Алгоритмический подход к квантовой физике

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Author: Ожигов

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y~y}zxx z|x α, β, γ xu|x~y}zxy ||zwyx}wxzx y ttu|~|w }wt~ux u|t z|}}twx |α|ttu|~|w}v | }yx|{u ||zwyx}wxz ~yy τ (α) u|~|{| z|z x(α), v(α) }|y}wt}w y}x z | zt| |{xw| }v|uu|v } z|uwt} |t τ (α) = λ(α) α uty}wt u|~yux {v zttxu|w x yuwt yywzx ttv|x}xw tw z|uwt |{xw }t}wtxw x {t s}w zttxu|w y|{ut ~|}wx xyvw}v ztwttty y{ty t| zttxu|w } wt~ut}w δ = r, r tut t{tyux y|{ut ~|}wx yw }ttwyw}wt|w (δ/) t{tyux z|uwt |{xw }y wx t{tyuxv |t{uvw z }t}wtvx x wt~yz x| C lδ + j, mδ + k, n + s y t|ty}{x u| ||ut t{tyuxy |zx }y z|uwt |{xw t }y j, k, s ∈ {0, 1, . . . , r − 1} u|tyuxv y|{ut t}wx wt~z| wt }~xw|y ~wt |{xw| ~|}wx zy C |u| X 3

δ, l,n,m

δ, l,n,m

λ(α).

u|t }yz| |w|zy t{|}wx tyy tu|~x |}}|wx|yt } ~ywt x zttxu|w x }ztt}wy y uxux {vy } wt~ut}w }y }yx|{u ||zwyx}wxz tyuw ~yy K }{x wyy |xz}xt|w u|~yuxy δ wtt ty yw t}z|w}v xuyz} ~|}wt ~x}{xw ty |}yy{yuxy wtttu|~|w t{y }ttwyw}wyy |{xw x{x t{ut uzx ztwt ~yy |Ψ i || ||~| }t}wtxw wt ~wt tyy{xw z|z xyuut yt|w}v }ztt}wx x |{x w z| x }wt{zutyuxv ~wt {v }y u|~yux t |Ψz| xux i {t yyuxy |uyuxv yxuy| } wt~ut}w t C(M )tδ M y}w ~x}{t }y t}wtvuutt |x{wtux|u| } wt wt~ut}w t{uv{t} |yu}wt }{t|x ~wt }{~|y δ, α: x(α)∈Cl,n,m

K ,δ

K(t) ,δ

3

$

i |ΨK(t) ,δ i ≈ exp − Ht , h

| }{~|y |x{wtux|u| |x}vyt tw yyux wt y |yu}wt ut }}{y tut{tx~y}zt z}tuyuw syt|t|uxy||zwyx}wxz z|~|}wx }wt{zutyuxv ttxxy y|zx x| v¯ , x ¯ , λ , ∆t ; v¯ , x ¯ , λ −→ v¯ , x ¯ , λ ; v¯ , x ¯ ,λ , y ∆t , ∆t yv tyyy }t yyux yyt }wt{zutyuxv ytt x wttt z|uw| |{xw } xx z | x t{ {|u t{ tyy{yuxx z|uwttt }t}wtvuxv y t~yy tyy{x t|t yt|w}v x z|zx wx ~x}{| x|w ~x}{| λ(α) yw }wt{zutyuxx z| yy{yuxy w|zx 1

1

1

1

1

2

2

2

0 1

2

0 1

0 1

0 2

0 2

0 2

i

λ0j = λj · e h ∆Sj , j = 1, 2,

y

∆S = L ∆t , L = E − E y}w {||ux|u z|uw| |{xw ~x}{yuu wt~zy {y|y u| }yyxuy wx }t y yytjt}wt{zutyuxv yux ty }ztt}wx z|uwt tut tyy{xw x}tv x wtt ~wt }t|yuxy }~xw|yw}v x y} ttu |x|uw }v|uuy } wy u|}zt{zt ztu}wz y}{x tux }z{||w}v x{x {xzx z wt tut }y{|w wxut xuwyyxw |{xw t{yλw ttx u| }{x|uxysy}{x }t|yuxy yuyy x x y |{xw xuwyyxw t}xy y}wzwxut }t|yuxy t{ut {xy z t wyy }y ~x}{| λ(α) {v }y z| α ztwty tyuw t u|tvw}v zy C st{~xyy}v u|~yuxy uzxx !Ê°±°Ã Ê°³ ·¯ Â®³È·³Æ µ³º²³¾Ã²Â °³´®± ¹³¾·³¹±¿ ËÂ·²Í¿ ÈÂ®¯° »¯°Ã º¾Ã·³ ¼±¸¼Ç¹·ÇÀ ¹®© j

