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Ïåðâûé îòâå÷àåò ñäâèãó íà÷àëà îòñ÷åòàíà âåêòîð −a, à âòîðîé ïåðåõîäó â ñèñòåìó, äâèæóùóþñÿ ñî ñêîðîñòüþ v = − mq .Íà ñîáñòâåííûå âåêòîðû îïåðàòîðîâ x̂ è p̂ îíè äåéñòâóþò òàê:T̂a | x i = | x + a i;è B̂q | p i = | p + q i.(55)Îïåðàòîðû ñäâèãîâ âûðàæàþòñÿ ÷åðåç îïåðàòîðû èìïóëüñà è êîîðäèíàòû x̂ è p̂:iT̂a = exp − (a, p̂).~èiB̂q = exp (q, x̂);~(56)rikÎíè íå ÿâëÿþòñÿ ýðìèòîâûìè, ïîñêîëüêó T̂+a = T̂−a è B̂+q = B̂−q.Íà âîëíîâûå ôóíêöèè â x- è p-ïðåäñòàâëåíèÿõ îíè äåéñòâóþò ñëåäóþùèì îáðàçîì:T̂a ψ(x) , h x | T̂a ψ i = h T̂+a ·x |ψ i = h x − a |ψ i = ψ(x − a);B̂q ψ(p) , h p | B̂q ψ i = h B̂+q ·p |ψ i = h p − q |ψ i = ψ(p − q).Òåîðåìà Ýðåíôåñòàab3.3.2(57a)(57b)Ïðèìåíèì ôîðìóëó (45) ê îïåðàòîðàì êîîðäèíàòû è èìïóëüñà îòäåëüíîé ÷àñòèöû: 2 ip̂p̂ẋˆ =,x = ;~ 2mmiṗˆ =[U (x, t), p̂] = −∇U (x, t).~(58a)(58b)Ýòî çíà÷èò, ÷òî â êâàíòîâîé ìåõàíèêå äëÿ ñðåäíèõ âåðíû óðàâíåíèÿ Ãàìèëüòîíà:èdh p i = −h ∇U (x, t) i,dt12d1h x i = h p i.dtm(59)3.3.3Ñîõðàíåíèå âåðîÿòíîñòåé3.4draftÏîëíàÿ âåðîÿòíîñòü îáíàðóæèòü êâàíòîâóþ ñèñòåìó îïðåäåëÿåòñÿ íîðìîé ñîñòîÿíèÿ,W = ||ψ||2 = h ψ |ψ i.
Äëÿ èçîëèðîâàííîé ñèñòåìû W = const.Îïåðàòîð ýâîëþöèè (46) óíèòàðíûé, òî åñòü Ŝ Ŝ+ = Ŝ+ Ŝ = 1̂. Ýòî ãàðàíòèðóåòñîõðàíåíèå âåðîÿòíîñòè â êâàíòîâîé ìåõàíèêå:W (t) = h ψ(t) |ψ(t) i = h ψ(t0 ) | Ŝ+ (t, t0 ) Ŝ(t, t0 ) | ψ(t0 ) i = h ψ(t0 ) |ψ(t0 ) i.(60)Â êîîðäèíàòíîì ïðåäñòàâëåíèè ñîõðàíåíèå âåðîÿòíîñòè âûðàæàåòñÿ óðàâíåíèåìíåïðåðûâíîñòè:∂ρ(x, t)+ div j(x, t) = 0.(61)∂tÃäå ïðîñòðàíñòâåííûå ïëîòíîñòü è ïîòîê âåðîÿòíîñòè îïðåäåëåíû òàê:ρ(x, t) = |ψ(x, t)|2 ,(62a)i~ ∗j(x, t) = −[ψ (x, t)∇ψ(x, t) − ψ(x, t)∇ψ ∗ (x, t)] .(62b)2mÏðåäñòàâëåíèå Ãåéçåíáåðãào@Ïðåäñòàâëåíèå Ãåéçåíáåðãà (.
