С.В. Петров - Лекции (1124220), страница 7
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Î÷åâèäíî,XXXXh v|u i =h v|m i h m| |n i h n|u i =h v|m i δmn h n|u i =h n|v i∗ h n|u i,mnm,n27nòî åñòü ñêàëÿðíîå ïðîèçâåäåíèå ñîîòâåòñòâóåò óìíîæåíèþ ñòðîêè íà ñòîëáåö. Äëÿ ïðîèçâîëüíîãî îïåðàòîðà BXXXB = PA B PA =|m i h m| B|n i h n| =Bmn |m i h n|,mnm,nãäå Bmn = h m| B |n i, à ìàòðèöà B íàçûâàåòñÿ ìàòðèöåé îïåðàòîðà B â áàçèñå âåêòîðîâ|n i .Îïðåäåëåíèå: ïóñòü âåêòîðû |u(1) i ïðèíàäëåæàò ïðîñòðàíñòâó E1 (dim E1 = N1 ), àTâåêòîðû |u(2) i ïðîñòðàíñòâó E2 (dim E2 = N2 ). E1 E2 = 0; òîãäà âåêòîðû |u(1) u(2) i,óñëîâíî ïðåäñòàâëÿåìûå â âèäå |u(1) i |u(2) i, ïðèíàäëåæàò ïðîñòðàíñòâó E1 ⊗ E2 , íàçûâàåìîìó òåíçîðíûì (êðîíåêåðîâñêèì) ïðîèçâåäåíèåì ëèíåéíûõ ïðîñòðàíñòâ E1 è E2 .Î÷åâèäíî, dim(E1 ⊗ E2 ) = N1 N2 , à, åñëè îïåðàòîðû A(1) è A(2) äåéñòâóþò â ïðîñòðàíñòâàõE1 è E2 ñîîòâåòñòâåííî, òî [A(1) , A(2) ] = 0.4.2.Îïåðàòîð óãëîâîãî ìîìåíòà.Äëÿ òîãî, ÷òîáû ïîñòðîèòü òåîðèþ, èíâàðèàíòíóþ ïî îòíîøåíèþ ê âûáîðó ïðåäñòàâëåíèÿ, íå áóäåì àïåëëèðîâàòü ê êëàññè÷åñêîìó îïðåäåëåíèþ ìîìåíòà êîëè÷åñòâà äâèæåíèÿ,à âîñïîëüçóåìñÿ ëèøü ïðåäâàðèòåëüíî âûâåäåííûìè êîììóòàöèîííûìè ñîîòíîøåíèÿìè.Êîììóòàöèîííûå ñîîòíîøåíèÿ: ïóñòü Jx , Jy , Jz êîìïîíåíòû îïåðàòîðà óãëîâîãîìîìåíòà J .
 êîîðäèíàòíîì ïðåäñòàâëåíèè Jx = ŷ p̂z − ẑ p̂y , Jy = ẑ p̂x − x̂p̂z , Jz = x̂p̂y − ŷ p̂x .[Jx , Jy ] = [ŷ p̂z , ẑ p̂x ] − [ŷ p̂z , x̂p̂z ] − [ẑ p̂y , ẑ p̂x ] + [ẑ p̂y , x̂p̂z ] = x̂p̂y [ẑ, p̂z ] + ŷ p̂x [p̂y , ŷ] = i~ Jz (ñì.îñíîâíûå êîììóòàöèîííûå ñîîòíîøåíèÿ â 2.1); àíàëîãè÷íî [Jy , Jz ] = i~ Jx , [Jz , Jx ] = i~ Jy .J2 = J2x + J2y + J2z ⇒ [Jα , J2 ] = 0, α = x, y, z; â ÷àñòíîñòè, [Jz , J2 ] = 0 ⇒ J2 Jz = Jz J2 .Îïåðàòîðû Jz è J2 êîììóòèðóþò, ïîýòîìó (ñì. 1, òåîðåìà î êîììóòèðóþùèõ îïåðàòîðàõ) îíè èìåþò îáùèé îðòîíîðìèðîâàííûé íàáîð ñîáñòâåííûõ âåêòîðîâ |λ, κ i, ãäå λ ñîáñòâåííûå çíà÷åíèÿ J2 (J2 |λ, κ i = λ|λ, κ i), à κ ñîáñòâåííûå çíà÷åíèÿ Jz (Jz |λ, κ i =κ|λ, κ i).
