2 Ψ0 (x) = p √ exp − 2 2x0 x0 π 1

¶

9 0 7=9: ?

.

M %!#"%-." ; S6- ;-* 3 2 6!# 2 ; X6-. 2 ( ] + 2 ,) !#6 ( ; EFE N) ^ -EF &("X N 4 *- E `"-! !#!#" # %!#;Q N-!# 7-n &(" ;-6 %!#$ 2 %!#' %!8 !8%!# 2 ! 2 "O**!# ; !7 0 =7 m6N7 4 a;# !#6- 2 L!#NQ +%! !8L"! cT7

f+(*,+ #)G G

B E > Di E=Ti GIKoE=G \w> R9EDTi H < R \ GME B=:=>[5 g;B i G 5 c BQ6 6lP9F G R R B4?> C K > E 4 G 5 E=G 4 EDG >`5 G 5fF G P E B9>on < i F^BA4?> R9G P9F EDG 5 F ]|> HJGwGYXME < R 8?N> E BQP 4 gY6_[5fg?B KoG 5fF^8 EDGYi GMX 6_!#XO-- 2 m E B7 , QO * + XE_"O- ! !#*!#"W %!#;Q>N-!8 9Y! 7!7 0 7 ?

)

)>

),!8- 2

=m

±a

1 }ω mω 2 a2 = E0 = ; 2 2 r } = x0 . a= mω

9 0 7=9 [?

b " N) ^ * L !#"O). ^ 8 X Q -E - 2 %!#X-- 2 !8-% )* >"-! !#*!#"W !#%;Q>N-!#>%!#'!#%!# 2 7 a - 2 #-E_ 2 %!#a;-%6* A P =

Z

+a

w0 (x) dx = −a

Z

=

+a

=

1 √

Z

+a

µ 2¶ x exp − 2 dx = x0

x0 π −a ¶ Z 1 2 x 2 e−t dt = erf(1) ≈ 0,8427. =t = √ x0 π 0

−a

µ

Ψ20 (x) dx

M EF!8- + 6- !#;-.) 6 !#Q!# !# ;* EF ( + 6-.Q$&("XB7 4 EF6 6 ! 2 )>6N*-%61c &("X\c ,6TN" 1J! 7 &(*- 9YK 7@? ? !# 0 3 7 < + "% )*. ^ ) *-.66 )*!#3aL ! !#E_!#XO-- 2 ^ a + !8E,7

? f+(*,+ #)G # G a

%

R9G I4 > R BDF^]A5 GMG F E=G 2> E B> E > G R > K >V6_> E=EDG 5fF2> i K l6 P g GYG R&K B=C E <MF T BBQ\ D8Y69]U5 < 4 5V698;:= 5 GIK[EDG \ > REDT \ 69B E > i=EDT \ H < R \ GYE BD:=> 5fg;BU\ G 5fC c UB 6 6lP9F G RG \ n x(y{z|y{}~yIB M(-E ( NE T% 6 4%!#. ; ] n^

h j6 ;kUl 9 0 7=9,9 ? hxi = 0 7VI

c !# 2 )*6E 8

#! !# ! !#6X(−1) E 8 !8%!# 2 2 -\cNQH!#!#E hpi = 0 9Y! 7)** S 1 !8 0/? 7M(] % h(∆x)2 i = hx2 i;

8

h(∆p)2 i = hp2 i

I O %!#XO-- 2 + ; X6- ; ) - 2 "O 6-. ;!#. E !8- '] Q8 !## Q8 - ! !#;-.)6* *E /- 9<; J 9O7 !8 0/? 7 a - 2 %!#XO-- 2 T 9 0 7=9em ?

hp2 i = mω 2 hx2 i. m

@ + QH!#E ^ ;;# -Uc !#66X6E 8>!#%!# 2 Q

9 0 7@? 5?

¶ µ hp2 i 1 1 2 2 . + mω hx i = }ω n + 2m 2 2

M(%! -,!#*!#+ 76T 2 9 0 7=9em ? hx2 i = x20

µ

1 n+ 2

¶

;

9 0 7@? 5? ;-%6* A

¶ µ 2 } 1 hp2 i = 2 n + , x0 2

n = 0, 1, . . . ,

+ x ;# - >9 0 7=9 1? 7 ^ !#; X&(*!#"QWO ;X/-. QL]0 + L;) --;-%. ] ) *-./6E N ) !#;-.) 6 2 * *"%+ + %6 2 7 IREF;- !#e 6T 2 ;# -%!# Q )8 *!8.$ 7 >R- + ^ %!# !#%!8 2 9 n = 0? ] !#,6T ; 53 ! 2 5fF RGWHJG > R E 5 F|4 G 7 @ %!8 2 2 ^ )! c 5\ !#, 6T ;# -%!# Q ^ ) EF% c! 2 g GIH > R > E F EDT \.B 7OM( '! - L3%!# !# !# 2 ,%!#XO-- 2 7 ¤ l &,&( " 0 6"O()EF% c! 2 )* 2 ^ ;! 8* 2 5T !#\!# 2 )*O QL)*6 2 T Q !#!#*QL;* )# Q!8* 6T+ ,]- "*!#"%+ ; - 2 7 ( T E !! ]*&,&( " 0 6"OP - 2 )*6 2 T + %!8XO-- 2 ;%!# 2 ]-"!#"%%';-7

f+(*,+ #)G #)#)G F G R K > i 5 F|4M8U>fF

)

GWKoEDGRGIKoEDG >j{d 6_>fg;F R =B :=>[5 g G > G 6_> E -< R DP NO> E C

: V(x)

E=0

V(x)

E>0 eE mw 2

x

0 x

0

2 2 - eE2 2mw

&UY=Y?

)

EDG 5fF^B E E -< R EDE=G > 4 KoG 69] G 5 B Ox n < i F^B d E > R&H BDB 5fF2< c B GME < RE=T Z 5 G C 5fF G P E B i B 5 GMG F|4I>fF25fF24M8;u3 ^B>.BQ\ 4 G 6 DE G 4 T >.@8 E g c B=B_n x y{z|y{}~yI 0 # + ( - 2 % !8XO-- 2 ( ]-"!#"%% ;- ( }2 d2 ΨE (x) + − 2m dx2

µ

¶ 1 2 2 mω x − eEx ΨE (x) = EΨE (x) 2

9 0 7@? 0/?

* L3%;%!#. ^ *!8- T;X6-.Q ] +WE -.T;-EFQW"%* A µ ¶2 9 0 @7 ? ? 1 e2 E 2 1 eE 2 2 2 − mω x − eEx = mω x − . 2 2 mω 2 2mω 2 M(%! -()* eE ; x→X =x− mω 2

e2 E 2 E→E =E+ ; 2mω 2

E ;Q8 '"a Uc −

0

ΨE (x) → ΦE 0 (X)

}2 d2 ΦE 0 (X) 1 + mω 2 X 2 ΦE 0 (X) = E 0 ΦE 0 (X), 2 2m dX 2

9 0 7@? Z? 9 0 7@? 1?

;>!# Q`!#%" G 6 EDG 5 F ]?u 5 G 4-< K <Mu3 2>V\.8 5w8 R E B9>V\ R > K B EH > R < 9 0 7 9 0/? K 6lP F|<Mg GWHJG NO> G 5 c BU6 6lP9F G R < X >Jk G 6lP 7:M(` +`&( % H- +" ; 2 . ^ ;%6-)6'9 0 7@? ? A ; X6-. 2 "% 2 %!#X- - 2 T; > Q!#*'7 6\ + 3]#- "!#"+ ;- 2 ; ; %3-\6\. < R < 6 6_>V6D] EDTi > R > EDG 5 9<!7 0 7 0*5? %&( P"Q ^ ;# - 2 c 5 2 !# ^ %!#66 ! 2 E >fBQko\ > E=EDGYi 7 b " N) ^ >%!#X-- 2 W!7 5 ! 2 ;-/ T ,*!# 2 H/- !#3] + B7Kl LX-"%%'6 6 ! 2 )*6 8 9 0 @7 ? Z? ^ P " % 9 0 @7 ? 1? a+ E 8W%! - 2 8 " * 9 %!#;%PU8 ? 7

[ U76T 9 0 7@? 1? -+ "% !#! 2 3%!86 9 0 7 9 ? ^ 9 0 7 9 Z? ¶ µ e2 E 2 1 − En = }ω n + ; 2 2mω 2 ! Ã eE x − mω2 1 , Ψn (x) = √ Ψn x0 x0

l

9 0 7@? Z? A

n = 0, 1, . . .

9 0 7@?9 ? 9 0 7@?,? ?

& &( " 0 6"O( - 2 %!#XO-- 2 P\;%!# 2 % ;- )*"O-c ! 2 3! , 5T W%!8 8 ] + *!8"Q8L QL) ,* "%!c - ^ !8 )* c 5\! c "O-! !#*!8" ;# -7 ¤ ( T ! ! 6%6 ! 2 !8-Q` O +>%!8XO-- 2 ^ c 5T + > + - E 8W! 2 8W; X6-.! c ] + \c

f+(*,+ K# G #\BG G5P Z ω ω

V (x, y) =

1 2 2 m(ωX x + ωY2 y 2 ). 2

9 0 7@?1: ?

RE=GIHJG G 5 c BQ6 6lP9F G R < m .:= R&H BDB 5fF2< c B GME < R9EDT Z X Y 5 GYG F24I>fF25fF24W8;u3 ^B>jBU\ 4 G 6 EDG 4 T >j@8 E g c B=B_n fF^] G :=< i g R 8 HJG 4 GIHJG G 5 c BQ6 6lP9F GR < ω = ω = ω n X Y x(y{z|y{}~yI6 0 8 + !(; X6- 9 0 7 ?

@?1: )

=?:

}2 − 2m

µ

T 4 H 6_V69] EDG 5V698?C

¶ ∂2 ∂2 1 2 2 x + ωY2 y 2 )ΨE (x, y) = EΨE (x, y) + m(ωX Ψ (x, y) + E 2 2 ∂x ∂y 2 9 0 7 [?

;!#"O 3)# - P; E 8>T "OE 8W"% 6 S8 A

@?

ΨE (x, y) = Ψx (x)Ψy (y); E = EX + EY ;

4

}2 − 2m }2 − 2m

d2 1 2 2 ΨX (x) + mωX x ΨX (x) = EX ΨX (x); 2 dx 2 d2 1 ΨY (y) + mωY2 y 2 ΨY (y) = EY ΨY (y). 2 dy 2

^ )/ S + `%!8XO-- 2 `!# ! 2 ;. " )*/ 6 18 E >Jk?
µ

1 nX + 2

¶

µ ¶ 1 + }ωY nY + , 2

nX , nY = 0, 1, . . . ;

9 0 7@?,m ?

m +

1 Ψ nX ΨnX nY (x, y) = √ x 0 y0

x0 =

s

µ

x x0

¶

9 0 7 : 5?

y y0

y0 =

s

} . mωy

5 0 0 5

7 77 7 77 7 77 7 77

Ψ nY

} ; mωx

¶

µ

,

U ! ! % !8- 2 7 0 7 ω 6= ω 7 C,6

%H) \c ] + #! !# W# !# !#%!8 2 X ^ /7 Y7] + *!8"Q`!#; "HN E > 4 T RG N K > EDEDT \ 7 woo

5

n nX ; n Y

7

5 5

0

5 0 0 5

7BI !8-3"+ + %!8XO-- 2 6T 2 9 0 7@?,m ? ^ 9 0 7 : 5 ? ;N) 1c! 2 A ωX = ω Y = ω

En = }ω(n + 1),

x 0 = y0

n = nX + nY , nX , nY = 0, 1, . . . ; µ ¶ µ ¶ 1 x y ΨnX nY (x, y) = Ψ nX Ψ nY , x0 x0 x0

^ >!# 9 0 7 : 0*? 9 0 7 : ?

] + 2 ;# #- 2 ! 2 \"O6

E )\"E 8>!##- n ^ n ; -.%!8 ^ F G 69];g G B Z 5 8M\\ GYi n = n + n 7OI *!#P! ^ !#X!# 2 Y 2 ^ !# !81 c 5T* V % 3 ) *X \c Yn ^ N % R <[k[69BD:=<WF^]U5P ) 6 2 n n 9<6N-X 0 7 0/? 7 b "'N6) ^ 3"+ % %!#X-- 2 ] + *X!#",%Y 4 TRG N K <Mu F25 P ! "!#.ec n + 1 7 ¤

b ; .

V"#v&2a*2"!D # # GFM( . * !#,\6 h∂ Hiˆ = ∂ E - 2 !86X6E 8 !#%!8 2 Qa!#XO-- 2 7 #)#)G M( . * !#eT6 h∂ Hiˆ = ∂ E - 2 !86X6E 8 !#%!8 2 Qa!#XO-- 2 3 7T 6 * ;- 9Y! 7; 0 7 00/? 7 #UBGU7\6 .3)*/ * ;>9 0 7 0/? - 2 8 + T%!#X-- 2 T! !8 6H + -E 8 %! 2 8 ω ^ ω ω 7

"%`"/%# _'

x

y

z

ω

ω

n

E

E

n

Z5

1 F24I>fF A/,! &( *!#"%%4%!#X-- 2

µ

3 En = }ω n + 2

¶

^

n = 0, 1, . . .

6"6

%!#.EF/

2 1 (n + 1)(n + 2) 7 3 #QV ∗ G a H!#XE 2! ! m1 m2 ^ !#;%!#NEF4!#\6 6. * % T ^ !8# E ;% Q 3*!#"%%!# κ 7 ,Q>] + 4!#66X6 E 8 !#%!8 2 QW6"%Qa!8!#*E a!# !81c 5T -EF &(%"X 7

L+ 2 v

a

EFQ )# - ;# ) ! 2 - 2 + -%N-+ ) % 2 ; ; % / T NEF.W;;15T 7 I_* )- 6+ ! 2 6-+ N6*!#"Q * 4 6T 2 )/ S>3-Q%'+*!#"%% %!#X-- 2 7 6;U6T* 0 # + , - 2 !#XO-- 2 W9 0 7=9 0/? TN ) )* O E 8 ;* E 8 7C( 6(T;*-.%! EF)a %!#X-- 2 E 8 # X 8 x }/x !# !8A 0

+

0

1 1 2 pˆξ ψ(ξ) + ξ 2 ψ(ξ) = εψ(ξ), 2 2

x0

;# - 2 ! 2 !#,6T *_9 0 7 9 1?

4

ε=

^

E ; }ω

pˆξ = −i

d . dξ

[ξ, pˆξ ] = i.

IR8 * 6"a) EF%6*EF XYG k?>fC G 9> R <MF GRT A iˆ p ˆb = ξ + √ ξ; 2

4

] E ^ ) !#; 2 3E "%% %6X%>!8,6\ Uc

iˆ p ˆb† = ξ − √ ξ. 2

9 0 7 : Z? 9 0 7 : 1? 9 0 7 :e9 ? 9 0 7 :? ?

!#+ -!# 9 0 7 :e9 ? - 2 c

9 0 7 :,: ? 9 0 7 : Z? PN) # ; 6A8 ;!8- ; N) 6Qa9 0 7 :? ? !R % 9 0 7 :,: ? ; L :A 9 0 7 : [? ˆb†ˆb ψ(ξ) = ε0 ψ(ξ), + ε0 = ε − 1 7M() 8 ˆb†ˆb N ] E '; 7 [ˆb, ˆb† ] = 1.

2

ZO0 M( 2 !# &()*!#"Q ! E !8- N) ; 7 M(# ;-/ ^ *!#.,!8N%!# 2 &("X 2 ; ˆb†ˆb ^ !# !#!c 5 2 !#N!# ψ(ξ) % ) * \c ε0 7M(* Q!#* 49 0 7 : [? !8- %3; ˆb† H; N6 ) * - ! c !#. !# !#4!9 0 7 :e: ? 7I ) *-./6P;- A 9 0 7 :m ? ˆb†ˆb[ˆb† ψ(ξ)] = (ε0 + 1)[ˆb† ψ(ξ)]. G() !8;!#6- 2 9 0 7 : [? 9 0 7 :m ? / a! -. EF* % ^ ˆb† ψ(ξ) F G N> P_4 6lP_>fF25Pt5 GYX 5fF24I> EDEDGYi @8 E g c B> i ˆb†ˆb ^ !8 !#! c 5TQ 3 + >!#N%!# %L) Uc ε0 + 1 9< O "O!#%'>&(%"X ? A 9 0 7 [5? ˆb† ψε (ξ) = C + ψε +1 (ξ). n 2O2 3 6\U8 ;8 E ! 5TQ8W! !#

2 8 ˆb† ˆb ^ ;"6)*. ^ 9 0 7 [0/? ˆbψε (ξ) = C − ψε −1 (ξ). n C(!#6E C ± `9 0 7 [5? ^ 9 0 7 [0/? N Q E_O 37 b " Nn6) ^ / \

. ^ T; 6 ˆb† N%- 2 ,%!#X- - 2 "% g G 6_> X <MF|>769] EDGMi ] + ^ EFQ 3NEF E 8 8 X 8 }ω 6" ) EF%6*EFQ @ GYEDGME 7 4 ; ˆb 6 '6"%Q T &( 7RM(] `; E ˆb† ˆb 6) EF% c ! 2 !#!# ; /

2 / T 2 &(O7 M(%!8-# 6#-.E Q!#* ] U8 ; * L;"O)*6. dMgU4MB K B95fF2< E F E=G 5fF^] !#; " !#XO-- 2 O7 IREF! -D ; . ] +\c %!#+ !#%!# 2 2 ^ " E * 3 ;8 - ) N) ; E,7 M(

%!# + )* ^ !#;" ε + !#) ` !#XO-- 2 + ; X6- ^ 6" H!8! 5T !# 4 /-. !8N%!# 3) ε !# !# 4&(%"X 2 0 7 (W!#+ -!#49 0 7 [0*? ^ ˆ ^ /7 7 * L ;%!#. ψε (ξ) ≡ ψ0 (ξ) bψε ∼ ψε −1 (ξ) W!#1 5T*!#! c 5T !#%!# 2 ψ (ξ) 7 a - 2 !# 2 + H;O 2 AN 8 ;-/ . ε −1 0

0

0

0

0

0

0

0

9 0 7 [? ! ! %6. ] _/ 3 !#_"O" ;# -$%!8+ !#%!# 2 2 9
2 ψ (ξ) )*6;U6T* % L9 0 7 [? 2 % ! % 9 0 7 : 1? 9 0 7 :? ? A ε ˆbψε (ξ) def = 0 0

0

d ψε (ξ) = −ξψε0 (ξ). dξ 0

9 0 7 [Z?

l L- Q, * &,&( X6-.( > R 4 GWHJG GR P K g < 7 6 ! 2 !#!T5 !# ;,5TT;a!# Uc !>9 0 7 : Z? 7 + >%6

Z 3# X 76T

9 0 7 [1? >NEFE 8`8 XA8`; 8* / X 9 0 7=9: ? 7 (;%!8# !# O 2 ; S!#6"O,9 0 7 [1%? 39 0 7 : Z? !8" ε = 1 9<- ε0 = 0? ] + Uc *- E 8H"%- N%Q4WN ))*E 8a# X 8 0 7 b 62" N) ^ ] + O *!#"Q4!#; "H%!#XO-- 2 TN ) )* E 8W# X 8 ψ0 (ξ) = π −1/4 e−ξ

1 εn = n + ; 2

ε0n = n;

2

/2

n = 0, 1, . . .