j

j

j

kin

pot

j

δ, l,n,m

δ=

yw wyy x{xyuxy u|}wtvy t{utt uzxx |}}|wx|yt ~|}wx |}yy{yut{ut}w wty}wxwy{ut }{~|yy}{x w|yzwtxx }y z|uwt |{xw }{~|ut|t t wt }{~|y ~x}{yuxy v| t{y tut x{xyuut y}w|xw z|z }xt|uxy ~x}y{ λ(α) t }y z| α t|x z }ttwyw}wx wt~zy wt Ö }{~|y y}{x{v}yzytu|~|{uy z| {x{||}t{tyu zy }ttwyw}wy wt~zy~x}{yuxv x }| u|~|{u |u|yxuxy t u||ty| ~x}y{ | tyuw wt| λ(α) |Ψ i yw x{xyuxy t{ wt y}w |}w u| x{xyuuty u|~yuxy t{utt uzxx ~|}wx tyuw t x }{txx ~wt |Ψ i v{vyw}v x{xyuxy yy t{utt uzxx tyuw t w|zt{| v{vyw}v x}zywu |u|{tt yu|ut}ztt xuwy|{| t w|yzwtxv t~ut}w wtt x{xyuxv yw wy y ~y yuy t t{ ~x}{| λ(α) }{x y t{txw ~wt |{xw| yux}w| wt y}w xyyw}v z|uw |{xw g |z}x|{u|v ttu|v yw x yw |}}twyu| wt~ut}w ~x}{yuxv wt| t}wxuw| w| }xw|xv |λ(α)| = g |||y ttvw t }{yy x}{yuuyz ztutxx z}yxyuw wt ~wt t{xu}wy }{~|yy{t}wt{zutxwy{u|v ty{ uyt{ xtxw y}}tx ~x}{xwy{u |y }{tuvyw yztwty tt yy |}}twx }x}wy ~|}wx u|tvx}v |twx~y}zt xyuxx }t}wtvw }{yy twyux|{t {v z|t v{vyw}v twyux|{ }wt{zutyux } }t}yuxx w|z ~wt yxu}wyuu ~|}wx|x }{x {xu| ty| ~|}wx y{xz| t }|uyux } x |y|x x tyy x ~x}{t y{xzt w|ztut}zttxyuxv }x}wy tut |}}|wx|w z|z |utxut}w |zt}ztt ty}}| | xyuut z|z tyuw ty{ }

s}w u(x, t) y}w {twut}w ~x}{| ~|}wx wt~zy x t wt| w| {twut}w t~xuvyw}v |uyux wy{tttut}wx |ΨK ,δ i

K(t2 ) ,δ

2

K(t1 ) ,δ

1

ut =

1 Cuxx , 2

y C t}wtvuu|v yyuxy ztwttt x u(x, 0) = δ(x − x ) xyyw x |}}tt zxt

Ö t }wttu |}}twx z|uwt t{x {v }ttut tutyut ~|}wx u|t ytu|~|{ut vy}v tzty x xyy |}}tt |}yy{yuxy |{xw ttxtu|{uty }{x t}~xw|w t t{y yy vtwt t{~xw}v t}~yw tut u|wx exp − x 0

1 x − x0 exp − u= √ . 2Ct 2πCt

α 2 2

K(x2 , t2 , x1 , t1 ) =

2π ih(t2 − t1 ) m

−1/2

exp

im(x2 − x1 )2 2h(t2 − t1 )

.

st}w|{vv wt |yuxy t{ x ~x}{x t{ut uzx }ttut ~|}wx tyu|wx {twut}wytvwut}wxtu|yuxvwt~|}wx wt~zy x tyuw t tu|t{~xw}v ttxtu|{ut ! exp −αx2

1

.