. .H ) îòëè÷àåòñÿ îò ïðåäñòàâëåíèÿ Øðåäèíãåðà (. . .S )òåì, ÷òî â Ãåéçåíáåðãîâñêîì ïðåäñòàâëåíèè ñîñòîÿíèÿ íå çàâèñÿò îò âðåìåíè3. Çàòî,êàê íàáëþäàåìûå â êëàññèêå, çàâèñÿò îò âðåìåíè îïåðàòîðû.Ñâÿçü ìåæäó ïðåäñòàâëåíèÿìè îïðåäåëÿåò îïåðàòîð ýâîëþöèè (ðàçäåë 3.1.3)| ψ(t) iH = | ψ(t0 ) iS = const.(63a)ÂH (t) = Ŝ+ (t, t0 ) ÂS (t) Ŝ(t, t0 ),(63b)Çàâèñèìîñòü Ãåéçåíáåðãîâñêèõ îïåðàòîðîâ îò âðåìåíè çàäàåòñÿ óðàâíåíèåì (ñðàâíèòå ñ ñîîòíîøåíèåì (45), âûâåäåííûì â ïðåäñòàâëåíèè Øðåäèíãåðà):ãäå ÂH (t0) = ÂS (t0).(64) êëàññè÷åñêîì ïðåäåëå ýòî ðàâåíñòâî ïåðåõîäèò â èçâåñòíûé â ìåõàíèêå çàêîí èçìåíåíèÿ âåëè÷èíû A(p, q, t):rik∂ ÂH (t)id ÂH (t)=+ [ĤH (t), ÂH (t)],dt∂t~dA(p, q, t)∂A(p, q, t)=+ {H(p, q, t), A(p, q, t)} .dt∂t(65)abÇäåñü {.
. . , . . .} ñêîáêè Ïóàññîíà.4Ãàðìîíè÷åñêèé îñöèëëÿòîðÃàìèëüòîíèàí è îáåçðàçìåðåííûå îïåðàòîðû äëèíû è èìïóëüñà:p̂2mω 2 x̂2~ωĤ =+=2m2222P̂Q̂+22!,ãäåP̂ = p̂ /p0èQ̂ = x̂ /x0 .(66)Äâà ïðåäñòàâëåíèÿ ýêâèâàëåíòíû è äàþò îäèíàêîâûå ïðåäñêàçàíèÿ, ïðè÷åì êàæäîå èìååò ñâîèïðåèìóùåñòâà.313Îñöèëëÿòîðíûå åäèíèöû õàðàêòåðíûå ìàñøòàáû äëèíû è èìïóëüñà:rx0 =√è~,mωp0 =(67)x0 · p0 = ~.~mω;11| n i = ~ω n +| n i,Ĥ | n i = ~ω n̂ +22Ïîâûøàþùèé è ïîíèæàþùèé îïåðàòîðû â+, â:1â = √ (Q̂ +i P̂),21â+ = √ (Q̂ −i P̂),2aftÏîñêîëüêó U (±∞) = ∞, ñïåêòð äèñêðåòíûé.