8* ! 2 F;-%>!# !#,! 9 0 7 9 ? 7 ; 6 ˆb†ˆb / L ;O . ! E ! - G X-- 2 ^ /7 "B7 + !#N%!# E ) * "%EF(! - A

9 0 7[ 9 ?

!8-\!#;%!86. 9 0 7 [ 9 ? 9 0 7 : [? ^ 9> R <WF GR <e:lB95V6_
9 0 7[ ? ? IREF! - ; .3EF"%!#6E H9 0 7 [5? ^ 9 0 7 [O0/? 7 a - 2 ] + ; N* ^ NE &("X 2 ψ (ξ) NE - ( P# X ; !8- ^ ψ (ξ) % TTn+1 O7 I %!#;-.) * ! 2 "%%!#"`N n ) * 2 A ˆb†ˆbψn (ξ) = nψn (ξ).

Z

+∞ −∞

=

2 ψn+1

|Cn+ |2

dξ ≡ hψn+1 | ψn+1 i =

|Cn+ |2

¯ E D ˆb† ψn ¯¯ ˆb† ψn =

hψn | ˆbˆb† |ψn i = |Cn+ |2 hψn | ˆb†ˆb + 1 |ψn i = |Cn+ |2 (n + 1) = 1,

"* C + = (n + 1)−1/2 7FK 6-+ EF` Q!# 2 !$9 0 7 [0*? Sc C − = 7 b 6n"_N6) ^ E ;-* !8-# !c 5T;O- L Q!# 2 Nn) n−1/2 ; 3 ;%!#6!# !#6X6E 8>!#%!# 2 Q>%!#X-- 2 A ˆb† ψn (ξ) =

√

n + 1 ψn+1 (ξ);

ˆbψn (ξ) =

√

n ψn−1 (ξ).

9 0 7[ : ?

@ + -!# %6 Uc 9 0 7 : Z? ^ ψ (ξ) ≡ Ψ (ξ) 76G,! 8* 2 )P9 0 7 :? ? a9 0 7 [ : ? ^ * L ;-.>!8-# !c 5T3!#n,6T n2 - 2 Ψ (ξ) ^ E > G 69]WWk 8MP_5 ]AB Z n P_4 EDT \ 4MB KoG \ 9 0 =7 9,9 ? `!8Q!8 9 > 7 1? A

ξΨn

(ξ) =

d Ψ (ξ) = dξ n

r

r

n Ψ (ξ) + 2 n−1 n Ψ (ξ) − 2 n−1

r r

n + 1 Ψn+1 (ξ); 2 n + 1 Ψn+1 (ξ). 2

9 0 7 [[?

ZZ @ ;,5T.,c ; 6/

2 &( -Q ( & %"X`%!# + a!8%!# 2 2 / L>;%!#.> -!c &( "XUc ; + W) N

+ L!#!# 2 2 !# !#H!L9 0 7 [ : ? A

Ψn

1 (ξ) = √ (ˆb† )n Ψ0 (ξ). n!

9 0 7[ m ?

G()3EF6 T 2 9 0 7 [ m ? * L>;-%.H& *-HU* +L - 2 ;-% h( NjE 6T ; kUl 6H9 > 7 Z? 7 l &,&( " 0 66"O - 2 )* 2 T + !#XO-- 2 ;%!# 2 * %']- "*!#"%%$;-,; NL; 0 7 00 7 4 !8 2 2 6T 2 )** S )*6"\- c6- !#.(\! + F*6- "% PP ;-/ T *!# 2 !## !#X " )*/ *NH!#XO-- 2 N );- 2 7:M( "O6 T* ^ H] 6a)** S>/ T HNEF.H7 6T !3!#;-.)6* 8O" N) ;67 M( QO * /-( 9 0 @7 ? 0/? "aN ) )* E ;* E A √ 1 1 2 pˆξ ψ(ξ) + ξ 2 ψ(ξ) − 2 γξψ(ξ) = εψ(ξ), 2 2

9 0 7@m 5?

+ γ = eEx /(√2 }ω) HEF) ' + ) N) ; E_;W6-+ H! 0 9 0 7 : [ ? A 9 0 @7 m 0/? [ˆb†ˆb − γ(ˆb + ˆb† )]ψ(ξ) = ε0 ψ(ξ), I # *$ ; .WEF,; E Bˆ ^ Bˆ † ^ !# 2 )*6EF\!PN) ; ˆb ^ 6T 2 ˆb† !#, 9 0 @7 m ? ˆb = B ˆ + γ; ˆb† = B ˆ † + γ. ( * %N# .%! 2 % ^ _/ T N %$N) ; ^ /7 7 - 2 U8 EF;- 2 ! 2 !#e 6T 9 0 7 :,: ? 7M( ;\6\* ; .49 0 =7 m 0/? ) EF(; E 9 0 @7 m ? A 9 0 @7 m Z? ˆ ˆ † Bϕ(ξ) = ε00 ϕ(ξ), B + ε00 = ε0 +γ 2 = ε− 1 +γ 2 7*M !# Q,!#%" 9 0 =7 m Z? O % \c 9 0 7 : [? !H2)*!#E 7 6T 1 ! 7(9 0 7 [ ? ? 3 7 M(] %$7 6\ 9 0 =7 m 5? * L)*;!. ; . !#) A εn = n +

1 − γ2. 2

I NEFE 8T# X 83\!8;* S6 !,9 0 7@?9 ? 7%G,!#;-.) 6EFQL&( 6-) N "%6Q H;- 2 7

Z61

(")w}2 }/ v2 b

C ">)*!# ^ !8-P%!#a; X6- 3%!##-. , !# 1 V (−r) = V (r)3 %!8. !#!# 2 Q N 4 + 6-% O 3 2 9Y! 7 ;V 9O7 :`!# 0/? 7 M ] % ; 6"U8$!8- 2 8';# !#6- 2 ! 2 X-!#N6) E # ;# - QW %!8 % T T,N 5\Q >76\ 2 !#6X+, 2 0 # + O7I$ % E 8 )/ SA 83!8- E 6 * + 7 @ -8 -. ^ E > k;fF EDG 5fF^] 5fF2< c B GME < RE=T Z 5 G 5fF G P E B i GIK[EDG \ > REDGWHJG @(B E BDF EDGIHJGtK 4MB=NO> E BUP GWKoEDG k E <W: EDG G R > K >V6lP_>fF25PB Z d E > R&H B> i 9<` 6N ? 7 ,;% ^ G 5 EDG 4 EDG > 5 G 5fF G P E B9> 4I5[> H K < P_4[6lP_> F25P :=>fF EDT \ 7 > ^BDF^]tk?< K <M:l8 R BU\w> R <tp;n 8;:lBDF T 4?< P 5 G[ZIR < E > E B>w:=>fF^C EDG 5fF^B_n x(y{z|y{}~yI& !8- /- "% ;*!#. !# # '; X6-.Q

f+(*,+ #)G #QVG %

('

2 E ^ ; X6- 9 0 7=? ? !#6 Q4&("XQa"% E,A V (x) =

(

− a2 < x < + a2 ;

0,

x 6 − a2 , x > + a2 .

+∞,

b ; .W!8-8 763. L9 0 7 : ? 4 ,N 5TQ>

! + E a!8- 2

Ψ(± a/2) = 0.

76

:

:=>fF EDT Z T Q 9 0 7 ? EF6

! 2 "%%!#%!# A Ψ(+) (x) = A(+) cos(kx),

+ k ;8 - 2 ! 2 9 0 7 05? 7M( (+ E 8 L!8-# !c 5T ;!#EF,) * 2 k A

4 ,N 5TQ> "*

kn(+) =

π (2n + 1); a

76

:

9 0 7@m9 ?

9 0 @7 m,? ? !8-Q 9 0 @7 m9 ? " 9 0 7@m,? ?

n = 0, 1, . . .

E >f:=>fF E=T Z T Q 9 0 7 ? EF6 6 ! 2 !8%!8% A Ψ(+) (x) = A(−) sin(kx),

9 0 @7 m 1 ?

9 0 7@m1: ? 9 0 7@m [?

9 0 7@m,m ? b " N) ^ E ;Q8 L" ) *!8%T )/-.6649 0 7 0 : ? ^ ;- W;V 0 7 7 ¤ kn(−) =

2πn ; a

n = 1, 2 . . .

Z

9

M )*"%$!#;8O 2 %!# %!8N']*&,&( "$; ( 76T aN- ,! -* LE 8 )*/ * 7

f+(*,+ #)G #!G ! R %2fFv@B E B=F EDG > K 4MB=NO> E B> 4`5fBQ\`C \w> F R BD: E=GMi R Pl\ G 8 HJG 69] E=GMi G F2> E=c B< 69] EDGYi Pl\w> H 698 X B EDT V B %^B R B EDT a n ) < i F^B d E > R&H BDBe5fF2< c B GYE < R9EDT Z 5 G 5fF G P E B i := * !8;-.) %6. O N 0

) * 2 ),; 0 7 1 7 R! - *6- "% W;%!#.HH!8 # ; X/-.Q 2 E ^ $;X/!#6 Q`&(%"X Qa"% E,A V (x) =

(

0, V0 ,

− a2

<x<

+ a2 ;

V(x)

E

-a 2

9 0 7 0*55?

II V0

I

III 0

a 2

x

&UY=o oo

x 6 − a2 , x > + a2 .

M ]% L 76 6. % 9 0 7 0/? ! ( ; X/-%_9 0 7 0*55? a+ E a%!8- 2 Ψ(±∞) = 0.

"a)EF%\) N% c 3; X6-aTHN- !# 9<!7 0 7 00/? 7 4 "%,!8-# !# !# %!#6P+ 6.%! 2 -."%PN-!# 2 I "%Qa) EF% x = + a 7 II GIK[EDGMi I N- !# II 7 6T 2 20 # + 9 0 7 0/? S! 2 EF6 3* 9 0 7 0/? 7I N- !# I + :=>fF EDT > 6T 2 c 9 0 @7 m,? ? ^ E >f:=>fF E=T > 9 0 =7 m [? 7 @ 6\%6 ] U8H 6T Q` ;QH"\) EF; " !8- ! c 5\'!#XO E %6 2 - 2 k A

a

(+)

kn(+) a

}kn ; = π(2n + 1) − 2 arcsin √ 2mV0 (−)

G

}kn = 2π(n + 1) − 2 arcsin √ ; 2mV0 >/ ;.HNB8 .A

8

l +

kn(−) a

}kn kn a = π(n + 1) − 2 arcsin √ ; 2mV0

n = 0, 1, . . .

n = 0, 1, . . .

9 0 7 0*5O0/?

8! 6XE 8 !#!# 2 Q EF! - 2 c! 2 ; "% 2 2 9 0 7 050/? H!# !# !>9 0 7 9 ? 7SU "%%O _!8. % >9 0 7 0*50*? ! ;-% E )*/ * 1 7

Z

?

I 6\* ; H!#;-.) 6>)"% !#A8 2 %!# ; !5\*!#HH% ^ 4; ! 6T% * LH76 .`!#!#* ) K 4W8 Z % Q ^ "O"a ; 0 7 1^ T :=>fF TR > Z 9Y! 7; 0 7 [? 7 M(8 -+`! %!# 2 -. ;%66-) 6. ] + *!#"Q!#; " ; 6-+a!(; % 0 7 1 7 ¤ ^ !# !#$!W; 2 Q$ !8-'!#XO-- 2 XQ *E O 6X Q Q ^ %!8. !#!# 2 Q %!8 + N ;# # - 2 .! 2 %!8,. c "%6+ 3!8- n A 9 0 7 0*5? Ψn (−x) = (−1)n Ψn (x). M(]% R B5 G ZIR < E > E BDB :=>fF EDG 5fF^B U6 G F EDG 5 F ]e4I> RG P9F EDG 5fF^B 4I5[> H K < X 8;C K >fF :=>fF E=GMi @8 E g c B> i g GYG R&K B E <WF 7OI !# !#W! %!#XO-- 2 XQL *Q ; !#; 8O $ !# G 5 E=G 4 EDG > 5 G 5fF G P E B9>A4I5[> H K < X 8 K > F :=>fF^C EDT \ 7 G() 9 0 7 0*5? / )*6"\- c._6"O 3 ^ _; !#A 86 %!# !## \) \"% E N *- E 9< !8;/ W!,X !# ? 7 a EFQ` * 7 6\ 2 ; F G 69];g GtK 6lP 5[4 Pk?< E=EDT Z 5 G 5fF G P E B i 7 I !#%!# 2 2 8 ! ;8 - E ;*-.%!#% %!#. E > 5 G[ZIR < E P_>fF25P 7 M( ] % ^ ; ^ S/ 3 + EF !8- 2 ",)*/ * ! !# 2 2 ; XO 6-.% N%. ("O) EF c! 2 E a%!8-. !#B7

V"#v&2a*2"!D

"%`"/%# _'

#!Gh,!#X ! !#E

m

#

O 3 ! 2 ; X6-.Q 2 V (x) =

8

,QW] + >!#6XE

(

+∞, 1 2

mω 2 x2 ,

!#!# 2 Q

x 6 0;

!# !#!c \ 5 -EF x > 0.

&(%"XB7 #U3G"h,!#X3 ! !8E m 8* ! 2 δ N6) Q ; X6-.Q 2 V (x) = 7 ,Qa] + 4!# 2 )*6E 8>!#6X6E 8>!#%!# 2 Q 7 −Ωδ(x) 9 F24?>fF AO# !# EFQa. E0 = − mΩ22 7 ? # ∗ GU7 6T. )** S(N %!8XO-- 2 ! 2}4)"% !#A 8 2 %!# 7 1 gU<[k?< E B> A, EF 7 6\ 2 % 2 9 0 7 : Z? !#"O. ψ+ (ξ) = 2 ^ EF 2 9 0 7 : Z? !#*!# " f+ (ξ ) EF/ 3 % + ; + Vψ−(ξ) *!#"%%= ξf9YK − (ξ 7 0/? 7 )3

Z

:

h

i d J D

* *

A

2 2 u

$#

0 # + 2 - 2 ! 2 %6 * +,; 2 "OP !8 E 8P;) * E 8 7 R! -3&() *!8" 2 !8!#* FN-/ S6 +T!#; 2 !#N E ^ ,)/ S 8/

2 !#N%!8E 8T&("XQ3L!#N%!# E 8 ) 6 Q4!#66! 2 + )# 3N-(* *"%Q>;L!#6 Uc ! % E !8-* 7 4 "%F6- +O-./R;8 - E 8P!#Q!# !# 6 ;) - 2 P 2 !8- P!#! 5\*!# N- + .(7 6T % 2 7

H/ T ,"O)*6.! 2 ^ S 2 )* (; E 8 ;# ,"L \c !# E 8 ;)* E 8 ! R <[k K >V6lP9u3 ^BU\.B5P > R >V\ > E=E=T \.B )*/ *!#N!# E 83&("XQ3!#N%!8 E 8,) Q3!#;/ S6 ! 2 ;3] %`!#"%-." )*/ H!, ,. 6T'!8-% ; * E 8 ^ /7 7N- (;%!# E 8 7 b "O 2 !#X 2 *!#;\! !8-# %( O 3 2 !#XE' !8 !#E µ X 6-. !# % ;- ^ "%+ \;X6-. 2 ] + 2 )*6 !#WW! !# 2 2 a!#O-+ LX r = px2 + y2 + z2 ^ TW/ ;- 2 / !8 " r 1 U (r) = U (|r|) = U (r)3 7OI ] % !8- !#6XO 0 8 + 9 7 0/? ˆ HΨ(r) = EΨ(r) ) )* !&( !#"%Q !# ; X6- N 7 6 6. a! &( *!8"Q !#!#*,"% 6L! +O-.6% +

2 ˆ2 p ˆ L r ˆ = H + + U (r), 2µ 2µr2

¶ µ 1 ∂ 1 ∂ pˆr = −i} , r = −i} + r ∂r ∂r r ½ ¾ µ ¶ 2 2 ∂ 1 ∂ 1 ∂ ˆ = −}2 L = −}2 ∇2θ,ϕ sin θ + 2 2 sin θ ∂θ ∂θ sin θ ∂ϕ

9 7 ? 9 7 Z? 9 7 1?

Z[

M( ; EF %

"/ 6WN66-.+ >V 6 9<; ^ Lˆ = [r, pˆ ]? 7 ! - +#* E 83 +O-./T9 7 ? 2 - 2 c ! 2 ; % " *!#"%QL] + ∇2 9 7 1? + - 2 !#. ; < ;-! T! &( *!#"%Q'!#!#* "%θ,ϕ *7M(%!#"%-." Lˆ 2 Q!# 4-\6\. H+ -EF ; EF ^ H"%% !,+#O-./6% 9 7 ?8^ ) ^ L2 2 - 2 ! 2 + 6- O T 2 7 G( + 6-% O T 2 4X 6-.% ;- ^ "O" -+ "% O . ^B2 - 2 ! 2 ;"X 2 N66-.+ `% 6`4E - 6;- 9<%!#. Oz ? 7 @ !#! c 5\Q ; WL! &(*!8"Q !#!#*,"% 6 A 9 =7 9 ?

ˆ z = −i} ∂ . L ∂ϕ

^ L2 L )* E !#*!# ^ ;!#"%-." U83; E "%% %1c/7 b " Nz)% ^ !#!8* ^ ;!8EF% 2 +-.6% 9 7 ? 9<*7 7!#X X 6-.%4;- ? ^ * 3 8* .%! 2 ,!#%!8 2 2 8T! ;# -QH] + Q ^ ;# - E $) * *$"%/ 63N66-.+ L 64 ;# #- E _) * ; "X'V 64HE - L/ ;- 7 IR-EF4&("X ] Q8 !#%!# 2 Q 2 - 2 c! 2 * !#N!# E H&("X 2 H!# 8W 8 jE 6T ; !8- E 8W; 67 @ N%!# EF&("XWa!8N%!# EF ) 2 ; Lˆ 2 Lˆ z + %NEF.Q E_)P 6T 2 Q ˆ 2 Ψ(θ, ϕ) = L2 Ψ(θ, ϕ); L

ˆ z Ψ(θ, ϕ) = Lz Ψ(θ, ϕ) L

# Q ! &(7 @ N%!# E 4) 2 ; 6 9 7 1? 2 - 2 c! 2 L2 = }2 l(l + 1),

l = 0, 1, 2, . . . ;

Lz = }m,

m = 0, ±1, . . .

9 @7 ? ? 9 7=9 ?

G _!#!#1cH! &( *!#"L&(%"X Y (θ, ϕ) 9Y! 7 ;O-* 3 I 7 ? 7 , G( "O! l )EF%c N66-.E ^ m + lmE a"%E P! - 7 b " "O",-% 2 &(%"X 2 !86X6 E 8,!8%!# 2 Q,!#XE ! ;# # - E ) 2 V L2 ^ L ;) -.%W;- ! &(*!8"Q !# - >* * aNEFz. !#N%!# Q &("XQ ; Lˆ 2 Lˆ ^ 4 2 9 =7 ? ? N 4% 6!#" * - 2 .%! 2 ^ *! - 7 6\ z %

2 0 8 + !(+#O-./_9 7 ? !#"O. A Ψ(r) = ΨElm (r, θ, ϕ) = fEl (r)Ylm (θ, ϕ),

9 7: ?