Çàìåòèì, ÷òîXX2 ≥ 0;h λ, κ| J2 |λ, κ i = λ =h λ| J2α |λ, κ i =h λ, κ| J+α Jα |λ, κ i = Jα |λ, κ iααJ2z |λ, κ i = κ2 |λ, κ i ⇒ κ2 = h λ, κ| J2z |λ, κ i = h λ, κ| J2 |λ, κ i − h λ, κ| J2x +Jy2 |λ, κ i = λ −h λ, κ| J2x + J2y |λ, κ i ⇒ κ2 ≤ λ, ïîñêîëüêó+h λ, κ| J2x + J2y |λ, κ i = h λ, κ| J+x Jx |λ, κ i + h λ, κ| Jy Jy |λ, κ i ≥ 0.Ââåä¼ì îïåðàòîðû J+ è J− : J± = Jx ±i Jy , íàçûâàåìûå îïåðàòîðàìè ïîâûøåíèÿ è ïîíèæåíèÿ ñîîòâåòñòâåííî.
[Jz , J+ ] = [Jz , Jx +i Jy ] = i~ Jy +~ Jx = ~ J+ ⇒ Jz J+ = [Jz , J+ ] +J+ Jz = ~ J+ + J+ Jz ; Jz J+ |λ, κ i = ~ J+ |λ, κ i + J+ Jz |λ, κ i = (~ + κ) J+ |λ, κ i . Òàêèì îáðàçîì, J+ |λ, κ i ÿâëÿåòñÿ ñîáñòâåííûì âåêòîðîì Jz , ñîîòâåòñòâóþùèì ñîáñòâåííîìó çíà÷åíèþ κ + ~, òî åñòü J+ |λ, κ i = C+ |λ, κ + ~ i îïåðàòîð J+ ïîâûøàåò íà åäèíèöó ~ çíà÷åíèåκ âåêòîðà. Àíàëîãè÷íî J− |λ, κ i = C− |λ, κ − ~ i .Îäíàêî κ2 ≤ λ, òî åñòü ∃ κmin , κmax : κ2min ≤ λ, κ2max ≤ λ, (κmin − ~)2 > λ, (κmax +~)2 > λ. Ýòî îçíà÷àåò, ÷òî J+ |λ, κmax i = |0 i, J− |λ, κmin i = |0 i (|0 i íóëåâîé âåêòîðïðîñòðàíñòâà êåò-âåêòîðîâ, òî åñòü íåðåàëèçóåìîå ñîñòîÿíèå).
Çàìåòèì, ÷òî J2x + J2y =11(J+ J− + J− J+ ), ïîýòîìó J2 = (J+ J− + J− J+ ) + J2z ⇒ J− J+ = 2 J2 −2 J2z − J+ J− ; êðî22ìå ýòîãî, [J+ , J− ] = 2~ Jz , òî åñòü J− J+ = J2 − J2z −~ Jz . J− J+ |λ, κmax i = J− |0 i = |0 i =(J2 − J2z −~ Jz )|λ, κmax i = (λ − κ2max − ~κmax )|λ, κmax i ⇒ λ = κ2max + ~κmax . Àíàëîãè÷íî28J+ J− = 2 J2 −2 J2z − J− J+ = J2 − J2z + Jz ; J+ J− |λ, κmin i = |0 i = (λ−κ2min +~κmin )|λ, κmin i ⇒λ = κ2min −~κmin = κ2max +~κmax .
Îäíàêî, êàê èçâåñòíî èç 2.2, ñîáñòâåííûå ÷èñëà îïåðàòîðàJz ðàâíû m~, m ∈ Z, ïîýòîìó κmax − κmin = N ~, N ∈ N. Òàêèì îáðàçîì, κmax = κmin + N ~è κ2min + 2N ~κmin + N 2 ~2 + ~(κmin + N ~) = κ2min − ~κmin ⇒ (2N + 2)~κmin = −(N 2 + N )~2 ⇒N~N~N~ N~⇒ κmin = −, κmax =, λ=+~ .2222Îïðåäåëèì òàêæå êîýôôèöèåíòû C± : J+ |λ, κ i = C+ |λ, κ + ~ i ⇒ h λ, κ| J++ J+ |λ, κ i =++22|C+ | h λ, κ + ~|λ, κ + ~ i = |C+ | . Íî J+ = Jx −i Jy = J− , ïîýòîìó J+ J+ = J− J+ =NNJ2 − J2z −~ Jz ; çíà÷èò, |C+ |2 = h λ, κ|(λ−κ2 −~κ)|λ, κ i = λ−κ2 −~κ =+ 1 ~2 −κ(κ+~).22NNÀíàëîãè÷íî |C− |2 =+ 1 − κ(κ − ~).224.3.Ñïèí.Çàìåòèì, ÷òî ïî ðåçóëüòàòàì 2.2 ñîáñòâåííûå çíà÷åíèÿ Jz öåëûå ÷èñëà â åäèíèöàõ ~;N~ N~îäíàêî â 4.2 áûëî ïîëó÷åíî, ÷òî κ èçìåíÿåòñÿ â ïðåäåëàõ −÷, ïðè÷¼ì N íå îáÿ22çàòåëüíî ÿâëÿåòñÿ ÷¼òíûì.