Ñîáñòâåííûå ñîñòîÿíèÿ ãàìèëüòîíèàíà| n i è èõ ýíåðãèÿ îïðåäåëÿþòñÿ íîìåðîì óðîâíÿ n = 0, 1, . . .ïðè÷åìh m |n i = δmn .(68)[â+ , â] = 1.(69)ïðè÷åìn̂ | n i = n| n i;drÎïåðàòîð íîìåðà óðîâíÿ n̂ = â+ â è åãî êîììóòàöèîííûå ñâîéñòâà:[n̂, â] = − â,[n̂, â+ ] = â+ .(70)Ìàòðè÷íûå ýëåìåíòû ïîâûøàþùåãî è ïîíèæàþùåãî îïåðàòîðîâ,è√h m | â+ · n i = δm n+1 n + 1.o@√h m | â · n i = δm n−1 n(71)Ïîíèæàþùèé îïåðàòîð ¾óíè÷òîæàåò¿ îñíîâíîå ñîñòîÿíèå, | â · 0 i = 0. Ïîýòîìó âîëíîâàÿ ôóíêöèÿ â Q-ïðåäñòàâëåíèè óäîâëåòâîðÿåò óðàâíåíèþ1â ψ0 (Q) = √2dQ+ψ0 (Q) = 0,dQîòêóäàψ0 (Q) ∝ e−Q2 /2.(72)Âîëíîâûå ôóíêöèè âûðàæàþòñÿ ÷åðåç ïîëèíîìû Ýðìèòà Hn(Q):ãäårik√2ψn (Q) = h Q |n i = (2n n! π)−1/2 Hn (Q) e−Q /2 ,(73)dn −Q2e ;dxnH0 (Q) = 1, H1 (Q) = 2Q, .
. .(74)hiÃàìèëüòîíèàí êîììóòèðóåò ñ îïåðàòîðîì èíâåðñèè, Ĥ, Î = 0; ñîñòîÿíèÿ è âîëíîâûåHn (Q) = (−1)n eQ2abôóíêöèè èìåþò îïðåäåëåííóþ ÷åòíîñòü:Î | n i = (−1)n | n i,55.1èÎ ψn (Q) = (−1)n ψn (Q).(75)Îðáèòàëüíûé ìîìåíòÎñíîâíûå óðàâíåíèÿÎïåðàòîð óãëîâîãî ìîìåíòà ÷àñòèöû:L̂ = [r̂ × p̂] = ~ · l̂;èëè, â x-ïðåäñòàâëåíèè,14l̂j = −iεjkl rk ∂l .(76)Îñíîâíûå êîììóòàòîðû:hi2l̂ , l̂ = 0;hil̂i , l̂j = iεijk l̂k ;hil̂i , x̂j = iεijk x̂k ;hil̂i , p̂j = iεijk p̂k .(77)è1(l̂+ − l̂− );2i22l̂+ l̂+ + l̂− l̂+= l̂z + l̂∓ l̂± ± l̂z = l̂z +.2l̂x =l̂draftÏîëíûé íàáîð âêëþ÷àåò äâå îäíîâðåìåííî èçìåðèìûå âåëè÷èíû: l2 è m = lz .
Ïîëíûéíàáîð ñîáñòâåííûõ ñîñòîÿíèé îïåðàòîðîâ ìîìåíòà l̂ 2 è l̂z : 2l̂ | l, m i = l(l + 1)| l, m i,ãäål = 0, 1, . . .(78a)l̂z | l, m i = m| l, m i,èm = 0, ±1, . . . , ±l.Îïåðàòîð l̂ àêñèàëüíûé âåêòîð (ñì. ñíîñêó íà ñòð. 10), ò. å. [l̂, Î] = 0. ×åòíîñòüñîáñòâåííûõ ñîñòîÿíèé çàâèñèò îò âåëè÷èíû l:Î | l, m i = (−1)l | l, m i.(79)Ïîâûøàþùèé è ïîíèæàþùèé îïåðàòîðû: l̂± = l̂x ±i l̂y ;21(l̂+ + l̂) ,2o@Êîììóòàöèîííûå ñîîòíîøåíèÿ:ihl̂z , l̂± = ± l̂± ;5.2l̂y =hil̂+ , l̂− = 2 l̂z ;(80b)(81)l̂+ = (l̂− )+ .Ìàòðè÷íûå ýëåìåíòû îïåðàòîðîâ ìîìåíòà(82a)(82b)(82c)(82d)2h l0 , m0 | l̂ | l, m i = l(l + 1)δl0 , l δm, m0 ;rikh l0 , m0 | l̂z | l, m i = m δl0 , l δm, m0 ;ph l0 , m0 | l̂+ | l, m i =l(l + 1) − m(m + 1) δl0 , l δm0 , m+1 ;p00h l , m | l̂− | l, m i =l(l + 1) − m(m − 1) δl0 , l δm0 , m−1 .5.3(80a)Ñôåðè÷åñêèå ôóíêöèèabÑôåðè÷åñêèå ôóíêöèè Yl(m)(θ, ϕ) = h θ, ϕ |l, m i ýòî ïðåäñòàâëåíèå ñîáñòâåííûõ ñîñòîÿíèé îïåðàòîðîâ l̂2, l̂z â ñôåðè÷åñêèõ êîîðäèíàòàõ4.