Z

m

+ f (r) / 6-. 2 - 2 &(%"X 2 ^ "% 2 ;# #- 2 ! 2 % ; ElX/-.Q ] + U (r) )*!# ) * Q ;-Q ] + E N66-.+ `% l 7VM(%!8"-." `;- ! &( *!#"%Q !8 ' E - E 8 ;- Q' ;%!#!# ^ / 6-. 2 &(%"X 2 f (r) N )*!# . ') * 2 + + '"+ ! - m 7FM(# !#6- Ψ(r) >9 7 : ? ;) - 2 L)# -.W; * EF r (θ, ϕ) % 0 # + >9 7 0/? 7M(* !#6- 2O2 9 7 : ? H9 7 0/? EF% 2 9 7 ? 9 7 1%?8^ - 2 &(%"XQ f (r) ;-* A El

}2 1 d − 2µ r2 dr

µ

d r2 dr

¶

·

¸ }2 l(l + 1) fEl (r) + U (r) + fEl (r) = EfEl (r). 2µr2

9 7 [?

n E !8-* - 2 76\ 2 %6 2 9 7 [? 2 - 2 ! 2 N% "% + 3) * 2 f (r) N-!# r 5 ∞ 7 4 NEF f (r) ;# !#6- 2 cTT f (r) = 1 R (r) ^ 3;* \" El

] "%6- ElQH&(P%6 2 9 7 [? A

El

r

El

· ¸ }2 l(l + 1) }2 d 2 REl (r) + U (r) + − REl (r) = EREl (r) 2µ dr2 2µr2

9 7@m ?

! N 2 )*6-.E + 6E %! -* R (0) = 0 7 l )E W %6 ! 2 / 6-.E * 0 # El+ 7 M( !# Q !8"% 9 7@m ? !#;* S6 ! % E * 0 # + L! ; X6-% }2 l(l + 1) Ueff (r) = U (r) + . 2µr2

9 7 0*5?

N6) ^ "%% 2 8%"O ^ ; N"O- ! !#*!#"%Q ^ ;) - 2 X 6-.%$;-T!#*!#4 8 1c )/ SW"4 % Q> N%- * X N +\!8- 6+*+ }2 l(l + 1) 7%M(L] 0 8 + ) "O 3X-EFQH X#-2µr .EF2Qa;6 l 7 , N!#"%%` 6- ,!#-+ X(+ !8- ;# - 2 ! 2 "" Qa;!#6"%Q )*/ * 7 b " ^ - 2 !# 2 ) E 8>!8%!# 2 Q 7 M( ]% &(%"X f (r) R (r) ^ fEl (r)|r→∞ → 0 REl (r)|r→∞ → 0 El El !8-# V%. 3# X !8- 2

b "

Z

∞ 0

2 fEl (r)r2 dr

= 1;

Z

∞ 0

2 REl (r) dr = 1.

9 7 00/?

M %!#"%-."\;W&("O!#%% l 6+ "% !8- m * 3 ( ; . ) * 2 m = 0, ±1, . . . , ±l 9 %!# + 2l + 1 ) * ? ^ "O6 3 ,)

15 #! 6XE 8 !#%!8 2 Q ! ;# - E ) * * l N 2l + 1 " EF/ 3 7 @ %!# 2 2 ^ %! 2 5\*! 2 " ) E ) * 2 l = 0, 1, 2, . . . ^ ; 2 N) 6. !8 !# !8; "%!8";*!8"3!#- s ^ p ^ ^ ^ ( 6- 6-&,%W; 2 "%7 b " ^ ; ^ !#!# 2 2 !V*- E d f g N66-.E '% % 9 l = 0? ) EF% c s!#!# 2 2 ^ !8%!# 2 2 ! l = !#!# 2 2 >H*7 :7 1 p 4 ; nOO-./ 9 7 ? "%% % ! ; %`;%!#!# O Q !8 Iˆ 1 IΨ(r) ^ 7 c 5T` %(!#N!# E 8 ) 2 ±1 7 ˆ = Ψ(−r)3 I !# 2 ) !T] _!#6XEF!8%!# 2 2 ! ! %* E 8 !8!#*_+ % NEF. )8 - E $ EF EF7M(_;X_ !# "% 6 r ` 2 ! 2 ^ %+ -EFa; * EFH; N) 1 c! 2 ; )*"% ^ ^ ;] % θ →π−θ ϕ=ϕ+π ˆ Elm (r, θ, ϕ) = fEl (r)IY ˆ lm (θ, ϕ) = fEl (r)Ylm (π − θ, ϕ + π) = IΨ

9 7 0/? G() 9 7 0/? !8-# ^ -EFH&("X ! -%aX 6-.E 8$;- Q 2 - 2 c! 2 !#N%!# E &(%"X 2 ;W !# 7BI !#L!#%!# 2 2 !> E l %! 2 ! 2 " E !#!# 2 2 ^ !#%!# 2 2 ! E l 2 - 2 c! 2 E `!#%!# 2 2 7 U ) N * *!#"%-."3; 7 f+(*,+`BG #)G <-DB5<WF^] G 9> R <MF GR g?B E >fF^BD:=> 5 g GMi d E > R&H BDB 4e5V@ > R BD:=>[5 g GMi = (−1)l fEl (r)Ylm (θ, ϕ) = (−1)l ΨElm (r, θ, ϕ).

f5 B95fF2>V\ > g MG GR&K B E <MF n x(y{z|y{}~yI ; T 4 "OE

\6

8

%" 6A8 *

*!# ^ !#;8 - - 2 ;) -.E 8W "E 8W; 6 Aˆ Bˆ A X 9 7 0Z? ˆ × B) ˆ 2= ˆ (Aˆ B ˆ − Aˆ B ˆ ) (j, k = x, y, z). Aˆ B (A j

k

j

k

k

j

jk

M( S!#6- 2O2 9 7 0*Z? %*!# Aˆ Bˆ !8 !# P;E r p L;O 2O2 ;W"%% %6X !#,6T [r , pˆ ] = i}δ 9<; -. j k jk ! !# 2 -. ?8^ ;-* A 9 7 0 1? ˆ 2 = [r × p]2 = r 2 p2 − (rp)2 + i}(rp). L

M %!8"-." r = (x2 + y2 + z2 ) 1/2 ^ ( % 2 "%% %6X !#,6T ;-% A

∂ ∂ ∂ ^ ∂ +y +z = r ∂x ∂y9 "O∂z )*6.L!∂r%!# 2 [r, pˆr ] = i} x

(rp) = −i}r

∂ = rpˆr + i}, ∂r

^ E -. ? ^

10 + P;

;# -a`9 7 Z?

9 70 9 ? C," * L>O .>)49 7 0 9 ? ^ ; 2 !#. 9 7 0 1? 6 r2 (p2 − pˆ2 ) 7SU ) r -N(!# 9 7 0 1? r2 ^ ;- pˆr

(rp)2 − i}(rp) ≡ ((rp) − i})(rp) = rpˆr (rpˆr + i}) = r2 pˆ2r .

ˆ 2 /r2 , p2 = pˆ2r + L

"*

a

2 2 ˆ2 p p ˆ L r Tˆ = = + . 2µ 2µ 2µr2

EFQ, ) *-.66FNE -,!#;-.) 6(T9 7 ? ;,;%!8 P+O-./ W! &(*!8"Q4!#!8* "% /7 ,;% ^ Lˆ 2 W! &( *!8"Q !#!#*,"% 6 ;# a`9 7 1? 7 ¤

f+(*,+YBG BG )

< i F^B^d E > R_H BDB f5 F2< c B GME < REDT Z 5 G 5fF G P E B i m d E > R&H >fF^B=:=> 5 g;B i 59>fg;F R r`BA5 MG G F24I>fF25fF24W8?u3 B9> UB \ 4 G 6 EDG 4 T >$@8 E g c B=B K 6lP RG 5fF R < E 5fF24?> E C EDGWHJGRG F2<MF G R E F G \ B E > Rc BDB I n x(y{z|y{}~yI* U63 -." R 6-.EF !# ; !#N E ^ "%

5

EFR N 8%" ) %6. %+ - θ ^ ϕ ! &( *!#"%Q\!#!#*E "% /7 a O T 6 2 - 2 ! 2 !#N E ^ ;] %`+-.6 * T;!#.T; 6-+ W! +#O-./ !#N + ;%!8%;-.+ T 2 ! )* Q p Lˆ 4 ! !#E µ I A ˆ2 L ˆ = H . 2I

@ ] +#-.66% !#6X6T% 0 # + - 2 66 3; 6 O 1 ˆ2 L Ψ(θ, ϕ) = EΨ(θ, ϕ) 2J

O-

ˆ 2 Ψ(θ, ϕ) = 2IEΨ(θ, ϕ). L

$9 7 0 ? ?T2 - 2 ! 2 %6 * - 2 !8N%!# E 8 !#N!# E 8 ) Q ; 6 Lˆ 2 ^ "%EFP') *!#E,A 2IEl = }2 l(l + 1),

78(*-*2 2 8(, 8 :1 , 1 1 ( , 1 --2', 7-,*--, 2-,7 , , *-

l = 0, 1, . . . ;

2IE

9 70 ? ? &(%"XQ

E 2.7- , 5-1

! 7 , *- -1 ,

1% Ψlm (θ, ϕ) = Ylm (θ, ϕ),

b "$N)

m = 0, ±1, . . . , ±l.

^ "O6

% ] + *!8" %\c 6 }2 El = l(l + 1) 2I

!# !81c !#%!# 2 2 ^ "%EFH)*/ c! 2 !&( *!#" &(%"X 2 9% . E EF/

>! "6 Ylm (θ, ϕ) %!#,. c 2l + 1 7 4 !#P!#%!# 2 L9 l = 0? 2 - 2 ! 2 lEF/ 3 E 7 ¤

f+(*,+ BG V G !
%

5 G I4 > R 2fF 5[4 GMXMGWKoEDG > K 4MB=NO> E B>`5 G R >fC K >V6_> EDEDT \ k E <M:=> E B9>7\OgU4I< KVR <MF|< G RX BDF2<[69] EDGIHJG \ G \w> E F2 EDGR \.BDC RG 4?< EDEDT > 4 G 6 EDG 4 T >.@(8 E g c BDB 5 F|< c B GYE < REDT Z 5 G 5 F G P E B i 5d E > R&H B> i E n x(y{z|y{}~yI @ N O 3 !# T!#- 2 ! 2 T;-(! ;%!# 2 E 9 - 2 µ

)

!5

;8 - !# *- E ? ; X6-% ^ "%EFQ 2 - 2 ! 2 !#E !8-* ;%!# 2 + T;- 2 7( M ] % %!#636Qa + *- 2 *- 7 6T P% 2 9 @7 m ? A R00 −

9 70 : ?

l(l + 1) R + k 2 R = 0. 2 r

*!8. k2 = 2µE/}2 7 G,!#;-.) 6T; S!86" R(r) = √rZ(r) !T# * Q ;*Q x = kr ;* 4% 9 7 0 : ? " % Uc - 2 &(%"XQ> *! !#- 2 9 a 7 0*5? !(;-X-E '; % A x2

2

"

µ

d Zl dZl 1 2 + x + x − l + dx2 dx 2

¶2 #

Zl = 0.

+ + *- 2 ,L*- 6T EF6

! 2 ! &( *!#"%Q4&(%"X Qp>R*! !#- 2 9 a 7 00/? 7 b " N) ^ %6!c -1c &(%"X\c !#6X + 4!#!# 2 2 !#N + O T 2 !3;8 - E ) * * "%/ 6 N66-.+ % 6L/ ;# S!86. !8-# ! c 5T'N) A 9 7 0*[? Ψklm (r) = jl (kr)Ylm (θ, ϕ). I ; 1 =7 93!8 0 NE -4;-% E -EF &("X !86X6 E 8 !#N + T 2 !;# #- E ;*-.%!#% 7 4 - U6 G 5fC g;B Z -L>!8-. -6-!#. 9 7 0*[? 7M(P6"+ )#- 2 \% ^ p L2 )* E !#%!#7IR-EF\&("X$9 7 0*[? + S/ ) EF c! * 2 5 @ > R BD:=> 5fg;BQ\`B -B7 ¤

1Z M -!#"O 2 - ;%!8"6 )-/ T \;>! &( *!#" - 9 7 0*[? 7 M(# * N )PEF* 3!#!#1c 5T!c &(*- A X 9 70 m ? il j (kr) Y ∗ (θ , ϕ ) Y (θ, ϕ), eikr = 4π l

lm

k

k

lm

+ + -E θ ^ ϕ )*/ c,;- F " k T! &( *!#"%Q !#!#* "% /7\o kk/- 9 7 0 m ? "%3O--\c(!# !#*!#1c_ )* !#. AV!#!# 2 ! G R > K >V6_> EDEDT \ ;/-.%!# !#. !#%;;) X 2 !#%!# p L2 2 Q ! R
L&%% 2&%%2&%% '

I 6 Q\6T QT)*/ SQ3 X 6-.+(;- 2T2 - 2 ! 2 )** S O

T )* 2 3Q !#XE "/-%!8" ;- ; 2 T 2 U (r) = Ze2 9<* ; NEFQ 6% ? 7 @ 6XH 0 8 + − ;a]r %' O :A ·

9 7 5?

¸ }2 2 Ze2 − ∇ − Ψ(r, θ, ϕ) = EΨ(r, θ, ϕ). 2µ r

!8-W! ! %. * ; NEFQ ^ \;* ! !#Q µ , 9 7 5? !8-# ; . ;# !c ! !# µ = me mp ]-" + (2 9 m /m ≈ 1836 ^ `;] µ ≈ m m?8^ e;+ mrp U8H%!#O -. ! !# 2 ^ e ;* p E ] +\c %!##-.e+ W T 2 7( ! ! ;T]% O T ! 2 !#N ^ ;%P + O 3 ;# !#6- 2 !# Q! *! 7 , !7 7 0 ;"6)*4; EFQ ]*&,&( "+ ; X6- 9 7 0*5? / 6-.+ %6 2 0 # + 9 @7 m ? >"*-%!#"% ;- ; 2 3 2 7 M( E < 0 O T `&( ^ /7 "B7 ]- " A 8 ! 2 ; O X6-.Q 2 8^ N) 6QL) !86 c 5T "*-!#" ; X6-% "%/ NEF% c 5T X N# LE '/-"* 7 R! -7 6T 2 2 9 7 5? !#"O.PPO 9 7 : ? ^ ;* EFF)8 - 2 c! 2 E';-*4!8-8 1 c 5T !#N%!# EF &(%"X ^ %EF !8-*_9 7 00/? A

fnl (r) = Nnl

µ

2Zr na0

¶l

µ

Zr exp − na0

¶

1 F1 (−n

** (-*8 $ /$ 1,,42-12! /:: ' : ,,*-*-*-8) ++ 9 *-, 1 2, 2*-1

+ l + 1; 2l + 2; 2Zr/na0 );

9 7 O0/?

/7 !: 7 , 18( 1-, *%08(, 1 2-

11 4!#N%!8EF() * 2 ] + A En = −

+

a0 =

}2 µe2

N!#"Q>* %! 9 - 2 ]-" Nnl =

µ

2Z na0

¶ 3/2

9 7 ?

1 Z 2 e2 , 2 n2 a 0

1 (2l + 1)!

s

a0 = 0.529

?

(n + l)! 2n(n − l − 1)!

&(%"X2 9Y! 7 ;EFQW-/ T* L-K,. 7 ? F ; # EF-*+

! 2 %!#+; 2 +- .% EF6*!#) "O2. 9 7 0/? )\;-%E < + ` 9 ! 7 ;O-/ T &n 7 ? n = 1, 2, . . . H 6_ "%6 ! -7*C,",O )(9 7 ? ^ ) * 2 ] + ;# - 2 -."%(] "%E 4!8-%` E > k; E F|< 7 b "%FEF/ 3 ;,N66-.% "%6\!8- l 9 !8-Q E / 3 ? N!8-- !#; X&("%Q4"*-%!8"+L;- 2 4WX 6-.E 8 ; X/- 8 KVR 8 HJGIHJG 4WB K < E > 4?5 F R >f:= F25 P 7 1 1

M @BDgU5fB RG 4I< E=EDG \ n !8- l ( ; ,) * 2 l = 0, 1, . . . , n − 1 9<;% ^ ; @BDgU5fB RG 4I< E=EDG \ 6+ _"% !8- m = l ^ *7 7 (2l + 1) ) * ? 7 0, ±1, . . . , ±1 b " N) ^ "6%!#.aEF/ 3 # 2 % 2 E n

gn =

n−1 X

(2l + 1) = n2 .

9 7 Z?

&UY#Wo @ ;" 9 7 ? + ) EF! 2 U O N + %!8" 7 h(!8- + %Q N*!8" 7 4 + 6% c 5T + !#1 8 *- ) _] + ] + 2 - 2 ! 2 F G :lg GMi 5 H 8 |> E BQP % Q 7 U / 6-.EF -EF &("X49 7 O0/? ^ "6" T - (NEF.PPX 6-. % ;- ^ )*! 2 3 + + T"%+ !8- m 7 4 [8"O ) 1 c! 2 + -E n N66-.E l "%E ! - ; ! 8V nl ^ ;' - 2 l / !8;-.) %6.N%" EFQ4!#; "%!#"%;*!#"Q`!#-B7 b " ^ 1s !#%!# 2 ) * 3!#%!# 2 ,! n = 1 l = 0 >/7 :7M(%!8"-." En 6-.; n = 1 ^ %!8E 2 - 2 ! 2 1s !#%!# 2 7 l=0

1

9

U ) N * *!#"%-."3; 7 f+(*,+ BG G s^5 G 69]WkW8MP 9 7 0/:? G 698;:lBDF^] _P 4 DE Ti W4 B K 4 G 6 EDG 4 GMi @ 8 E g;C c =B B GYE < 5 G 5fF

G 5 EDG 4 EDGWHJG 5 G 5 F G P E QB Ph4 GIKoGRGIKoG GIKoGWXMEDGWHJG M< F G \w K B E B c 8Qn T DB95 <WF^] G 6 E 8;u 4 G 6 EDG 4M8;u @ 8 E g c BDu 1s C G P E BUP&n

x(y{z|y{}~yIG()W9 7 0/? ! -# ^

b

¶ 3/2 ¶ µ Z Zr 7 + S f10 (r) = 2 exp − a0 a0 ¶ µ Z Z ∞ Z 1 ∞ 2 −t 2Zr 4Z 3 ∞ 2 2 2 dr = r exp − f10 (r)r dr = 3 t e dt = 1, a a 2 0 0 0 0 } | 0 {z µ

` 8 * ! 2 #! + -!# !`9 7 00/? 9<;* -!#.H)* a; *E 8 t = 2Z 2Z ? 7 r; dt = dr a0 M(%!#"%-.a"0 !#%!# 2 ^ m 6" TT *-cT7SM(- 2 - s l=0 % 2 &("X 2 8 ! 2 !# !8H!L9 7 : ? A 2!