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1+

φ(y, t) Ct

α2 h2 t2 m2

t

φ

u

x x+y u(x, t + t1 ) =

R

u(x − y, t)φ(t1 , y)dy t1

|ux|v t{ x Ö |z{~|y~wt ty {twut}wx xyw |}}txut {twut}w tut}zx ~|}wx

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wt tt{vyw ytw }wty |wxw }{xzt

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2

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K ,δ

0

K ,δ

Ψ(k)i = A

Z

Ψ(r)e−irk dr.

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α

α

−irα kα

st}w|{vv Ö |yuxy {v t{utt uzxx x x x}t{v t{~|y Ψ(k0 )i = A

X

µα eirα (kα −k0 ) .

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α

Ψ(k0 ) ≈

X

0

$

µα .

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α: |kα −k0 |<

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α

β

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j∈J

λj |Ψj i,

J

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λ0osc j

=

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M (X(t0 ) − X(t0 )) = + ω 2 X(t0 )2 + g 2(t0 − t0 )

xj (t0 ) − xj (t1 ) 0 0 0 , xj (t ), t Xj (t ) dt, t0 − t1

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0

3

3

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=

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+

particles

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z|uwt xyyw x λ0α δS1,α S2,α

i

= λα · e h δ(S1,α +S2,α ) , 2 0 a2,α (t0 )−a2,α (t0 ) 2 1 a1,α (t )−a1,α (t0 ) 2 2 2 2 =2 + − kα c (|a1,α | + |a2,α | ) (t0 − t0 ), (t0 −t0 ) (t0 −t0 ) √ x (t0 )−x (t ) x (t0 )−x (t ) 4π = (2π) a1,α 1,β t0 −t11,β 1 + a2,α 2,β t0 −t12,β 1 eij·xβ (t0 − t0 ), 3 sign(eβ )

y x , x zttuyuw |{tyuxv yzwt| x t{ p x pu|tx~y y}{x z|uw α tyxuyu }{||yy ztwy } xx z|uw|x ~|}wx wt {yyuw y}wxv ~|}wyt|xw }ttwyw}wxy x yt|t|uxy x |}}twx wyy |{xwu ~|}wx~utt z|uw| yt β s}w u|xy tu tyxuyu ztwy } α x } z|uwt }wt{zutyuxx } z|uwt wtt y wx| γ |{xw t ~|}wx t| |{xwu|v ~|}w yw yt|t|u| w|z 1,β

2,β

β

λ0β δS3,α

k,1

k,2

i

= λβ · e h δ(S3,α +S2,α ) ,

=

msign(eβ )(xβ (t0 )−xβ (t1 ))2 2

+

eβ eγ |xβ (t0 )−xγ (t0 )| )

dt,

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aj1 , aj2 , . . . , ajl ; qk1 , qk2 , . . . , qkl −→ aj10 , aj20 , . . . , ajl0 ; qk10 , qk20 , . . . , qkl0 ; Sr1 , Sr2 , . . . , Srl

y a , q|x{| ttu|~|w }tyxty v~yzxttu|~|w tty|yt t t{tzt x }t}wtvuxy t t{tzx t x t}{y xyuyuxv |x{| yuyuxv wxt|uuy }xt{ wy y y{x~xu ttu|~|yw yyyyuxyztwtty u|t u| t{tztx xux|yw tut |~yux wt }tyxw x u| S t }x |t }x {yt x uytxu| tut u| tut x wt y u|ty {yuw |}wxw uy}zt{zt |xux }t}wtvuxv xu|x~y }y{|w |x{|{v ux |x}vxx uy wt{zt tw }tyxt ttty|yw}v t{ttz t{tz|x v~yyz {yuw |xuut|x{| x tw wttz|zxx {x w| x{x xu|v v~yz||uut wx }{tuyuuy {vvw w|z z|zx |xu jt

kt

rt

aj1 , aj2 , . . . , ajl ; qk1 , qk2 , . . . , qkl ; x ¯ h1 , x ¯h2 , . . . , x¯hl −→ aj10 , aj20 , . . . , ajl0 ; qk10 , qk20 , . . . , qkl0 ; Sr1 , Sr2 , . . . , Srl

y x¯}y t{ttz | x|ztwt ztwy tty|x }t}wtvx x v~yz uty t{tzx uty }ttwyw}wy y |xu

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