Îíè ÿâëÿþòñÿ ñîáñòâåííûìèôóíêöèÿìè îïåðàòîðîâ (çäåñü r = (r sin θ cos φ, r sin θ sin φ, r cos θ)):2l̂Sph = −2l̂Sph Yl∂1 ∂21 ∂sin θ−;sin θ ∂θ∂θ sin2 θ ∂ϕ2(m)(θ, ϕ) = l(l + 1) Yl(m)(θ, ϕ);Sphl̂zSphl̂z= −iYl(m)(83a)∂.∂ϕ(θ, ϕ) = m Yl×åòíîñòü ñôåðè÷åñêèõ ôóíêöèé îïðåäåëÿåòñÿ çíà÷åíèåì l, ñì. (79):(m)(m)(m)Î Yl (θ, ϕ) = Yl (−n) = (−1)l Yl (θ, ϕ),ãäå n = rr .(m)(θ, ϕ).(83b)(84)Îïðåäåëåíèÿ ñôåðè÷åñêèõ ôóíêöèé â ðàçíûõ èñòî÷íèêàõ ìîãóò îòëè÷àòüñÿ çíàêàìè èëè ôàçîâûìè ìíîæèòåëÿìè.4155.3.1Ñôåðè÷åñêèå ôóíêöèè äëÿ l = 0, 1.Óãëîâûå çàâèñèìîñòè côåðè÷åñêèõ ôóíêöèé ðàçäåëÿþòñÿ: Yl(m)(θ, ϕ) = eimϕ Θ(m)(θ).lÏðè l = 0, 1, ñôåðè÷åñêèå ôóíêöèè ðàâíû:5.3.21=√ ,4π(0)Y1r=3cos θ,4π(±1)Y1Ïîñòðîåíèå ñôåðè÷åñêèõ ôóíêöèé (∗ )r=±(85)3sin θ e±iϕ .8πaft(0)Y0Ñîîòíîøåíèÿ (82) ïîçâîëÿþò íàéòè ñôåðè÷åñêèå ôóíêöèè äëÿ ëþáûõ l è m Âíà÷àëåíóæíî ðåøèòü ñèñòåìó óðàâíåíèé, àíàëîãè÷íóþ ñîîòíîøåíèþ (72) äëÿ ãàðìîíè÷åñêîãî îñöèëëÿòîðà.