Ψ1s (r) = Ψ1s (r, θ, ϕ) =

+ P ^

s

1 7 Y00 (θ, ϕ) = √ 4π

f+(*,+ BG 3G )

µ ¶ Z3 Zr exp − , πa30 a0

¤

%

R9G P9F E > i 2>[>h8 K <[6_> E B>,d{6_>fg;F RGYE < 4 5 G 5fF G P E B=B G F P VK R < 5k?< R P K[G \ Ze n x(y{z|y{}~yI

0 #! ;%!#N 7 @ + -!#H9 7 : ? ^ ;- 2 - 2 &("X 2

9 7 61?

1s

1s

C

!#%!# 2 2

Ψ1s (r) = f10 (r)Y00 (θ, ϕ),

+ / 6-. 2 &("X 2 f (r) ;# - F39 7 0/? 76M %!#"%-." Y (θ, ϕ) = 10 00 # ! % 8 ! ) ; / 7 7 ! ( & # ! " '! # ' ` 1 ^ 2 ^ ^ √ 1s )*4π ! 2 ^ *!8-`;W-!#a+ ->;!#4 + 7 GP!#; -.) 2 !8- 2 " 9 I @7 ? ?8^ 9 7 00/? ^ ;-* A Z

2 3

|Ψ1s (r)| d r =

Z

∞ 2

r dr 0

Z

2

|Ψ1s (r)| dΩ =

Z

∞ 0

f 2 (r)r2 dr = 1. | 10{z } w10 (r)

9 7 9?

1

?

9 7 9 ? N%! -- * ^ (! &( *!#"%Q\!#!##

"% ]-* aN * d3 r = r2 dr sin θ dθ dϕ 7BM(%!#"%-."H"%/ *- 2 -Q &("X 4 2 %!#EFQ !E !8- ^ N8 L&(O + %Q\!8"N"%QPT9 7 9 ? !8-# F!! %.F"6",;-%!#. 2 %!8 8/

2 ]-"3 ! &(!#"$!8-P/ ! r \- 5TQ dr A 9 7 ?? 2 2 w10 (r) = f10 (r)r2 = R10 (r) 1 ;V ^ 7G()W9 7 0/? !8-# ^ R(r) = r−1 f (r)3 ,6- / - 2

r2

µ

2Zr w10 (r) ∼ r2 exp − a0

N) ^ )** S )*"O-c ! 2 ;!#"\ "O!# &(%"X w (r) 7 n &("4&(%"X w10(r) ;# !#6-10 a !7 7 7 `"O!# 6-. ) * w ^ "O"L*!8-/ ,;"6)*. ^ %!8+! 2 ;10 7 l !#. 6 2 \Q 6\ a0 * r0 = 6-Z,]- " !#O-+ X 7 I % H6 2 ! 2 N !#"%% / !#7

b "

¶

.

##

&UY I W

!#;%!#N 7 R! - %*!# &(%"X !8;-.) %6. R (r) = r−1 f (r) ^ &(%"O f10 (r) X 2 R (r) N* - 2 . * %% 10 \c 10 0 8 + 9 7=m ? 7BG,10! 8* 2 ) 2 %!#+ a! E !8- -Q &("X ^ ;Q8* " 9 7 ? ? a;-* T\! EFQ> /7 ^ 2 %!8. N6 3 2 ]- " ;) -. 6-PFX 1s !#%!8 2 P!# + 2 - 2 ! 2 /- QB7 4 6" ; N-,. 6\U8 r ,. 6 6 ! 2 ] "O!#; X6-. ^ 6" >6-.EFQ )* % TN 3" E 7 ¤

f+(*,+$BG G )

< i F^B2P_4 EDTi 4WB K G 6 EDT Z 4 G 6 E=G 4 T Z @8 E g c B i 5 G 5fF G P E B i 5 G k E <M:=> E B9>V\ H 6_ K B E B c 8Qn x(y{z|y{}~yI M( ! 241 )-E H!#%!8 2 2 A n = 2 ^ l = 0 ^ m = 0 n=2 9 9 9 ? ^ ^ ? ^ ^ ? ^ ^

8

Ψ200

n = 2 l = 1 m = 1 Ψ211

n = 2 l = 1 m = 0 Ψ210

n=2 l=1

1 9

m = −1 Ψ21−1

:

? 7 ,3!#%6 9 7 : ? A

+ ^ !8+ -!#H9 7 O0/? ^

Ψ200 (r) = f20 (r)Y00 (θ, ϕ)  (2s) Ψ211 (r) = f21 (r)Y11 (θ, ϕ)    (2p) Ψ210 (r) = f21 (r)Y10 (θ, ϕ)   Ψ (r) = f (r)Y (θ, ϕ) 21−1

21

9 7 :?

1−1

µ ¶ 3/2 ¶µ ¶ Zr Zr Z exp − 1− ; a0 2a0 2a0 µ ¶ 3/2 µ ¶µ ¶ 1 Z Zr Zr f21 (r) = √ exp − . 2a0 a0 2 6 a0 1 f20 (r) = √ 2

µ

N F( ^ ;%!8-# FW&("X-c! 2 -."% ! &( *!8"`&(%"X 2 ^ %E > # X ^ !#+ - !#H9YI 7@? ? 7 M("O6 T* ; . ^ a / 6-.EF\!# 9 7 : ? 6" T %E # X7G,!#;-.) 2 9 7 : ? ^ ;-%6* A

4

Z

µ ¶2 ¶ Z µ 1 Z3 ∞ Zr Zr = 1− exp − r2 dr = 3 2 a0 0 2a0 a0 0 µ ¶ µ ¶ Z Z Zr 1 Z3 ∞ Z2 2 2 1 − r + 2 r r exp − = dr = 2 a30 0 a0 4a0 a0 ¶ ¶ µ µ Z 1 1 1 2 −t 1 ∞2 2! − 3! + · 4! = 1; t 1 − t + t e dt = = 2 0 4 2 4 µ ¶ Z ∞ Z ∞ 5 Z ∞ 1 Z Zr 1 2 2 4 f21 (r)r dr = r exp − dr = t4 e−t dt = 1. 5 24 a0 0 a0 24 0 0 {z } | ∞

2 f20 (r)r2 dr

I NU8 !8- 2 W 8 ;* -!#.T)* 3; * E 8

7 Zr =t a0

4!

¤

f+(*,+ B G X G ) > G 5 R > K 5fF24?> EDE=T \ R fF G \ G g / \ c T A Ψ (r) 100

200

I=

Z

Ψ∗200 (r, θ, ϕ)Ψ100 (r, θ, ϕ)r 2 dr sin θ dθ dϕ

=

Z

∞

f20 (r)f10 (r)r2 dr = 0. 0

1[ G,!#;-.) 2 9 7 61? ^ 9 7 : ? Z

- 2 )* ;* E 8

3Z ^ r=t 2a0

;-* A

¶ µ ¶ µ √ Z3 Z ∞ 3Zr Zr exp − r2 dr = 1− f20 (r)f10 (r)r dr = 2 3 a0 0 2a0 2a0 µ µ ¶ ¶ Z ∞ 8√ 1 1 8√ 2 −t = 2 2 2! − · 6! = 0, t 1 − t e dt = 27 3 27 3 0

∞

2

0

H N%/-!#.\ "O)*6.7

¤

f+(*,+ BG g G

T l: B5769BDF^] 5 R > KoE > >jk E <M:=> E B9> 4 GIK[G RGIKoG GIKoGWXME=GIHJG M< F G \w
r−1

4 G 5 EDG 4 EDG \ 5 G f5 F G P E B=B

0 !#;%!#N 7 G,!#;-.) 2 9 7 61? ^ * A r−1

Z3 =4 3 a0

Z

∞ 0

µ

2Zr r exp − a0

¶

Z dr = a0

Z

∞

t e−t dt = 0

Z . a0

#! ;%!#N 7 IR%!8;-.)* ! 2 Q /- 7 @ + - !#>]Q * ^ 2T = N U ^ + T U !# !# !8# 2O2 "!#" 2 L; X/-. 2 ] + 4!#!8*E a;# ;-+6 ! 2 U (r) ∼ rN 7 I % U (r) = −Ze2 /r ∼ r−1 W * ,6- P; 6 A 9 7 [? 2T = −U RG Blk?4 G 69] E=G \ !86X%'!#%!# 2 7 I 1s !#%!8 2 9 7 m? 1 e2 T + U = E = − Z2 .

U76 2 2 9 7 [? H9 7 m ? !#%*!#%!#-. T U ^ ;-* A e2 ^ " −1 Z7 U = −Ze2 r−1 = −Z 2 r = a EFQ * a0 N> ;%!## Sa!80+ WEF! - 2 +6- a N >;a! ! a) N% 3 E 8>!#!# 2 Q 7 ¤ 1s

f+(*,+ BG G )

2

a0

< i F B 8 RG 4 E B d E > R&H BDB := DE c B[< 69] EDMG i Pl\ > R < K BD8 5 < a n X >[5fg MG E f> : EDG|H 96 8 MX G g GWi G 2

µ

4 5 @ > R BD:=>[5 g GMi

m 1

x(y{z|y{}~yIM(X/-. 2 ] + 2 T SQa)/ S( A U (r) =

(

0,

r 6 a;

+∞,

r > a,

/7 7:;- X 6-.!8 7 M(] %` %!#6476\.`# 9 @7 m ? 7 b "_"O" ; X6-. 2 ] + 2 $) EFEFQ 8%" ^ 7 6T P% 2 9 7@m ? !8-# !#"O. - 2 0 6 r 6 a - 2 r > a. R(r) = RI (r) R(r) = RII (r) I !8 2 ) _!4 ^ U (r) = ∞ ; r > a ^ R (r) ≡ 0 ^ 9 =7 m ? ;N 8 >7 6 6.a4N- !# 0 6 r 6 a ! UII(r) = 0 +E !8- 2 V R (0) = 0 R (a) = 0 7 IR+ !8- ! -# H) !#6 E 8 I!8-Q ^ I a) !8- 2 ; EF%!# R(r) a"% 7 I ) *-./6 - 2 &(%"X R (r) ;-* % 9 7 0 : ? 7 + r =a 7 6T PQ ; 7 Z 7 (AN I8 VT-\6T.T ;--. ; N%6. ^ NE_N6O-!#.3/-."% r = a 7 o\EFQ 8%" O T 2 !#XE ;* " "%%6Uc ] + *!#"Q8_% Q ^ !# !#! c 5TU8 )*/ % N66-.% "%%L!8- l A 2 9 7 Z5? }2 Xnl , E = +

Xnl > 0

Qa*-. ! &( *!#"%QH&(%"X>R! !#- 2 nl

n

jl (ka) = Jl+ 12

2µa2

Ãr

2mE a }2

!

jl (x)

= 0.

A

9 7 Z0/?

%!# ! 8 79 a )7 0/0* L? nE 83=) 1,*2, .. Q. ^ !82 X !#6N!*-c 5TQ8%6R <[E,k[697 BD: EDT \ ;^ ,!8# ^ !#;* S c T 5 Q8 /7 b " XN) ^ !#; X&(!#"% - 2 "*-%!#"%+ (n,H;l) nl

nl

X6- !8-Q EF/ 3 P3 ! -%(!#!# /7 I !#%H!8-% l = 0 9 s !#%!8 2 ?8^ EF% 2 ^ j (x) = sin x ^ !# 9 7 Z0/? ;-*$%6 sin(ka) = 0 ^ " S ka = 0πn 9 n = x1, 2, . . .? 7 b + S!#; " s !#%!# 2 Q> A Ens =

π}2 2 n 2µa2

(n = 1, 2, . . .)

9 n = 0 P* - 2 &() *!#"%Q>;!#6"%()** S ? 7

¤

95 "%`"/%#_'V"#v&2a*2"!D # #UXG ,Q cos θ cos θ s !#%!# 2 >;!#!# + 67B9, 4I>fF A 0 ^

2

17? 3

#Qg G ,6Q cos θ p!8%!# 2 4;%!8!# + L6W! 9 F24?>fF A 3 ^ 1 7 ? # G M("O)*56. 5 +/-.%!8. E 8 &("XQ Ψ 2

4

7 N# .%! 2 + 6-.!#

NΨ210 2 !#. ;7

B G ,6Q

F^C

7

m = 0, ±1

^

Ψ210 Ψ100 7 ^ f21 (r) f10 (r) f21 (r) 200

f20 (r)

!## T) rn 1s !#%!8 2 `* ; N+ 6% 9 n > −2? 7:9 F24I>fF A (n + 2)! ³ a0 ´n 7 ? 2n+1 B#)G ,6Q !## T) rZn 2s !#%!8 2 `* ; N+ 6% 9 n > −2? 7:9 F24I>fF A 1 ³ a0 ´n (n2 + 3n + 4)(n + 2)! ? B)BG ,Q !## T) 8*Z rn 2p !#%!8 2 ` ; N+ 6% 9 n > −4? 7:9 F24I>fF A 1 ³ a0 ´n (n + 4)!? BV G ,QT!8# ) 24* Z 2 p2 r−1 ;)-.a!#6X%a!#%!# 2 ; N+ \% !F+ -E "%E !8-% n 7S9 F24I>fF A Z 2 e2 −1 Z 7? 2 p =µ r = n2 a 0

n2 a 0

B\ G ,Q !#" EFQ !8; "> - 2 ]- "3;- U (r) = − Ze + α 7 r r 9 F24?>fF A E = − 1 Z e ^ + µ = αµ 7 ? 2 (n − µ ) a } (l + ) B)3 G ,Q'!## Q ; X6- ]- "%!#6*!#"%+ 4;- 2 ^ !#)# 6*+ ; NE H%4 1s!8%!# 2 ^ "O"&(%"XUc ! !8 2 2 ( 2

∗

2

2

2

nl

l

2

0

l

2

1 2

∗

+ 2 ! )* 2 Z > 1 Z=1 9 F24?>fF A Ze e − ϕ(r) = r r

Z

76I * !#%!#;X;6-. - !8-% 2

¸ µ ¶¾ ½ · 2 2Zr 2Z 2 2Z r + r + 1 exp − − 1− a20 a0 a0 µ ¶ µ ¶ 2Zr Ze 2Ze + 1 exp − . − a0 a0 a0

90 ; X6- ϕ(r) !86! 2 "%" Q!#1c 5T 7 ? B ∗ G 4 ;# -.`% ] + - 2 T 2 !#XE ! !#E µ ! l = 0 ! &(*!8"Q ; 2 + -.QL;X/-.Q 2 V"% Q + -%NE,A M(

Z=1

U (r) =

(

0,

r 6 a;

M("O"%QL 6-.Q + -N 2 ;!#%!8 P-U6T.,* . 9 F24?>fF A π2 6 2µU20a 6 9π2 7 ? B)X ∗ G ,Q 4 ] + }H!#6X4E 8 !#%!# 2 Q H!# !#!c 5\ - EF &("X> - 2 ! &( *!#"%+!#XO-- 2 L!, ! !#Q µ 4!#Q ω 7 U7 6T ,;!#> ! &( *!#"%Qa!#!8*,"% /7 9 F24?>fF A E = }ω¡n + 3 ¢ n

l −ξ 2 /2

2

U0 ,

r > a.

n−l 3 2 Ψnlm (r) = Nnl ξ e 1 F1 (− 2 ; l + 2 ; ξ ) Ylm (θ, ϕ) p ? ξ = r/r0 ; r0 = }/µω; n − l = 2nr ; nr = 0, 1, . . .

9

Dw

D*

2% "b 2 O}( * S -Q &("X Ψ (r) 9 a N'"E 8 !#- ? "%&(+ %6X%`;!#!#.1 "% 6 r \+ Ψ (r)3 2 - 2 ! 2 8 !# E 7)3 o "*!#" - 2 K < E=EDGIHJG !#!# 2 2 !#!5T!# E 8 !#- a 9 "!\!8%!# 2 2 ? 7 @ $ 3 2 - 2 ! 2 -\6\.WNa"%6E > W-QL&("X Ψ (r) ;# S!#66- 2 !#NQ3-U6T.

a

a

*!#"%T) N6 T a + !#%!# 2 2 a 7SI *!# Ψ (r) - 2 ;O a ! 2 !#%!# 2 2 a / LP!#;-.) 6.("%]&,&(X E c (G )-/ T a n) 2 Ψ (r) a X 9 Z 7 0/? Ψ (r) = c (G )Φ (r)

; G 6 EDGYi !#!8*P!#N!# E 8 &(n%"XQ Φ (r) 69u XYGIHJG ] % ; 6 G Gˆ 9 GΦ ˆ G = Gn ΦG ?8^ Q!#!c 5\ + F G \ NO> RG 5fF R < E 5fF24?>#^ " % ;# -E &(%"X Ψ (r) 7Ul ! -# )+ ^ 3

Ψ (r) a >6N% "%]&,&(X c (G !) #!5T*!8 T)*63* ) P!#a !#A)*/ c (G ) a) n*\;# - 2 Ψ (r) ;&(*-9 Z 7 0/? ^ a 3) Ψ (r) ;)a- 2 n 3Q 4?5[> c (G ) A a

a

n

Gn

n

n

n

a

a

ca (Gn ) =

n

Z

Φ∗Gn (r)Ψa (r) d3 r.

Z

ca (G)ΦG (r) dG;

9 Z 7 ?

; 2 EFQ N c (G ) ) EF ! 2 -Q'&(%"X Q$!#%!# 2 2 a G R > K 5 F| E B=aB 7 nIR#- |c (G )|2 99 Z 7 0/? )* 2 ! 2 n + %* A Ψa (r) =

9 Z 7 Z?

9Z ca (G) =

Z

9 Z 7 1?

Φ∗G (r)Ψa (r) d3 r.

U ! ! a ; . !#EFQT! -%Q ^ "%+ Ψ (r) ;/ ! Q )T!#N%!# E 8>&("XQH; Gˆ ^ 6; ^ Φa (r) 7 b + S )>9 Z 7 ? G -# ^ m

ca (Gn ) =

b "

Z

Φ∗Gn (r)ΦGm (r) d3 r = δGn Gm = δnm .

9 Z 7=9 ?

N6) ^ !8N%!# 2 &(%"X 2 ; 6 Gˆ G ;# !#6- TO δ !#-W9 - 2 !#" +!#; " ? δ &("X 9 - 2 ; EF+ L!8; " ? 7 4 ;! !8%!# 2 2 ! ;, 5T,. c 6) EF%! 2g GYG R&K B E <WF EDT \ R > K C Ψa (r) 5fF2 E B9>V\ 9 r ;# S!86-* ? 7 R!8- "O**!# ; Gˆ !#;-.) ! 2 ; V;*-.%! pˆ ^ ; N6) 6 9 Z 7 ? ,-1 c_&(%"XUc !#%!8 2 2 BQ\ D8M69]U5 E=G \ R > K 5fF2 E B=B 9 ;a# !#6- ? 7 ,6;% ^ !#;"p ; p 7 5T*!# EFQ ; EF EFQ ^ ;) -. !#N!# & UY ?o ) * \c p !# !# !#N%!# 2 &(%"X 2 µ ¶ 9 Z @7 ? ? 1 i Φ (r) = exp pr .

p

M S!#6- 2O2 9 Z =7 ? ? 9 Z 7 1? ^ ;- &(*- ; 8* " 6+ ( ;# !#6- 2 a " ;*-.!#SA ca (p) =

Z

Φ∗p (r)Ψa (r) d3 r

(2π})

=

3/2

1 (2π})

3/2

}

Z

¶ i exp − pr Ψa (r) d3 r. } µ

9Z 7: ?

K + O p ] Q`& "X 2 - 2 ! 2 ; EFQ>-Q 7I O ^ ; 8 W "% + ;8 S!#6- 2 "W;/-.%!#% 2 - 2 ! 2 ^ ;3!# > - ^ )*!#E '; N) 6*$o\%.P-Qa&("XB7 R! - ; % ˆ 2 - 2 ! 2 +VO-.6 ˆ 9<;8 ;- 6+ ! 2 ^ 3)*!# *G ?8^ >; N)%6a9 Z H7 ? d E > R&H >fF^B=:=> 5 g G > R > K 5fF2 E B9> -QH&(%"X 9 E ;# S!86- ? 7

91 f+(*,+ V G #)G

G 6 E=G 4 GMi <MgU>fF ?k < K fF25 Pe@(8 E g c B > i Ψ(x) =

)

1 1/ 1 x0 2 π /4

x2 exp ik0 x − 2 2x0

R > K 5fF2 E B> K :=< i r;n a G 698;:lB=F ] R K >V6_> E B> G BQ\ x(y{z|y{}~yI* G,!#;-.) *H&(*- 9 Z 7 ? 7

/ -. c(p) =

@?

(2π})

-3/2

1

1 (2π})

)* 2 ! 2

1/2

=

1/

x0 2 π

1/4

Z

1 (2}x0 )

1/2 3/4 π

¶

.

< DE EDGWHJG 5 G 5 F G P E BQP m GWKoE=G w \ > 9R EDTi 5V698?C D8M69]U5 < \ 4 <MgU>fF2>on

,;% ^ * %%4!8-

(2π})

-1/2

A

¶ x2 i exp − px + ik0 x − 2 dx = } 2x0 ¾ ½ ³ Z +∞ ´ x2 p − k0 x − 2 dx. exp −i } 2x0 −∞

+∞ −∞

µ

µ

9 Z 7 [?

I E -' ;"O)*6#-(] "O!#; E 9 Z 7 [? ;-EFQ>"%/ A

³ ³p ´ ´o x2 1 n 2 2 p + i x = x + 2ixx = − k − k 0 0 0 2x20 } 2x20 } ½ ³p ´ ´2 ³p ´2 ¾ ³p 1 − k0 − x40 − k0 + x40 − k0 = = 2 x2 + 2ixx20 2x0 } } } ´o2 x2 ³ p ´2 ³ 1 n 0 2 p − k0 + − k0 . = 2 x + ix0 2x0 } 2 }

'6"O 3 ^ ;H;) -.Q "%%;O- "O!#Q>"!#6 Z

+∞

e

−(t+α0 )2

dt =

Z

+∞

2

e−t dt =

√

π.

9 Z 7@m ?

α0

9 Z 7 0*5?

M * !#6- 2O2 9 Z 7@m ? 9 Z 7 [? T%EF% 2 9 Z 7 05? ^ ;-*aV;*-.%!# ;# ( !#6- (-+3;"% 6A −∞

c(p) =

r

−∞

½ 2³ ´2 ¾ x0 p x0 √ exp − − k0 2 } } π

9< "%% * ! %!8 2 -. ;* -. !# !#!c T 5 EF"O- / " ? 7 U !#;# - P;;/-.%! ½ 2 ¾ x0 x0 2 w(p) = |c(p)| = √ exp − 2 (p − }k0 ) } } π 2

9e9 W+#*! !#a&(H!,; }k 9Y!## Q ;/-.%! ? }2 9 6TO 0 2 ? 7n &("\;"O)* !67 Z 7 0 7/M(# - 6+4! %!# 2 -.%2x N# 0 .%! 2 EF;-a!8- 2 " Z +∞ w(p) dp = 1 7 ¤

f+(*,+ V G BG )

−∞

K >V6_> E B9>5 G BU\ D8M69] 5 <[\ 4 G 5 EDG 4 DE G \ 5 G 5fF G P E B=B 4 GIK[G RGIKoG GIKoGWXME=GIHJG B GYE < 5k?< R P KoG \ Z n x(y{z|y{}~yI I - 2 &("X 2 %!#+ !#!# 2 2 9 ? %" % 1s

;# !#6- a O

Ψ1s (r) =

s

µ ¶ Z|r| Z3 exp − , πa30 a0

9 Z 7 00/?

+ a N !#"QW* %!67M(* !#6- 2O2 9 Z 7 00*? H9 Z 7 : ? ^ ;-% -1c &(%"0XUc 1s !#%!# 2 2 ;*-.%!#%';# S!86-BA 1 c1s (p) = 2 π

a

µ

Z 2}a0

¶ 3/2 Z

¾ ½ i Z|r| − pr d3 r. exp − a0 }

9 Z 7 0/?

- 2 EF!8- 2 +6- 9 Z 7 0/? ; %!8. Oz -.\ " p ; QO * 3! & !#"1c !#!#*"% 9 T] % !8-%6 + - θ !# EF% ! 2 " p pr = pr cos θ ^ d3 r = r2 dr sin θ dθ dϕ? A 1 c1s (p) = 2 π

µ

Z 2}a0

¶ 3/2 Z

¶ Zr × r2 exp − a 0 0 µ ¶ Z 2π i dϕ. × exp − pr cos θ sin θ dθ dr } 0

∞ 0

Z

π

µ

9 Z 7 0*Z?

M %! -# Q +6- _9 Z 7 0*Z? 2π 7 G( +%H; + - θ !#% ( 5T*!8- 2 ! 2 !P!#;-.) 6*') E cos θ = t 9<;`] % sin θ dθ = −dt ?A cos 0 = 1 cos π = −1 Z

π 0

µ

¶ Z 1 ´ i i} ³ −ipr/} − }i prt ipr/} e exp − pr cos θ sin θ dθ = dt = e −e ; } pr −1

¶ ¾ µ ¶−2 ½ µ Z ∞ 7 i i Z 1 ±ipr/} Z e dr = ∓ p r dr = ∓ p r exp − r e a0EF}6 3 2 9 Z 7 0*Z? ; a0-}* A M(0%! -,6-+ Nr*!#" ; N0)6Q )P

Z

∞

2 −Zr/a0

U8

9?

=−

4} πp

¸−2

¸−2 )

·

Z Z Z i i + p − − p = 2}a0 a0 } a0 } µ · ¶ 3/2 ¶−2 ³ a p ´2 ¸−2 Z Z i 8 ³ a0 ´ 3/2 0 Im 1+ . + p = 2}a0 a0 } π 2}Z }Z

2i} πp µ

c1s (p) =

¶ 3/2 (·

µ

U !#;# - P; ;*-.%! _9<;O-%!#. 2 !# ? ! 2 EF6 T * · ³ a p ´2 ¸−4 8 ³ a0 ´3 0 w1s (p) = |c1s (p)| = 2 1+ . π }Z }Z 2

M(# -+#*>!%!# 2 -. N# .%! 2 EF;- P!8- 2 " R aQH 2 QU6T () * P;/-.%! O7 ¤ w (p)d3 p = 1

f+(*,+ V G V G

1s

I4 < E F G 4I<[Pj5 B5fF2>V\ < E < ZDGWK BDF25 Pj4`5 G 5 F G P E BDB G DB5 T 4?V\ G \ 4 G 6lP R9EDT Z g GMG R_K B E <MF|< Z 4 G 6 EDG 4 GMi @8 E g c B> i Φ(ϕ) = √ 1 (3 + cos ϕ) n <MC DB95 <WF^] >[> 4$d E > R&H >fF^B=:=> 5 g G \ R > K 5fF2 E B=B GX <[kWB5f8 Q619π G 5fg GWHJG RG F2<WF GR < 5\ G \w> E F G \ B E > Rc BDB I B G P_5 E B=F ]j57\ T 5V6 4I>V69BD:QB E |c(E )|2 n m x(y{z|y{}~yI U )-/ L &(%"X ; % N ) ! # E $ ( & % " X 2 ';O-%!#"%+ 3 Φ(ϕ) 6 B # ! ; " ! # " F E B Q A eimϕ 9 }2 m 2 ? 7 √

\c

M( 2O2 &(*-Ol Ψm (ϕ) =

m = 0, ±1, . . . 2π QO- ^ ;-%6* r

Em =

A

2I

r 1 2 2 Ψ0 (ϕ) + {Ψ1 (ϕ) + Ψ−1 (ϕ)}, Φ(ϕ) = 3 19 2 19 r r /7 7 c(E ) = 3 2 ^ c(E ) = 1 2 7 I !#66-.E !8- 2 9 |m| > 1? 0 ±1 19 7 % +

! 8 % ^ -219 2 &(%"X 2 E ;# S!#66- \) N c(Em ) = 0 6

! 2 N*!8" E !#-NX 7 |c(E )|2 > 2 %!#. w N% m

T 2 !#%!8 2 2 Ψ (ϕ) a!#!# 2 Ψ(ϕ) 7 I !#%!# ^ Tm Q`)*/ * m

8

a

8

6 )( 8

;a"O6

% )* a] + aN ;-.%! 2 ) Q ] + A E = 0 9<%!8T!8%!# 2 ? - E 9<; ) N

\!# ±1 !# 2 ? 7IR0 2 !#aQ8 N% T 2 w0 = |c(E0 )|2 =

18 ; 19

?7 C,">4!8-# %6-3/ O . ^ EF;- 2 ! 2 !8-(" w±1 = 2|c(E±1 )|2 =

1 9 %"PEF/

19

P

wm = 1

7

m

¤

9,: "%`"/%#_'V"#v&2a*2"!D # Bg G h,!#X ! !#E m 8 ! 2 N!#"% + -%N"%Q ; 2 +-. Q ; X6-.Q 2 dT 6 Q a 7 ,Q -EF &(%"X !#66 XE 8 !#!# 2 Q ;*-.!# ;8 S!#6- 9 F24I>fF A c = √

(−1) e −1 ? n = 1, 2, . . . (pa)2 − (π}n)2 n −ipa/}

πa}

n

7

QEFQ3+6*!#"QT%!#XO-- 2 a9Y ! ! m ^ !86 ω ? 8 ! 2 W%!# !#%!# 2 7 ,Q`;*-.%!#\;# S!#6- \-Q &(%"XB7V9 F24I>fF A c (p) = p 1 e−ξ /2 ξ = p p2 = m}ω 7 gU<[k;< E B9> A √ 0 p0 0 p π 0 ! 7; Z 7 0/? 7 V G h,!#X ! !#E m 8* ! 2 N!#"% + -%Nr"%Q'; X6-.Q 2 6\Q a ;# !#!# 2 Ψ(x) = 30 x(x − a) 7 ,Q ` + L] + *!#"% ;# !#6- ^ !## ) a] 5 + ^ 6"O T √ & -"X\c ] + 9 F24I>fF A C = − 240 1 + (−1)n+1 hEi = 5 }2

B G<

2

n

p √ }2 7 h(∆E)2 i = 5 n = 0, 1, . . . ma2 ∞ P ?7 (2k + 1)−2 = π 2 /8

π3

gU<[k?< E 9B > A

n3

∞ P

ma2

(2k + 1)−4 = π 4 /96

k=0

V #)G ;! . -1c &(%"X\c !#XE _X 6-.% ;-$ L ;# !#6- 7:9 F24?>fF AOU / /-. 2 &(%"X 2 f (r)? 7 VKBG 6s;! .L! &( *!8"1c &("XUc Y (θ, ϕ) L ;# !#6- B7B9, F^C 4I>fF A 2l + 1 (l − |m|)! P (cos θ) ^ + P (x) ;!## EFQ ;-% <

2? 7 (l + |m|)! 2 ^ ' $+"'

k=0

2

El

lm

m l

z

m l

, 2 ! _!#;-.) 6E N) * -Q &("X

"% %W;# !#6- , # "% !#;-.) ! 2 # a O "%%$!#"%N N) A 9 Z 7 0 1? Ψa (r) = hr |ai . M( 2 !# !E !8-'N) 2 9 Z 7 0 1? 7 @ + -!# a " ^ \- cNW!#%!# 2

Ψa (r)

9[ a "%Q !#!#*E V* L ;! 6. 9< )*!# EFN;# !#6- 2? "%Q 6* 6!#"%Q3"%!#%"X Q ^ "% 2 ) EF%6 ! 2 gU> F "%W N) 6 ! 2 !#- 7/I !8-8 S!# ;X; !#%; O |ai ;) X "% "E / L !#"O- / EF%. /

. "%%;O-"!#EF !#"O6- 2 EFF-E ;-.,EF "% "E,7 @ ";!#.P%!8 8 )*/ E 8 "% "aN) 4N%!8"";- "O!#L " ;%!8!# N*!#"% + !8- ) Q ^ "%H) EF%6 ! 2 H BQ69]?C X > R F G 4 T \ R9G 5fF R < E 5fF24 G \ 7 C,6

% "% " / `!#;%!#6. 6-.EFQ " !#%!# 2 2 N ^ "%EFQ N) *6 ! 2 !#- ha| !# 2 )* ! "% "% ; X Q ] % !#; 2 T 2 A ha| = |ai† 7LM(] % - c N !#%!# 2 W"Q !#!#E * L ;! 6.`"O" "% "% ^ 6" N " 7 C( N "E_7 c )#-! c ;* ^ ;] >Q8 -.) 2 !#"-/ EF%.7 @ -# %6-. ^ `+ % NEF. ) NE !#$7 5T*!# 1 c !# 1 c !# 7 l '";- "O!#EF -E %!#N+ >* 7 a Q!#; "% " %!8! 5T*!8- 2 ! 2 !8- ;4'; , 6T Uc "'] !8; 2 T \c ! ! %! 2 "O" ;) 8 ^ /7 7*!8- ˆ |ai , (|bi)† = hb| = (G ˆ |ai)† = (|ai)† G ˆ † = ha| G ˆ†. |bi = G b " N6) ^ K > i 5fF24MB> G > R <MF G R < E < gU> F qC 4I>fg;F GR 5V6_>[4I< E < R E F DE GAK > i 5fF24WBDud R \.BDF G 4 G 5 G R PDNO> EDE=GIHJG G 9> R <WF GR < E < K 8 <[69] E=Ti XR < qC 4I>fg;F G R 5 R [4 G 7 @ "O6- 2 P;) # "% " !# ! 2 ; V/ T

2 |bi- 2 ! 2 NEFE "%% * hb| |ai A hb |ai 7 @ "O6- 2 >;)# |ai ;O- "O!#E '!8-%'a - 2 !#,6T \c hb |ai = ha |bi∗ 7 IR "E ^ !# !#!c 5T !#!# 2 2 &(+ O T 2 ^ * L %. !8- ha |ai = 1 7 > )!#EF "E -Q+ ] %D; Gˆ 9 Gˆ |Gni = ? N - 2 . !8-\c " Gn |Gn i 9Z 70 9 ? hGn |Gm i = δG G = δnm . @ Q!# 9 Z 7 0 9 ? );! > - 2 !#" + !#; "7 I !8-; EF + L!#; " δ !#-a)* 2 ! 2 δ &("XQB7 C(!#"X 2 Fˆ = |biha| ^ T- hb |ai ^2 - 2 ! 2 G > R <MF G RG \,^ *7 "B7 ;a + T Q!#H3 "a;-%! 2 EFQ> "BA n

Fˆ |ci = |biha |ci ;

m

8,++

hc| Fˆ = hc |biha| .

,+

*- -46

4642! , 82*-) & 46, 1 1,1 1 : 1 8( 1(/ ( 1:9 4"9 42 -/22*-, --2', 7 , *--, ' *8 :,10/1(46 64 46

9m a

- 2 ;-QW%Q !#!#*E_ "\EF;- 2 ! 2 #! ,6T ^ 8U5V6 G 4MB>V\ G 6 EDG F T A X 9Z 70 ? ? |G ihG | = ˆ 1, n

n

+ ˆ1 # EFQ; 7 a - 2 N) ! ^ !# !81c 5T +P ; EF >!#;" ^ !# %(`9 Z 7 0 ? ? )* 2 ! 2 + * 7 @ e 6T 49 Z 7 0 ? ? ) EFQa NW - 2 )-/ T 2 ;)-. + " |ai ;N%6) !# A n

|ai = ˆ 1 |ai =

X

|Gn ihGn |ai =

X

9Z 70 : ?

c(Gn ) |Gn i .

; 6 Pˆ = |G ihG | 9 Z 7 0 : ? ) EF%6 ! 2 RG >fg c B GME=EDT \,^ *7 " 73 ;) - 2 ; n-%.'n"%]&,n&(XE )-/ T 2 " |ai ;'N%6) !8SA 7 c(Gn ) = hGn |ai M(%!8. N%6) ! ; Gˆ )/ S6 ! 2 * 3*!8 " |Gi 7 b + S %; 2 EFQ N "%]*&,&(X )-/ T 2 "%+ n " |ai ; N%) !# ; Gˆ 1 ! 7 9 Z 7 0 : ? 3 ; 2 D)EF%. G ;# !#6- !#%!# 2 2 |ai 7 a - 2 + 3H ! 2 6"%!#"%>N ) * hG |ai 7 @ - "% "`)EF%! 2 B EK >fg 5 G \ 5 G 5 F G P9C E BUP^ N n " B E9K >fgU5 G \ R > K 5fF2 E BQP 7 a + !8-% ^ G ;# !#6- (!#%!# 2 2 |ai ;# !#6- 2 3!#NQ * 3*!8T%!8 8L +T; "XQ 4!#%!# 2 2 !;8 - E ) * 2 -E G 7 4 a 6 2 EFQ " |ai ^ NEFQ - 2 )- E 8'EF!8- QB7 a 6 3%

3; 2 !# 2 a! E ! - N) 2 9 Z 7 0 1? AS) * T-Q &(%"X Ψ 4"%>!" 6Q r 4; "X$!8%!# 2 2 a !#%!8 2 ! a"% Q r 7 M(-.) 2 !#.H "%%!#"%Q 8O"Q ^ - +" ;-* ;- ; 8* F ;8 S!#6- 2 -Q &(%"XL!#%!8 2 2 |ai " G ;# !#6- \c\7 a - 2 ;%!#E !#;"'; Fˆ Gˆ ;# ;-+#* !8" E 7 , %!#6 9 Z 7 0 ? ? * A X X 9 Z 7 0*[? ∗ hG |ai = hG | ˆ 1 |ai = hG | |F ihF |ai = hF |G i hF |ai .

4

m

m

n

m

8

n

n

n

n

n

m

n

n

,N$"%]&,&(X ; S N) F ;# !#6- H!# hFn |Gm i !# 2 2 |G i J1 ! 7 6"O 3a9 Z 7 ?8^ 9 Z 7 1%? 3 7 NN \ 9 Z 7 0*[? W!8-Q` ; EF+ L!8;m "3 O 7

f+(*,+ V G Ga

4 e5

G 96 8;:lB=F ]^4 K B R <Wg G ?4 5 g G \ @ G R \w< 69Blk[\w>3 R R > ZDGWK < G F g GYG &R K B E W< F EDGWHJG R > K 5fF2 E BUP 5 G f5 F G P E BQP |ai gjBQ\ D8M69]U5 E=G \.8Un

?5 x(y{z|y{}~yII &(*- 9 Z 7 0[? )* hp |ai =

Z

^

Gm → p Fn → r

A

9Z 70 m ?

∗

d3 p hr |pi hr |ai .