Ïîñêîëüêó l̂− | l, −l i = 0, â êîîðäèíàòíîì ïðîñòðàíñòâå ôóíêöèÿ(−l)Yl (θ, ϕ) óäîâëåòâîðÿåò óðàâíåíèÿì:Sphl̂zYl(−l)(θ, ϕ) = −eYl(−l)(θ, ϕ) = −i−iϕ∂∂− i cot θ∂θ∂ϕYl(−l)(θ, ϕ) = 0;∂ (−l)(−l)Yl (θ, ϕ) = −l Yl (θ, ϕ).∂ϕÄàëåå, ïîñëåäîâàòåëüíî äåéñòâóÿ îïåðàòîðîìôóíêöèè äëÿ äðóãèõ m.Sphl̂+= eiϕ (∂θ + i cot θ ∂ϕ ),(86a)(86b)ïîëó÷àåìo@5.4drSphl̂−Îïåðàòîð êîíå÷íûõ âðàùåíèéÒåîðèÿ ñïèíà Ïàóëèab6rikÝòîò îïåðàòîð îòâå÷àåò çà èçìåíåíèÿ ñîñòîÿíèé è âñåõ íàáëþäàåìûõ âåêòîðíûõ èòåíçîðíûõ âåëè÷èí ïðè ïðîñòðàíñòâåííûõ ïîâîðîòàõ ñèñòåìû. Îí î÷åíü ïîõîæ íàîïåðàòîðû ñäâèãà è áóñòà èç ðàçäåëà 3.3.1.Îïåðàòîð R̂n(α) ïîâîðîòà âîêðóã îñè n íà óãîë α ìîæíî âûðàçèòü ÷åðåç îïåðàòîðóãëîâîãî ìîìåíòà l̂. Òàê æå êàê îïåðàòîðû ñäâèãîâ, îí íå ýðìèòîâ:R̂n (α) = exp −iα(n, l̂),ïðè÷åì R̂+n(α) = R̂n(−α).(87) êîîðäèíàòíîì ïðåäñòàâëåíèè îïåðàòîð ïîâîðîòà âîêðóã îñè z äåéñòâóåò íà âîëíîâóþ ôóíêöèþ òàê:R̂0z (α) ψ(ϕ) = h ϕ | R̂0z (α) · ψ i = h R̂+ 0z (α) · ϕ |ψ i = h ϕ − α |ψ i = ψ(ϕ − α).(88)6.1Ìàòðèöû ÏàóëèÌàòðèöû Ïàóëè è åäèíè÷íàÿ ìàòðèöà 1:σx =0 11 0,σy =0 −ii 0,σz =1 00 −1.(89){σi , σj } = 2δij · 1.(90),1=1 00 1Ïðîèçâåäåíèå, êîììóòàòîð è àíòèêîììóòàòîð ìàòðèö Ïàóëè:σi σj = δij · 1 + i εijk σk ;[σi , σj ] = 2i εijk σk ;Ôîðìóëà Ýéëåðà äëÿ ìàòðèö Ïàóëè (n åäèíè÷íûé âåêòîð):exp(i ϕ (n, σ)) = cos ϕ · 1 + i sin ϕ · (n, σ).16(91)6.2Îïåðàòîð ñïèíàÎïåðàòîð ñïèíà (σ = (σx, σy , σz )):Ŝ = ~ · ŝ =~· σ,2ïðè÷åì2Ŝ =3· 1.4(92)χ+1/2 =χ−1/2 =1001≡ α ≡ |+iäëÿ≡ β ≡ |−iäëÿP = hPiχ = h χ |σ| χ i;~/2;(93)hSz i = −~/2.(94)hSz i =drÂåêòîð ïîëÿðèçàöèè (íàïðàâëåíèå ñïèíà)aftÑïèíîâûå ôóíêöèè:ïðè÷åì∀h χ |,P2 = 1.(95)Îïåðàòîð êîíå÷íûõ âðàùåíèé R̂Spin(α), äåéñòâóÿ íà ñîñòîÿíèå | χ i, ïîâîðà÷èâàåònâåêòîð ïîëÿðèçàöèè âîêðóã íàïðàâëåíèÿ n íà óãîë α.