@ "%N"O hr |pi !# !# "% $;# !#6- \c !#%!# 2 2 ! ;*-.%!#% p 7/C,6",) *!8 ^ hr |pi = (2π)−3/2 eipr/} 7 U "%% O *4!#. 9 Z 7 0 m ? !9 Z 7 : ? 7 ¤

f+(*,+ V G 3G s

[5 5V6_> oK G 4?<MF^] MG X ^8;u 5fF R 8;g?F^8 R 825 GMX 5fF24?> E=EDGMi @(8 E g c BDB G 9>fC R <MF G R < Gˆ 4 5[4 G >7\ 5 GMX f5 F24?> EDEDG \ R > K 5fF2 E B=B 5 R > K G 6_< H < P > HJG 5>fg?F R K B5fg R > F =E T \ n 7 x(y{z|y{}~yI - 2 ;# -%!# Q ;# !#6- !#%!# 2 2

a

I #! !8 ! ;O- ;-% 2 G*) / +a;# !#6- 2 !#|G mi a 6"%!#"!c !#"%N"SA 9 Z 7 5? hGn |Gm i = δG G = δnm . I ; EF%$!8; " δ !#-HC( "% T) 2 ! 2 δ&(%"X Q a O "O7 b 6" N) ^ 5 GMX 5fF24I> EDE <[P @8 E g c BQP 69B E > i=E=GIHJG d R \.BDF G 4?< G > R <MC

n

F G R <.4.5[4 G >7\ 5 GYX 5fF24I> EDEDG \

m

R > K 5 F| E BDB2BU\w>[>fF

C GYX R [< k E 8;u 5 F R 8;g;F^8 R 8 7

U "%% *'6"O 3\!8.49 Z 7 5? !9 Z 7=9 ? 7 I "O**!#(; 3;%!#$!#N!# !c &("X\c ; T"% E `"% %_;8 S!#6- B7M(!#.` " |r0 i )*/ `!#%!# 2 `!>"% Q r0 7 C," ) !# ^ rˆ = r ^ + !#; " ;EFEFQ ^ ;] %W !# !8H!L9 Z 7 5? hr |r0 i = δ(r − r0 ) 7 ¤ ( T %! cR >)*%!# 2 ) .)- E 8>;# !#6- Q4N S6%.%! 2 "%%!#"%'&( 6-)*7

$# 229% - $),_ / 2

% #

δ

2 v}(

2

+"/

- 2 "%"E 8 EF!8- Q N;8 !#;-.) 6. "% ;# !#6- F"O"3 - 2 " !#%!# 2 Q ^ 6"L3 - 2 ; 7M( N !8-% c ! " ^ G ;# !#6- ; Fˆ 2 - 2 ! 2 %; 2 E N "%]&,&(XT+ 3)-/ T 2 ; N) !8; Gˆ 7 I "%%!#"%Qa&( ] N%) ! ;# !#6- 2 !#NQ ;1 c "%O !#"XUc |G ihG | ^ 6"aL)-/ T (EF+ - 2 L!8-8 1 c 5T$N6) A

a

k

n

Fˆ =

X kn

|Gk i Fkn hGn | .

9 Z 7 O0/?

?0 I EF6 T - 2 "%]&,&(X !#%3" 9 Z 7 0 9 ? A

;-%6 ! 2 )9 Z 7 0/? %!#!#QO

Fkn

9 Z 7 ?

Fkn = hGk | Fˆ |Gn i .

C(!#%"X 2 H;Q !# 9 Z 7 ? ) EF%6 ! 2 \w<MF R DB : =E T \ d{6_>V\ > E F G \ G > R <MF G R < Fˆ 4 G C R > K 5 F| E B=B 7

f+(*,+ V G G

s 5 5V6_> [K G 4I<WF^] 5fF R 8?g;F^8 R 8 \w<MF R B Dc T 96 B E > iDE=GIHJG d R \`BDF G 4I< 4 5[4 G >7\ 5 GYX 5fF24I> EDEDG \ R > K 5fF2 E B=B

G > R <MF G R < Gˆ x(y{z|y{}~yIIR.3+ ! 2 ! -% !#" + !#;"OA

ˆ |Gn i = Gn hGk |Gn i = Gn δkn . hGk | G

&(%"XUcT7 <Mg;BQ\ MG XR <[k G \ `4 5[4 G V> \ 5 GYX 5fF24I> EDEDG \ R > K 5fF2 E BDBL\w<MF R B c <`69B E > i=E=GIHJG d R \.BDF G 4I< G > R <MC F G R < X 8 K >fF K B< JH GME < 69] EDGYi 7 4 N

P 7 ¤

I !8- ; EF+ !8; "4;-

f+(*,+ VG X G&a

δ

G 698?:lBDF^] R i 5fF24WBUP G > R <WF G R < Fˆ E < 4I>fg;F GR |ai 4 C > 5fF2 E BDBn G R K x(y{z|y{}~yI M(%!8. ˆ 7 * L ] 4!8 T !8- % N) !8

EFQH "

hGn |

A

,6

a

|bi = F |ai

hGn |bi = hGn | Fˆ |ai = hGn | Fˆ ˆ 1 |ai =

X m

hGn | Fˆ |Gm ihGm |ai

9 Z 7 Z?

NEF(;- / T 2 6XE 3 "B7 I ; EF%$!8; " hG |bi =

Z

dG0 hG| Fˆ |G0 ihG0 |ai .

l L + 6-.P; N) 6,!

9 Z 7 61?

2 % hG| Fˆ |G0 i 7 I !8- + 6-.+ $; 6 !# <9 + ? !# 6 ! 2 7 ¤ G,!#;-.) 6EF 6 ; E'P" %`;# !#6- 6"

T + NEF. )*6;! E Q & A hr| rˆ |r0 i = r δ(r − r0 ) *7 7RM( ]% + %6 9 Z 7 61%? ; hr| p |r 0 i = δ(r − r 0 )(−i}∇r ) !8 ! 2 δ &("X Q 7 d3 r 0 0

? M 6- ; *!# 6 + ]-* 6 )a + ;# !#6- 2 ( %+ - + "% EF ! 2 ),!#Q!#\;-E 9 Z 7 0 ? ? ^ ; ^ - 2 ; 8* T ξ ;# S!86- 2 " G ;# !#6- \c ; Fˆ * A hGk | Fˆ |Gn i = hGk | ˆ 1Fˆ ˆ 1 |Gn i =

X

hGk |ξ 0 ihξ 0 | Fˆ |ξihξ |Gn i .

9Z 7 9 ?

- 2 ; 8 SW>"% + ;# !#6- 2 K B< HJGME <[69] E=GIHJG ; 6 Fˆ " G ;# !#6- Uc %!#%6L9 Z 7 9 ? ;-%6*>!8-8 1c 5T!c$&( *-SA Z 9Z 7 ? ? hG | Fˆ |G i = Φ∗ (r)Fˆ Φ (r) d3 r,

a

k

+

f+(*,+ V G g G )

ΦGn (r) ≡ hr |Gn i

n

ξ ξ0

Gk

Gn

7

< i F B G > R <WF G R g GMG R_K B E <MF T 4$d E > R&H >fF^B=:>[5 g G \ R > K 5fF2 E BDB G X i==E GIHJG H < R \ MG E BD:=> 5fg GWHJG^G 5 c BU6 6lP9F G R < 5`:=
4

+/-.E ^ 6"\ - 2 EF! - 2 E 8,]-* / %! ;-.) 6.%! 2 9 Z 7 ? ? 7 ,6;% ^ W; N) !#E `&("X 2 !8!#*E ; c! 2 !8N%!# EF4&("X 4+#-.667 G 8$" ;# !#6- ! 2 EF6 3*_9 0 7=9 Z? 7 IREF! - + ,]-*6P; 6 "% E %!#15T*!#- 2 ! 2 ; &(*- 9 Z 7 ? ? ! )*Q x = ξx 7OM() 8 ξΦ(osc) (ξ) ;N ) ! 2 !,;%, 5\,. c "+ !#0, 6T 2 9 0 7 [[? 7 a 6n-. \Q 6T, + % %!#! 5\*!#- 2 ! 2 !L;%, 5Te. c !#, 6T 2 " 9 0 7 9? ? 7 M(# -+ EF;-.>%!#3;%# LEF3EF!8- 2 !%!# 2 -.7M(* )8 *!#. "%6-.EFQ> A xmn ≡ hm| x |ni = x0

,;% ^

6EF

(r

n δm,n−1 + 2

r

n+1 δm,n+1 2

)

.

9Z 7 : ?

} 7 ]#-*mω E x * L )*;! 6. *!#"%-."% A mn r r n n + 1 ^ ;* ) % 2 %!# %!#66-. hn − 1| x |ni = x0 hn + 1| x |ni = x0 EFP]-* E_2E >*- \7M(8 * 2 2EFQ> `&(+* 6TN*!#"% O x0 =

r

c

?Z Q4 XE

xmn

A  √ 0 1 0 0 ... √  √  1 0  3 0 . . .   √ √ x0   =√  0 3 0 5 . . . .  2 √  0 0 5 0 . . .   ........................ 

xmn

4 N 6P3 E > K B< HJGYE <[69] E 8;u !8"%7 f+(*,+ V G G <-DB5 <MF^] G > R <WF GR Fˆ = cos θ m 5 @ > R B=:=>[5fg <[P

¤

5fB95fF2>V\ <2g GYG R C R G 5fF R < E 5fF24?> EDE=GIHJG

K B E <MFre4td E > R&H >fF^B=:=> 5 g G \ R > K 5fF2 E B=B G X <[kWB5f8 RG F2<WF G R <;n x(y{z|y{}~yI 6 ) !# ;%!#!8+ 6 2 - 2 ! 2 ! &( *!#"

>

c &(%"X Y (θ, ϕ) ^ 8") !c 5T! 2 N66-.E l + E m "%E \!8-H9 ! 76; 7 ? 7 a - 2 ! &(*!8"Q8T&(%"XQT) *!#6 , "% !#e6T ,9YI 7 [?8^ /-+F!#eT 6 \c 9 0 7 [[? - 2 %!#XO-- 2 E a 8 &("XQ 7M(] % %!#PEF!8- 2 * LEF;-.; 6-+ !T;# E 1T 5 ;% 7 !8-* "` - 2 ! &( *!#"U8a&("XQaN 49YI 7@? ? 7M(# * ) *-./6A lm

a

hl0 m0 | cos θ |lmi = δm0 m

(s

(l − m + 1)(l + m + 1) δl0 ,l+1 + (2l + 1)(2l + 3) s ) (l − m)(l + m) + δl0 ,l−1 . (2l − 1)(2l + 1)

9 Z 7 [?

+ 4!8-% ^ *- E VH E 4]-* 6HN hl + 1 m| cos θ |lmi =

s

(l − m + 1)(l + m + 1) ; (2l + 1)(2l + 3)

hl − 1 m| cos θ |lmi =

s

(l − m)(l + m) , (2l − 1)(2l + 1)

/7 7 6X ; 6 gU4I< E F G 4 G \.8j:lB5V698 7

f+(*,+ V G # G F T

z

cos θ

"6) EF%6 ! 2 K B< HJGYE <[69] EDGMi

G \w< HJE BDF EDG \.8 ¤

T l: B5V69BDF^]^\w<MF R DB : DE T > d{6_>V\ > E F T K > gU< R F G 4 GYi g YG G R&K B E <MC \w> N K 8 5 F|< c B GYE < REDT \.B 5 G 5fF G P E UB Pl\.B 4 GWKoG R9G?KoG GIKoGMMX EDGIHJG M< F G \w<e5

?1

H 6_^:lB5V6 G P K R < Z \wfg;C F RGYE < µ n x(y{z|y{}~yI I ! &( *!#"%Q !#!#* "% 7 C( z = r cos θ ;# !#6- >-E '&(%"XQ hr |2lmi = f (r)Y (θ, ϕ) 1 ! 7 9 7 ? 3 7 + *!#"Q . ! EF " 2l EF/lm

`; N66-.

8

l

% H 6+ % "%E !8- E 8W]#-* 3 0 ?7 4 "%T%!8-# E 8>]-* .,6 ! 2 n=2

h210| z |200i = h200| z |210i = = h10| cos θ |00i

Z

:

8

7 M(] L!8- !#"%%E 8 O !#(!# a!8-3 *-E 8 K 4M8 Z A

∞ 3

0

f20 (r)f21 (r)r dr = −3

a0 . Z

9Z 7 m ?

U "%% * ! !# 2 #-. EF!8-./ /-.EFQ> + 6-B7 4 ;# - 9 Z 7 ? ? ;+* \L - 2 ; Gˆ ! ; EFE !#;"¤ 7I ] ! -% "!8E !#%!# 2 2 G G0 X hG0 | Fˆ |Gi N % ;N +6.3 ; EFEFQa 2 4) * QB7 M(a&(*-a9 Z 7 61%? ) *!#EF3 ; 6E + % NEF.>; # E ;*-.%!#H;# !#6- ^ !8- "O**!#HN%) !# E 8 &("XQ N % ) 2 E Φ (r) = (2π})−3/2 eipr/} 7

f+(*,+ V G #)#)G"a p

G 698;:lB=F ]jBQ\ D8M69]U5 E=G > R > K 5fF2 E B9> G > R W< F G R < g GYG R&K B=C

E <MF T n x(y{z|y{}~yI ,P%!#T9 Z 7 61? ;- *6- EFQ ]-* T;

"% E W;*-.!# ;# !#6- ^ ! 8* 2 )T + WO L "% %';# S!#6- BA r pp0

· ¸ · ¸ Z 1 i i = hp| rˆ |p0 i = exp − pr r exp p0 r d3 r = 3 (2π}) } } · ¸ Z i 0 1 0) exp (−i}∇ (p − p)r d3 r = −i}∇p0 δ(p0 − p). = p 3 (2π}) } {z } |

F :=> ^ F 6DB94 D8Y69]U5 EDG \

(2π})3 δ(p0 −p)

G M4 B [K E < K B< HJGYE < 96 ] E < P f5 F R 8;g?F 8 R <`\w<MF R B Dc T g GYG R&K B E W< F T 4 BQ\.C R > K 5fF2 E B=B 7

I ;*-.!# ;# S!#66- ; 6 " E Q!# &(%"XUc a!#!# ! ;O- 9 Z 7 61%? ^ *7 7S )T+ /-.\; N)%6P! 2 % r A pp0

?,9 I o

4

; C( 6 rˆ G,;*-.%! p

4V ;*-.%! ˆ L C( *!#"O 2 ] + 2 Tˆ M( X/-. 2 ] + 2 Vˆ b(p) =

b "

Z

0

r

r

;# S!#66-

r pp0 a(p ) d p = −i}

Z

Q p

;# !#6-

r

i}∇p

−i}∇r

p

−i}[r × ∇r ] }2 2 ∇ − 2m r V (r)

3 0

p

i}[∇p × p] p2 2m V (i}∇p )

∇p0 δ(p0 − p) a(p0 ) d3 p0 =

= i}∇p0 a(p0 )|p0 =p = i}∇p a(p) = rˆa(p).

N) ^ rˆ = i}∇ ^ ; !#%"a6-+ 4; ;*-.%!T"% 6% ;p# !#6- ^k?<2B5fgM69u :=> E B9>7\vk E <MgU< 7 ¤ ( T,;# 36N-X Z 7 0 - 2 "%E 8 ; 3L"% %'a;*-.%!#%';# S!#6- 2 8 7

f+(*,+ V G #UBG

< =B5 <MF ] G > R W< F G R ˆ G f5 8 2>[5fF24 l6 P9u3 B i R G 5fF R < E C Ta 5fF24I> EDEDTi 9< R < 6 6_>769] E=Ti 9> R > EDG 5 g 4?< E F G 4 GMi 5 B 5fF2>V\ T E <^4I>fg;F G R a 4 BQ\.C D8Y69]U5 EDG \ R > K 5fF2 E B=Bn x(y{z|y{}~yI @ + -!# ;# - ; ˆ ^ T

Uc

a

9 Z 7 Z5? I EF!8- 2O2 EFQ\]#-* T9 Z 7 61? !#N%!# E 8P& "X 2 8(; ;*-.%! ! %_9 Z 7 Z5? ^ ;-%' XW; Tˆ ;/-.%!#% a ;# !#6- A Tˆa ψ(r) = ψ(r − a).

µ ¶ ¶ µ Z 1 i i 0 hp| Tˆa |p i = exp − pr Tˆa exp p r d3 r = 3 (2π}) } } ½ µ ¶Z ¾ ¶ µ i i 0 i 0 1 0 3 0 exp exp − p a r(p − p) d r = δ(p −p) exp − p a . = (2π})3 } } } | {z } 0

H)# !#.F + 6-.!#. XE,7 4/ -. ^ A8 2 5TQ! 2 ;%!8- &("X ^ "O" ) WN* ; ! +P\;*-.%!#% ;# S!86-# δ

4

(2π})3 δ(p0 −p)

?e? A

9 Z 7 Z0/?

µ

¶ i Tˆa = exp − ap . }

I "% ;# !#6- ; Tˆ = exp µ− i aˆp¶ ^ *7 7\; a #! "% &( 6-. !#;* S6 _! 9 Z 7 Z0/? 7 4 "% _] } !8- pˆ = ^ /7 7B; Tˆ EF/ 6 ! 2 );1c ] "O!#; ^ −i}∇r ;*-.%!#%>;# !#6- aW9 Z 7 Z0*? Tˆ NEFEFQ\/ -. ^ 2 - 2 c 5TQ! 2 a &(%"X Qa;*-.%! p 7 ¤

f+(*,+TV G #QV G )

K 5fF2 E BDBn

F > DE c B9<[6_< G2

4 BQ\ D8M69]U5 E=G \ 2 x(y{z|y{}~yIIR%!#;-.)* ! 2 !#6 E $;-%_9 Z 7 /1? A Z

e2 hp| VC |p i = − (2π})3 0

+

7

eiqr

e2 VC (r) = − r

d3 r , r

9 Z 7 Z?

G( + 6- 9 Z 7 Z? EF! - 2 ! 2 ! &(*!8"Q_!#!#* " $! %!#.ec Oz -. q 7 G( +%W; + -E ; E ;* ! 2 ;>6-+ !,; % Z 7 7 4E ;# - 6+*$%!8;) *!#4 +>!%!# 2 -.7M(aEF!8- >/ /-.E 8W +6- q = (p0 − p)/}

I± =

Z

∞

e±iqr dr.

) "O ! 8 %!#.7l 6 ! 8 %!#. ^ * "% ^:2 - 2 ! 2 &( 6-. Q ^ ;%!8"-." !# )* ; %* 3 " *-%!#"%+ ; X6- " *-\cTA )* 8 c 5TQH/ -. e−ζr H!# - ζ a I± = lim

Z

∞

0

¯∞ ¯ i 1 e(±iq−ζ)r ¯¯ = ± . e(±iq−ζ)r dr = lim ζ→+0 ±iq − ζ q 0

I EF;U6T*')8 *!#. ) *-./6 + %6 2 ζ→+0

0

Z

e

iqr d

3

r

r

=

4π , q2

9 Z 7 ZZ?

!#;-.)*EFQ H! !8 2 2 a"Q a;- 2 7 G,!#"%%EFQL EFQL]#-*P;- ! 2 ;* !#6"%QH9 Z 7 ZZ? 9 Z 7 Z? A hp| VC |p0 i = −

e2 1 . 2π 2 }3 |p0 − p|2

?: C(*-!#"Q>; X6-H;*-.%!#%';# S!86-a8 + 6- B7

f+(*,+ V G #!G

W< F R DB : =E T > {d 6_>V\ > E F T G 9> R <MF GRG 4 R > K 5fF2 E B=B Blk?4I>[5fF EDT n < i F B hGn | AˆBˆ |Gn i n x(y{z|y{}~yI I !#;% 2 %!8-(;-E 9 Z 7 0 ? ^ * A

)

?

0

ˆ |Gn i = hGn0 | Aˆˆ ˆ |Gn i = hGn0 | Aˆ hGn0 | AˆB 1B

G(6" ^

ˆ |Gn i = hG | AˆB

X

X n00

Aˆ

B

ˆ B

4

¤ G

C

ˆ |Gn i . |Gn00 ihGn00 | B

9 Z 7 Z1?

ˆ |Gn i , hGn0 | Aˆ |Gn00 ihGn00 | B

!# !# NEF% ;nO- ;*/ T 2 6X 7 o\*- 9 Z 7 Z61? !#;# -%4 - 2 ; EF 2 c 5TU8%! 2 "O!# !#!# 2 2 ;a] % !# %()* 2 ! 2 + %* 7 ¤ n0

00

"%`"/%#_'V"#v&2a*2"!D # V)V G ,Q ; "% E ] + !#"%%H;8 S!#6- ; N%) O !#W!#XE N*!#"% Q>; 2 + -.Q ; X6-.Q 2 T 6 E a 7 ( ) 9 F24?>fF A hn| x |ni = 1 a hn | x |ni = a (−1) − 1 − (−1) − 1 0

VG

π2

2

?7

n0 −n

n0 +n

(n0 − n)2

(n0 + n)2

<9 ;- 2 E 8 "% 6 8 ? ] + *!8" ;# S!86- ; N%)!# ;-%!#"%+ 67 9 F24I>fF A 1 ^ O- hm ± 1| cos ϕ |mi = 1 7 ? hm0 | cos ϕ |mi = (δm ,m+1 + δm ,m−1 ) VK3G ,Q ] 2+ *!#"% ;8 S!#6- ; x2 ; 2N%)!# - Q+ +*!#"%+ %!#X-- 2 ! ! !#Q m !#Q ω 7 9, F^C ¶ µ 4I>fF A hn ± 2| x2 |ni = x20 p(n + 1 ± 1)(n ± 1) hn| x2 |ni = x2 n + 1 n = 0 2 2 2 7 ? 0, 1, . . . x = }/mω n, n0 = 1, 2, . . . n 6= n0

,Q ;

0

Fˆ = cos ϕ 0

$# 229% - 2 & /2 0

b %.-' \

2

+"/

M 8 4 + H;8 S!#6- 2 " %+ % )*+ a - ( E 84&(%"XQ``; 7 )*E ^ "% ^ %!#6c ! 2 !8-# !c 5T -E_4!#e 6T 2 A

?[ k

"OT-E 8 &("XQ k + 6-.!#. -E 8 &("XQ k " %6XEFL!#, 6T 2 ; 9Y ^ ) * ^ !8,6\ 2 ;# #- !#Qaa + 6-E O 3 2? k !#N%!# EF ) * 2 ; 7 b " N) ^ O ;# !#6- 2 T- 2 ) 2 6N-cR S6*E 8 8%" !#" !!8-# *Q !#!#*E,7 b ^ S EFN ;# !#6- P;) - 2 3) *-. ;!#.L7 6T P)** S 7

f+(*,+ VG #\3G)

|@8 E g c BDB 5 F|< c B GYE < REDT Z 5 G 5fF G P E B i GIK C \ V (x) = −F x EDG \w> RE=GIHJGjK 4MB=NO> E BQP :=
9 Z 7Z 9 ?

¶ pˆ2 − Fx ˆ Ψ = eΨ 2m

; pˆ ) ! 2 3"/ ^ xˆ 3; !c !# ; .7M(] L%6 \9 Z 7 Z 9 ? ,"% ;# !#6- 2 - 2 ! 2 *`+ ; 2 "O ^ ;*-.%!8% ;+ ^ !8-# %-. ^ )** S * N 763. \;*-.%!8% ;# S!86-B7 39 Z 7 Z 9 ? ; ]% ;VTO 9Y! 7 6N-B7 Z 7 0*? A · 2 ¸ 9 Z 7Z ? ? p d Ψ (p) = EΨ (p). − i}F 2m

E

dp

E

M(%!8"-."`H * % ;-3 O 3 L%!8 + &( ^ ] + # !#"Q4!#; "H ; EF 7 M( )*/ Q ] + E ^ !#;-.) 2 * 6#) #- 2 ; * E 8 ^ 7 6T `9 Z 7 Z ? ? 9Y!3%!#.,c H;) -.+ ;!# 2 + H/ - 2? * LEF;! .L!#6) SA ΨE (p) = C exp

½

ip }F

µ ¶¾ p2 E− . 6m

9 Z 7Z : ?

- 2 ; 8* " " % ;# !#6- \c N;8 Q &(%.# N6) &(%"X 9 Z 7 Z : ? 7 M(!8-H)* E p = z √2m}F !H%!#.,c ;!# 2 + * L#- 2 ;-* A

a

3

ψ(x) ∼

Z

+∞

Ψ(p) exp ∞

µ

¶ i px dp = }

?em µ ¶ ¸¾ ½ · E p3 p x+ + dp ∼ = exp −i − } F 6m}F −∞ ( " r #) Ã r ¶ ¶! µ µ Z +∞ 2 z E E 3 2mF 3 2mF exp −iz − ∼ + dz ∼ − , x + x + }2 F 3 }2 F −∞ Z

+

(ξ)

+∞

&("X 2

l Q ^ 7c 53 2

1 (ξ) = √ π

Z

∞

+ 6-.P;# S!86-

µ ¶ z3 cos zξ + dz 3

9Y! 7;O-* 3 a 7 ? b " N) ^ -E F&("XT T 2 P* * %`;- P" % ;# !#6- `EF6 c! 2 )3!#; X6-.1c &(%"XUc &(%"XUc l Q 7\I ;*-.%!#% T ;# S!#66- 76T 2 E 8 &(%"X 2 8 9 Z 7 Z : ? 7 b 6"_N) ^ 0 8 + W ! 2 a]-* 6 K < EDE=GMi )*/ *P;*-.%!8P;# !#6- ,"O) EF6 ! 2 ;# ;-. "% +7 U "%% * ! %!# 2 -.>.>-EF&("X Ψ (p) ;O 6T"O6-L] + Q ^ /7 7BQ EFQ / L-. C ^ ! 8 E2 ) !8- 2 R +∞ Ψ∗ (p)Ψ (p) dp = δ(E 0 − E) 7 ¤

f+(*,+ V G # G) −∞

n

E0

0

E

R > K 5fF2 E B9>.5 @ > R BD:=>[5 ;g B Z @ 8 E g c B i

2 - 2 c! 2 #! N%!8E W&(%"O X 2 ; Lˆ 2 ^ !# !81c 5T !#N!# E ) * 2 }2 l(l + ^ "% % ;# S!#6- BA 1)

Ylm (θ, ϕ)

x(y{z|y{}~yI @ &( *!#" &(%"X

Ylm (θ, ϕ)

9 Z 7 Z[? b "%%a 3T Uc 9 Z 7 Z[? N @ W - 2 .>!#N%!8EF3&("X ;*-.%!#% ;# !#6- 7 -# W)*. ^ H!#N!# EF ) 6 2 ;a] 'P)* 2 ! 2 7 @ 6 ; 3 %!# + $ "O E 8 "%%; Lˆ )- E 8 i ;# !#6- 2 8S7 I "% 6% ;# !#6- A ˆ 2 Ylm (θ, ϕ) = }2 l(l + 1)Ylm (θ, ϕ). L

ˆi = L

X jk

εijk xj pˆk = −i}

X jk

εijk xj

I ;*-.%!8% ;# !#6- 9 ! 76N-B7 Z 7 0/? A

∂ . ∂xk

:5 ˆi = L

X

εijk x ˆj pk = i}

jk

X

εijk

jk

= −i}

X

∂ pk = (εijj = 0; εijk = −εikj ) = ∂pj εikj pk

X ∂ ∂ εijk pj = (j À k) = −i} . ∂pj ∂pk

G 6" ^ "% $;*-.%!#% ;8 S!#6- 2 8'!#"% "%%; ( Lˆ O 7 4 ' 6- ^ [ˆx , pˆ ] = i}δ -\cN';8 S!#6 - ^ i!8-8 -. ^ !#"% Lˆ 2 4"%ij ij% ^ 4;/-.%!#% ;# !#6- 2 8H "%%O7M(] %H!#N%!# E &(%"X 2 Lˆ 2 W ;*-.%!8%$;8 S!#6- 46"O 3\N* W!&( !#"T&("X Y (θ , ϕ ) 7 4 -LH"% + a;8 S!#6- 2 !8%!#a4% ^ `lm%+ -E p θpp ; .L)/ S c ;- p ^ T r 7 ϕp ^ \ 1 c )*/ *L/ TNE -TNE 7 6T. ^ EF!8-\-\6\. &(%.# N6) Y (θ, ϕ) ^ * 6"3 6\*$!8-%6P] + T -.P% L7 ¤ lm 0 8 + (Hˆ − E)Ψ(r) = 0 ] + *!#"%%$;# S!86 - 9<)*WN%)! "%+>EFN6E !#N!# EF &("X "%+ +6 O-./ Hˆ ^ -+ Hˆ ^ ! !#N%!8E ) * 2 E (0) ? L- QQ !#!0 8*E Q%!##-.,"%]*&P&(X Pk)-/ T 2 ;N%)!# A X nD

jk

jk

¯

(0) ¯ ˆ E n0 ¯ H

¯ o E ¯ (0) ¯En − δn0 n E CEn(0) = 0.

@ N!# EFW) * 2 ] + 2 - 2 c! 2 "% 2 %8 6" !#*!8"+ % 2 n

¯D ¯ ¯ ¯ E ¯ (0) ¯ ˆ ¯ (0) ¯ det ¯ En0 ¯ H ¯En − δn0 n E ¯ = 0.

0 # + 3;*-.%!#%';# S!#66- a*!#.

+

µ

¶ Z p2 − E ΨE (p) + hp| U |p0 i ΨE (p0 ) d3 p0 = 0, 2m ½ ¾ Z 7 i 1 exp (p0 − p)r U (r) d3 r hp| U |p0 i = 3 I ;"*!8(2π}) " ;O-* 3 }2 N- `!#'!#;-.) !

Q8

8

2 "% ;# !#6- 7Ul TN!8-- * ^ T] + 2 )*V Q!# 2 EF6

! 2 &("XQ "% a!8X 4"% %_;# S!86-# L!#;/ P!F!8 !#!c 5T`; 6% 7C *!#"O 2 ] + 232 - 2 ! 2 ;!#Q &(%"X Q\;*-.%! 7/M(] %P ; "%

:0 ;# !#6- ,6"O T ;%!#Q, 7M(,! !8-# %6 !#!#* ^ !#%!# 2 5TQ84) !8- 6N )** Q!#!c 5TQ8a!#X ^ !#W; 2 ! 2 ;*-.%!8 ;8 S!#6- ^ \*!8- !#;-.) ! 2 2 )*15T Q ^ 3] + O *!#"%7

V"#v&2a*2"!D

"%`"/%# _'

#

V G < QEFQ +*!#"Q %!#X-- 2 9Y ! ! m ^ !#6 ω ? 8 * ! 2 !86X% !#%!# 2 7 ,Q ;*-.%!#W;# S!86- -Q &("XB7 9, F24?> F A c (p) = p 1 p ξ = √ H (ξ) e n

2 n!p0 π 7 ? 2 p0 = m}ω n = 0, 1, . . . VKXG @ !#* HV* T H 8 .%! 2 -\6T.a> 18 !#%!# 2 EF$]-* E ; Hˆ ) *!#E,A H = 11 7 , Q _! 8 % N # ! F E 4 ) * ] + 71 2 ∗ H21 = c p 7 b ± (a − b)2 + 4c2 }3 V)g G 6;! .\% 0 # + T! "*-%!#" α ;*-.!#';8 S!#6- 7

r

1 F24I>fF A

V G ,Q

µ

n

n

−ξ 2 /2

2 S8 A

p0

7 6

Ψ1 Ψ2 a H22 = b H12 = F|4I>fF A E = 1 {a + ± 2

; X/-%

V (r) =

¶ Z 73 α p2 1 0 3 0 − E ΨE (p) + 2 3 Ψ (p ) d p = 0 E 2m 2π } |p0 − p|2

#! N!# EF\&("X4; 6 "% 6E W"% ;# !#6- 3! -# !c 5T`!#;%!#N A/!8N%!# EFR& "X\; " E U5T%! 2 ;*-.%!8%';8 S!#6- ^ ) *-./6; ! 2 "% (;# S!86-7

-}\}2 +($#

M 8 ,* +,;# !#6- 2 " + 2 - 2 ! 2 !#E ! -%6 ( * 6"L6) EF%+ 8 E BDF2< REDGWHJG R > GYX R <[k G 4I< E BUP^ "% %8! !5T*!8- 2 ! 2 ! ;, 5T,. c "+ 36+ ; 6 Uˆ A |a0 i = Uˆ |ai .

,;% ;# #- 6+ T;A !F;O-% ; N6) 6 2 9 Z 7 Z ? - 2 4I>fg;F GR <

m

9 Z 7Z m ?

7I Uˆ† = Uˆ−1

#! !8 ;# - * ;-%

: ;- ; N)%6 2 - 2 G > R W< F G R < A 9 Z 7 1%5? ,F%!86W9 Z 7 Z m ? ^ 9 Z 7 15? ;"O)*. ^ ;\66E 8,; N)%6 2 8 %!8 c! 2 ) E `!8-# 1c 5T,"%!#%"XBA k !## P) 2 &() *!#"Q8 - k !#N%!# EF ) * 2 ; k !#"/- 2 EF3;)# 2 " >+ O-.N ;!#!#H9Y ) ^ + 6-.%!8.3H"O ? k 6EF ]#-* E ; k " %6XEFH!#e 6T 2 ; 9Y ) ^ !8, 6\ 2 ;# #- !#Qaa + 6-E O 3 2? 7 M( % 6 E 8 ; N) 6Q 2 - 2 c! 2 ! + ;E !#!#*E "% H9Y! 7)*/ * 61 9Fh,!# 0/? 7 6EFL; N) 6 2 6 LE ;% ^ HH 2 W! -%a *O TEFN ; N)%6 ;) - 2 \!#! 5\*!# \%;%!#.T7 6\ )*/ 7 Fˆ 0 = UˆFˆ Uˆ−1 .

f+(*,+ V G #UXG >%^BDF^] k?< K <M:l8 R BQ\ > R < p;nqp;p 4A@ GR \w< 69Blk[\w>28 E B=F2< RE=T Z R > GMXR ) ;6A8S7K> * !#"O.

6"% 6>;N) 6 ^ "% %/ - QEF>; ˆb ^ ˆb† !8-+6*EF7 U )-/ L *6- -!c &(%"XUc %!#+H!#%!# 2 2 ! 75T O + 4%!#X-- 2 ψ (ξ) ;`N%) !# -E 8 &("XQ *! 75T+ 4%!8 X-- 2 χ (ξ) A 0 n

∞ X

∞ X a √n (ˆb† )n ψ0 (ξ) = f (ˆb† )χ0 (ξ). ψ0 (ξ) = an χn (ξ) = n! n=0 n=0

9 Z 7 10/?

I !# !# !P;# -* %!8+ !8%!# 2 2 9 0 7 [Z? - 2 &("X * A ψ0 (ξ) 9 Z 7 1? ˆ Bψ0 (ξ) = (ˆb − γ)χ0 (ξ) = 0. l ,N ( *H - 2 %!#+ P!#%!8 2 2 ! 7 5T + (%!#XO-- 2 7

*

28, /2:2/-, 10/7- -: *2-1, *2)

:Z M(* !#6_9 Z 7 10/? `9 Z 7 1? A G,!#;-.) 2 / 3 !#

[ˆb, f (ˆb† )]χ0 (ξ) = 0.

∂f (ˆb† ) † ˆ ˆ ; [b, f (b )] = ∂ˆb†

;-%6* ; EFP%6 2 - 2 76T P"%+ :A +

∂f (ˆb) [ˆb† , f (ˆb)] = − , ∂ˆb f (ˆb† )

A

∂f (ˆb† ) = γf (ˆb† ), † ˆ ∂b ˆ† f (ˆb† ) = c0 eγ b ,

I

*!8; * -ˆ# N5\-Q >(; #N E - 2 Uc- 2 ; %!8! 2 2 EFQ4/ L-.7

c0

9 Z 7 1%Z?

f ( b† )

9 Z 7 11?

9 Z 71 9 ? EFNW6" N) ^ NE ;

2 ˆ ˆ† ˆ ˆ U(γ) = f (ˆb† ) e−γ b = e−γ /2 eγ b e−γ b

1 ! 7 9 0 7 [Z? 7 / L- 2 N)%63 ˆ NE - 6E c07

)-/ T L9 Z 7 10/? b ; .LU(γ)

* L;# S!86.

9 Z 71 ? ? IR8 * ;. ; /-+'! 9 Z 7 1 ? ? !8 U6T 4N,5T W ; N6) 6

!#"%%E _76T* ψ(ξ) %6 2 0 # + 9 0 7=m 0/? aQa&("X Q χ(ξ) 9 Z 71 : ? ˆ ψ(ξ) = U(γ)χ(ξ) a; S!86_9 Z 7 1 : ? `9 0 @7 m 0/? A 9 Z 7 1%[? ˆ ˆ [ˆb†ˆb − γ(ˆb† + ˆb)]U(γ)χ(ξ) = ε0 U(γ)χ(ξ). R! - L 9 Z 7 1%[? ;* Q!8%6.W!8- %; 6% ˆ−1 ^ +#-.6 ; ; %6W; N)7 ,Q UEF(γ) Q +#-.67 M( +* %6% ; N) 6\c Uˆ−1 (γ) >R) ; E ; -.%!8 7 a - 2 ] + /- ; N) * 9 Z 7 1 9 ? !# !#`! /

*!#%$IR QO- 2 9Y! 7)*/ * 00 h,!# 0/? ˆ ψ0 (ξ) = U(γ)χ 0 (ξ).

ˆ† ˆ ˆ U(γ) = eγ(b −b) ,

:1 )** !#;-.) ! 2 / 3 !#% R> "% AkOC ]*;N-- ;k(! &P_9Y! 7 ' )*/ 05 h,!# 0/? 7I + P;- A 9 Z 71 m ? ˆ ˆ Uˆ−1 (γ) ˆb† U(γ) = ˆb† + γ; Uˆ−1 (γ) ˆbU(γ) = ˆb + γ, "* ˆ ˆ Uˆ−1 (γ) ˆbU(γ) ˆ Uˆ−1 (γ) ˆb†ˆbU(γ) = Uˆ−1 (γ) ˆb† U(γ) = (ˆb† + γ)(ˆb + γ),

6"aL%6L9 0 7@m 0/? ; NEFQ O ˆb†ˆbχ(ξ) = ε00 χ(ξ),

+ ε00 = ε0 + γ 7I "% % + ] + *!#"Qa!#; ">N* 3!#;* S6. ! ;-% E 6)# #- 0 7 97 M(%!#_; . -E W&("X'!#6XE 8 !#%!# 2 QB7I !# !#`!9 0 7 [ m ? ^ I !#!#H!L9 Z 7 1 : ? ^

1 χn (ξ) = √ (ˆb† )n χ0 (ξ). n!