Ôîðìóëà àíàëîãè÷íà (87):77.1iα(n, σ)).2o@SpinR̂n(α) = exp(−i α (n, ŝ)) = exp(−(96)Çàäà÷à äâóõ òåë.Ðàçäåëåíèå ïåðåìåííûõ çàäà÷å î ñâîáîäíîì äâèæåíèè äâóõ òåë ñ ïàðíûì âçàèìîäåéñòâèåì U (r1, r2)U (|r1 − r2 |) ïåðåìåííûå ðàçäåëÿþòñÿ òàê æå, êàê â êëàññè÷åñêîì ñëó÷àå:~2 ∆1 ~2 ∆2−+ U (|r1 − r2 |) = ĤR + Ĥr .2m12m2rikĤ = −=(97)Öåíòð ìàññ (ÖÌ) äâèæåòñÿ ñâîáîäíî (R êîîðäèíàòà ÖÌ, M ïîëíàÿ ìàññà):ĤR = −~2 ∆R,2MãäåR=m1 r1 + m2 r2,m1 + m2èM = m1 + m2 .(98)abÇàäà÷à îá îòíîñèòåëüíîì äâèæåíèè òåë â ñèñòåìå ÖÌ ýêâèâàëåíòíà çàäà÷å î äâèæåíèè ÷àñòèöû â ïîëå íåïîäâèæíîãî öåíòðàĤr =~2 ∆r+ U (r),2µãäår = r1 − r2 ,Âåëè÷èíà µ íàçûâàåòñÿ ïðèâåäåííîé ìàññîé.17èµ=m1 m2;m1 + m2(99)7.2Öåíòðàëüíîå ïîëå öåíòðàëüíîì ïîëå (99) ñîõðàíÿåòñÿ îðáèòàëüíûé ìîìåíò âðàùåíèÿ:hiĤr , L̂ = 0,èh2Ĥr , L̂i(100)= 0.aftÑîñòîÿíèÿ îïðåäåëÿþòñÿ íàáîðîì 3-õ êâàíòîâûõ ÷èñåë: | E, l, m i, ãäå m = lz .
Îíèíîðìèðîâàíû óñëîâèåì(101)h E, l, m |E 0 , l0 , m0 i = δl l0 δm m0 δ(E, E 0 ).Ïðè E > U (∞) ñïåêòð íåïðåðûâíûé, à ïðè E < U (∞) äèñêðåòíûé. ñôåðè÷åñêèõ êîîðäèíàòàõ Ãàìèëüòîíèàí (99) èìååò âèä:∂1 ∂21 ∂sin θ++U (r);sin θ ∂θ∂θ sin2 θ ∂ϕ2{z}|dr~2~2 ∂ 2 ∂r−Ĥr = −2µr2 ∂r ∂r 2µr2(102)− lˆ2 , ñì. (83).èëè, ÷òî òî æå, (Ûcf òàê íàçûâàåìûé öåíòðîáåæíûé áàðüåð):î÷åâèäíî, ÷òîo@~2 1 ∂ 2Ĥr = −·r + U (r) + Ûl ,2µ r ∂r22~2 l̂Ûl => 0.2µr2(103)Ãðàíè÷íûå óñëîâèÿ íà âîëíîâóþ ôóíêöèþ (M > 0 ÷èñëî),|ψ(r → 0)| < M,èψ(r → ∞) = 0.(104)rik ñôåðè÷åñêèõ êîîðäèíàòàõ ïåðåìåííûå ðàçäåëÿþòñÿ, è ñîáñòâåííûå ôóíêöèè èìåþòâèä:1(m)(105)ψE l m (r, θ, ϕ) = h r, θ, ϕ |E, l, m i = χE l (r) Yl (θ, ϕ),rïðè÷åì ðàäèàëüíàÿ ÷àñòü χE l (0) = 0 è |χE l (r → ∞)| < M 0. Óðàâíåíèå äëÿ ðàäèàëüíîé ÷àñòè:~2 ∂ 2~2 l(l + 1)−χE l (r) + U (r) +χE l (r) = EχE l (r)2µ ∂r22µr2(106)abÏðè r → 0 â ôèçè÷åñêèõ çàäà÷àõ âîëíîâàÿ ôóíêöèÿχE l ∝ rl+1 .(m)×åòíîñòü ðåøåíèé îïðåäåëÿåòñÿ ôóíêöèåé Yl (θ, ϕ) è çàâèñèò òîëüêî îò l, ñì.
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