1 ˆ ˆ ˆ U(γ)(ˆb† )n Uˆ−1 (γ)U(γ)χ ψn (ξ) = U(γ)χ 0 (ξ) = n (ξ) = √ n! 1 ˆ ˆ† ˆ−1 1 n = √ [U(γ)b U (γ)] ψ0 (ξ) = √ (ˆb† − γ)n ψ0 (ξ). n! n!

*!8. χ (ξ) = π−1/4 e−ξ /2 7 ¤ 0 C,!#"; N) 6 2L2 - 2 c ! 2 "-! !#*!#" 6-+ 6 E 8 ; N6) 6QB7 2

V"#v&2a*2"!D # G a "O)*6. N,\5 ,!#Q!8%\%6E 8 ; N) 6Q 7 #)G a "O)*6. 6%!#.; N)%6 2 9 Z 7 0*[? 7 "B G a "O)6.3*

*!#%H9 Z 7 1%Z? 7

"%`"/%# _'

9 gU AB %_ 6* 6!#"%Q %"X EF!8-.a"%% 6E 7? ^ [ˆb, (ˆb† )n ] [ˆb† , ˆbn ] KV ∗ G M( . %6%!8.; >9 Z 7 1 9 ? 7 ) ∗ G M(-.L!#, 6T 2 9 Z 7 1 m ? 7 ∗

:9 2% "b $# +"_'V 2 v 2, 2 (*) v 2-'V b 2 ;8 S!#6- Q> *!#"%-."!#; "7I ;# E 1T5 U8W) - A8 !! %6-!#.`)-EFH!#;%!#NE EFN> *!#"%Q';

QBA%" E ^ ;*-.%! ^ ] + /7 :7 !#. E ! ! O N- F!#;!# EF ;# S!86- 2 )*6!8%!8; 6PL- E 8 &(%"XQ `* B7I -W`%!# 8 ;# E 15TU8 !8- 1 ! 7 ^ ; ^ 9 Z 7 0*[? 3 ^ !#!#1 c 5TL6EF ; N)6 2 - E ; . 2 !# %

. )*!#V!#.3 * B7

f+eZP1!) , +e,,+ R! - !#;">!#N%!# E 3 8 ) * QL; ()* 2 ! 2 !F O * * ^ N`;-.) %6.%! 2 6"$; ^ * *!8" 2 &( "%E > 8 )!8L* B7I ] !8-,)* T!#%!# 2 2 F* ,EF6

! 2 -."% T 5 * "F!#%!8 2 2 7 b "% ;# !#6- ( "!8%!# 2 Q>a; 3) EF%6 ! 2 R > K 5fF2fC E B9>V\ R > K B EH > R < 7 4 ;E ,-EF &("X,F;# S!86- 0

+ ,N * 6. "!8% S 7%M(;-* EFQ ! -%Q ^ *7 7" + SL * )! 2 L-."%W; E ^ W\-EF\&("X ^ / ) EF ! 2 ;# !#6- * n% Q) N +9Y! 7O T ? 7 b "%% ;# !#6- \c N L!#;%!#6- > "O! H 7 4 ;# -'; >] \- cXH!#%!# 2 2 Uˆ (t) !#,6T 9 Z =7 9 5? ΨS (ξ, t) = Uˆ (t)ΨH (ξ); Uˆ (0) = ˆ 1, + Ψ (ξ) ≡ Ψ (ξ, 0) ^ ;* !#6 -! c &(%"X\c $&('9 Z =7 9 5? L*H ,%S6 0 # + !,+#O-./% Hˆ 7 I ) *-.66 ;-%'% O 3 2 - 2 ; \] \- cX A

∂ Uˆ (t) ˆ Uˆ (t); =H ∂t

+ 76\ P LO :A i}

Uˆ (0) = ˆ 1.

¶ i ˆ . Uˆ (t) = exp − Ht } µ

9 Z 7=9 0/? 9 Z 7=9 ?

6%!#. Uˆ (t) ;"*!#"H 7 ,;% ^ L; 6-+ H!(; N) 6* ] -\cX>L"O-! !# !#"%QW 86"% O 3(* L\! ! %. "O" ; EFT!# \6 / *("6*!#"% ; N) 67 M(#

,a%! cR !#;-.) /-!#.3;# !#6- 0 # + 7

:? f+eZP1!) , p",_01)+ A a R > K 5fF2 E B9>_> i k?> EDX > R&H < ,T 6 \c " ;# !#6- Uc 0

2 - 2 ! 2 ;;-* LE !8-%6* ; 8 + ^ /7 7 "E !#!# 2 Q %!#66 c! 2 ; O LE ^ !#%!# 2 2 )* 2 c ! 2 3* >-."%3)* !8 ; 7 4 O ^ ;H; 8* P ;# !#6- A2 0 # + "H;# S!86-# \c n% Q) N+#- 2 &("X 2 N W; N) EF6.! 2 ; )*6" ^ N% 9 Z =7 9 5? A

ˆ −1

ΨH (ξ) = U

(t)ΨS (ξ, t)

=

¶ i ˆ Ht ΨS (ξ, t). exp } µ

I ; 8 ; 2 ! 2 k;V\w> E B A FˆH (t)

= Uˆ

−1

(t)FˆS Uˆ (t)

=

µ

¶ ¶ µ i ˆ ˆ i ˆ exp Ht FS exp − Ht . } }

9 Z 7=9 Z? 9 Z 7=9 1?

M N6) 6\9 Z 7=9 1%? ;) - 2 PEF*!8 O 3 2 - 2 ; ( T;# !#6- n% Q) N +OA dFˆH 1 ˆ ˆ = [FH , H]. dt i}

M(8 S!#6- %n Q) N +\ N3"Q a;- 2 7

9 Z =7 9,9 ?

f+eZP1!) , O)01!(*',_ Z PQK I "%Q 8 6"% # "%`;Q8* ! 2 !!8-8 .`!#!#*E ^ !# !# 2 \ 5 ) *!#"%-."Q83) Q!#1c 5TQ8T; S!#!#* 7%I']% !8-F+6 O-./4/ ;# S!#6. !8 E ˆ =H ˆ 0 + Vˆ , H

+ Vˆ ; 6 )* Q!# 2 ^ Hˆ +#-.66 N ) % )*/ * Q!8 2 7 I 6"Q8 !8!#* 8 !# 0 - 2 ;! 2 * Q ] -\c X !#;-.) ! 2 R > K 5 F| E B>24ok?<MBQ\ GWK > i 5 F|4MBQP 7BM( 8* a-Q &(%"X`W;# S!#66- 0 # + L"`-Q`&("X`W;8 S!#6 - )*V Q!# 2 Ψ (ξ, t) %!8! 5T*!#- 2 ! 2 !>;%, 5T,. c 6+ ; N6) 6 2 ¶ µ 9 Z =7 9e? ? i ˆ H t Ψ (ξ, t), Ψ (ξ, t) = exp

"%P-! 2 49 Z 7=9 5? )* Q

}

0

S

ˆ →H ˆ0 7 H

:e: M * !#6"49 Z =7 9e? ? W* 0 8 + - 2 Ψ (ξ, t) ( ! +O-./% Hˆ ; * "4!8-# 1c 5T*W% \c_ - 2 Ψ (ξ,S t) A 9 Z =7 9: ? ∂ i} Ψ (ξ, t) = Vˆ Ψ (ξ, t), ∂t + µ ¶ ¶ µ 9 Z =7 9 [? i i ˆ t Vˆ exp − H ˆ t . Vˆ = exp H

I !# !#!,9 Z 7=9e? ? ; 8* (;# !#6- 220 # + " ;# ' !#6- \c n% Q) N +\W;3%!#15T*!#- 2 ! 2 ;;-

h,!8E

Fˆ

}

0

}

µ

0

¶ ¶ µ i ˆ i ˆ 0t . = exp H0 t FˆS exp − H } }

9 Z =7 9em ?

!8-*_9 Z 7=9em ? 2 - 2 ! 2 9 Z 7=9 [? 7 O T 2 - 2 ;\;# S!86-a)*V Q!# 2 EF+ - 2 L!8-# ! c 5\'N)% A

9 Z 7@? 5? K 6-)39 Z 7 9,: ? 9 Z 7@? 5 ? ;) - 2 ,! - 6.,EF L % ^ \;# S!86-# P) Q!# W 2 2 - 2 ! 2 ;% E 3 0 # + %!#" n% Q) N +!#" 7 dFˆ 1 ˆ ˆ 0 ]. = [F , H dt i}

"%`"/%#_'V"#v&2a*2"!D # "3G @ !#* !aH)*! 2 5T +-.6% *6-.EFQ % $* 8 * O- !8. !#%!# 2 _!`-Q &("X Q Ψ(ξ, 0) 7 ,Q>-!c &("XUc ]Qa!#!8*E_3;%!8-8 1c T 5 ,%E_ Ψ(ξ, t > 0) 7-l + *!#"Q !8; " N%6) ! !#!#*E ;# ;-+ c ! 2 ) *!#E B7 * S T 6 . ! !#;-.)6*_6+ `; N)/ 2 7 G M(-%.% 2 9 Z 7=9 0*? ^ 9 Z 7=9,9 ? ^ 9 Z 7=9: ? 9 Z 7@? 5? 7

:[

*R2 % 2

2 a'V v/2

# 9

b#

$#

U ! ! 6">6) EF%EF/ 3 +; + % !#"%(%6 A 9YK 7 0/? d2 y dy x 2 + (b − x) − ay = 0, + a b )** SEF "%dx %;O- "O!#EF "%dx!#6E,7 + R > H 8Y6lP REDG > 4 E 8M6_> 7 6T (;# !#6- 2 !#NQa!%!#,. c ;!# 2 + * L- 2 E / 3 1 c + ; + % !#"1 c &("XUcTA a x a(a + 1) x2 y (x) = 1 F1 (a, b, x) = 1 + + + ... b 1! b(b + 1) 2!

9YK 7 ?

M( X #-E 8 ;-* L-.E 8 a P; 5 6 ! 2 \;-% #! ; −a 7 M( |x| → ∞ T ! -# !c 5T !#;*!#"%P;# !#6 - A 1 F1 (a, b, x)

∼

*!8.

Γ(b) (−x)−a 2 F0 (a, a − b + 1, −x−1 )+ Γ(b − a) Γ(b) −x a−b e x 2 F0 (b − a, 1 − a, x−1 ). + Γ(a)

x x2 + a(a + 1)b(b + 1) + ... 1! 2! PN) - 2 Γ &(%"XB7

2 F0 (a, b, x) = 1 + ab

I # L!86O

2 2' "!D" ' - M(-%E ( h NEj6\ ;k\l 6 2 - 2 c! 2

&,&( X6-.+3 2

y 00 − 2xy 0 + 2ny = 0,

+ *- 2 E 76T 2

n = 0, 1, . . . ,

:m U8 ; 2 "B7 a / )# *!#. *!8"-."%3)-E 8 Q8>;# S!#66- QBA

+

n

0/? &(/-TU* +OA

n x2

Hn (x) = (−1) e

5 '!# ; 2 ? 6)-* 3P;NEF%c T Hn (x) = (2x)n −

9 > 7 Z?

dn −x2 e ; dxn x

A

n(n − 1)(n − 2)(n − 3) n(n − 1) (2x)n−2 + (2x)n−4 − . . . 1 1·2

Z? )PEF/ 3 1c + ;+ % *!8"1c &("XUcTA

µ ¶ (2m)! 1 2 H2m (x) = (−1) ; 1 F1 −m, , x m! 2 ¶ µ (2m + 1)! 3 H2m+1 (x) = (−1)m 2x1 F1 −m, , x2 . m! 2 m

1? "% 2 &(*- A

Hn+1 (x) = 2xHn (x) − 2nHn−1 (x);

2/ 2 w

o\%"X 2 <

H0 (x) = 1;

H1 (x) = 2x.

9 >7 1%?

O (;+ P SL9<;!## E ;-% <

? ) EF c! 2 + *- 2 EF 4"A8 ±1 76T 2 &,&( XO 6-.+ 3% 2 ·

¸ m2 (1 − x )y − 2xy + l(l + 1) − y = 0, 1 − x2 2

M(

00

0

4!#6 2 ! 2 ;-% <

O 7 M(# * )# *!#.L&(/- U* +\ - 2 ] Q8H&("XQBA l = 0, 1, . . . ;

m = 0, ±1, . . . , ±l.

m=0

|m| Pl (x)

l+|m| 1 2 |m|/2 d = l (1 − x ) (x2 − 1)l . l+|m| 2 l! dx

M(!## EF ;-%E !#"U8a&(%"XQ A Ylm (θ, ϕ) =

s

<

6O \8 2 TL!8"%>! &(

2l + 1 (l − |m|)! |m| Pl (cos θ) eimϕ , 4π (l + |m|)!

9YI 7=9 ?

[5 2 - 2 c 5TQ8%! 2 + *- 2 E H3# Q4! &((76\ 2 H%6Q ·

µ ¶ ¸ ∂ 1 ∂2 1 ∂ sin θ + + l(l + 1) Ψ(θ, ϕ) = 0; sin θ ∂θ ∂θ sin2 θ ∂ϕ2 ¸ · ∂ + m Ψ(θ, ϕ) = 0 i ∂ϕ

aN) 1c T 5 >T Qa;-!c 1c !#!#V

∞ X

Z

0 l X

2π

dϕ

Z

N) !6A

π 0

Yl∗0 m0 (θ, ϕ) Ylm (θ, ϕ) sin θ dθ = δl0 l δm0 m ;

∗ Ylm (θ0 , ϕ0 ) Ylm (θ, ϕ) = δ(cos θ 0 − cos θ) δ(ϕ0 − ϕ).

9YI 7@? ? 9YI 7 : ?

@ &( *!#" &(%"Xa - 2 c 3 " %a!#,6T Uc l=0 m=−l

cos θ Ylm (θ, ϕ) = s s 2 2 (l + 1) − m l 2 − m2 = Yl+1,m (θ, ϕ) + Yl−1,m (θ, ϕ), (2l + 1)(2l + 3) (2l − 1)(2l + 1)

9YI 7 [?

;# !#6- 2 c 5TH!8NQa)-* T , +- Qa!#4;LN%) !#>! &( !#"U8a&(%"XQ 7 M(# * 2 EFQaO 4 "%E 8a! &( *!#"Q8a&(%"XQ A 1 Y00 (θ, ϕ) = √ ; 4π r 3 cos θ; Y10 (θ, ϕ) = 4π

Y1±1 (θ, ϕ) = ∓

$9%.2}2R 2 2}(' M(!## EF`;-%E < +

76T 2 &,&( X6-.+ T 2

<

(α)

r

2

Ln (x)

xy 00 + (α + 1 − x)y 0 + ny = 0,

3 sin θ e±iϕ . 8π

9YI 7@m ?

2 - 2 c ! 2 + *- 2 E

n = 0, 1, . . .

a; 8* 2 NEFEF(;-%E < + 7 a / "%EF 2 EFTEF6 T 2 - 2 ;!#8 E 84;-% + A

M(

α=0

[O0 0/? &(/-TU* +OA Ln(α) (x)

1 −α x dn n+α −x x e [x e ]; = n! dxn

? EF/

2 +; + % !#"O 2 &("X 2 A

1 Γ(n + α + 1) 1 F1 (−n, α + 1, x). n! Γ(α + 1)

Ln(α) (x) =

L $_#

+ '; 2 "O ) EF% c! 2 + *- 2 EF 76T 2 X - *!8"+ 3 &,&( X6-.+3 2 A 9 a 7 0*5? x2 y 00 + xy 0 + (x2 − ν 2 )y = 0. a / "%EF 2 EFPEF6 T 2 - 2 &("XQ>R*! !8- 2 A 0/? 6)-* 3P 2 :A

o\%"X 2 T> *! !##- 2

ν

Jν (x) =

∞ ³ x ´ν X

2

k=0

(−x2 /2)k ; k!Γ(ν + k + 1)

? EF/

2 +; + % !#"O 2 &("X 2 A

µ ¶ 1 e−iz ³ z ´ν Jν (x) = 1 F1 ν + , 2ν + 1, 2iz . Γ(1 + ν) 2 2

> *! !#- 2

@ &( *!#"O 2 &("X 2

jl (x) =

r

π J 1 (x), 2z l+ 2

l = 0, 1, . . .

EF6

! 2 )P]-* 6EF ^ 6; ^ sin x ; j0 (x) = x

sin x cos x j1 (x) = 2 − ; x x

o\%"X 2 l Q

(x)

µ

j2 (x) =

¶ 1 3 3 − cos x. sin x − x3 x x2

2 - 2 ! 2 + *- 2 E '76\ *'% 2

aEF6 6 ! 2 ) &("X>R*! !8- 2 ; 2 " y 00 − xy = 0

+

(x) =

1√ x [I−1/3 (ζ) − I1/3 (ζ)]; 3

Iν (ζ) = i−ν Jν (iζ)

ζ=

2 3

x3/2

7

(−x) =

9 a 7 00/?

± 13

A

9 a 7 0/?

1√ x [J−1/3 (ζ) + J1/3 (ζ)], 3

[

2

5 EDG 4 E < P

a E HK 7 @ 7C( 2 %8 "OaK 7 @ 7 a E 7 7 A ,*%" ^ 0 m1: Z 7 : 51 !7 7j> -A8OX a 7 G 7 4 !#E "%Q 8%" a 7 G 7> -;8OX 7

7 A ,*%" ^S0 m [Z 7 ?,? 1 !7 Z 7jn /-X"Q I7 7 / S ; "Q_ 8%"%'I7 7 nO6-X"Q ^ > 7 7C,6"% ^ I7JG 7C(+# 7 7 A ,"O ^S0 m,m 7 [[5 !7 1 7 @ N"`)*/ *4;W !#"%Q &() "% < 7@n 7"n "% ^ I 7 G 7 @ +#6 " ^)4 7 W o 7 b % !# a B7 7 AIRE ! W6 7UT6 " 7 ^ 0 m [61 7 Z0 3m !7 07

G G 6 E BDF2>V6D] E < P

O * < 7 a 7 b *!#"O 2 &() "O7 A I 0*5 /7 < 7 a 7 < S ^ 7 7 < & 6\X 7 7 A,o )* -*7 b 7 Z 7 AC(%6 2 8%"O7 ( - 2 !#!#"O 2 2 7 55O0 7 [ 5Z !7 7 < I @7 n 7WC(%! *!8"Q &() "B7 AWI /7 I7@n 7 < ^ 7 K 7I ^ I7JK,7 2 - 7 7 A ,*%"O7 b 7 7 0 m: 0 7 m Z ?3!7 /7 7Do,-c++ Z 7 o - c + + V7 * S ; "%Q 8%"%7 API M(* L# :7%K,7JK,7 @ "-O7 h( ; X A H "QQkOM(U @F@ 7 b 7 0 7 555 7 Z10 7 07

<

[Z @ %!86-BADC(;EFHG(+ .LI !#-. C( LK - "O!#Q @ 6 ! - U# "

b U8%% 4

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