Èòàê, â êâàíòîâîé ìåõàíèêå âîçìîæíû ñîñòîÿíèÿ, â ïðèíöèïåíå îáúÿñíèìûå ñ òî÷êè çðåíèÿ êëàññè÷åñêîé ìåõàíèêè.Ýêñïåðèìåíòàëüíîå ïîäòâåðæäàíèå ýòîãî ôàêòà áûëî ïîëó÷åíî â õîäå îïûòîâ ØòåðíàÃåðëàõà; ïó÷îê àòîìîâ âîäîðîäà â ïîñòîÿííîì ìàãíèòíîì ïîëå ñ íàïðÿæ¼ííîñòüþ H ðàñùåïëÿåòñÿ ïî ýíåðãèè, ïðè÷¼ì âåëè÷èíà ðàñùåïëåíèÿ ñîñòàâëÿåò 2µB , õîòÿ ìåõàíè÷åñêèéìîìåíò l äëÿ ýëåêòðîíà â àòîìå âîäîðîäà ðàâåí íóëþ, à ïîòîìó è ìàãíèòíûé ìîìåíòe→−l = 0.
Ðàñ÷¼òû (ïðèâåä¼ííûå íåñêîëüêî íèæå) ïîêàçûâàþò, ÷òî òàêîìó ðàñùåïµ =2mc1ëåíèþ ñîîòâåòñòâóåò íàëè÷èå ó ýëåêòðîíà ñîáñòâåííîãî ìåõàíè÷åñêîãî ìîìåíòà l =2âïåðâûå ïîäîáíàÿ ãèïîòåçà áûëà âûñêàçàíà Óëåíáåêîì è Ãàóäñìèòîì. Ñîáñòâåííûé ìåõàíè÷åñêèé ìîìåíò ÷àñòèöû íàçûâàåòñÿ ñïèíîì è ìîæåò ñ÷èòàòüñÿ ðåçóëüòàòîì âðàùåíèÿ÷àñòèöû âîêðóã ñâîåé îñè. Íåîáõîäèìî, îäíàêî, èìåòü â âèäó, ÷òî â äåéñòâèòåëüíîñòèíèêàêîãî âðàùåíèÿ íå ïðîèñõîäèò, à ñïèí ÿâëÿåòñÿ îñîáûì, ÷èñòî êâàíòîâûì ñâîéñòâîì÷àñòèöû.3~Èòàê, N = 1, λ = ~2 , κ = ± ; âûáèðàÿ âåêòîðû42 3 13 110 ,è , −→4 2 → 014 2â êà÷åñòâå áàçèñíûõ, çàïèøåì ìàòðèöû îñíîâíûõ îïåðàòîðîâ (äëÿ ñïèíà îíè îáîçíà÷àþòñÿáóêâàìè S):3 2 1 0~ 1 02S = ~., Sz =0 142 0 −1Èñïîëüçóÿ ïîëó÷åííûå â 4.2 âûðàæåíèÿ äëÿ C+ è C− , íàéä¼ì0 10 0S+ = ~, S− = ~.0 01 0Îòñþäà 01Sx = (S+ + S− ) = ~ 21211 0 − 2 , Sy = −i (S+ − S− ) = i~ 2 .2100229~σα (α = x, y, z) íàçûâàþòñÿ ìàòðèöàìè Ïàóëè :20 10 −i1 0σx =, σy =, σz =.1 0i 00 −1Ìàòðèöû σx , σy , σz : Sα =Îïðåäåëåíèå: ñïèíîâûì êâàíòîâûì ÷èñëîì íàçûâàåòñÿ âåëè÷èíà ñïèíà (òî åñòü ñîá-1ñòâåííîãî ìåõàíè÷åñêîãî ìîìåíòà) ÷àñòèöû, äëÿ ýëåêòðîíà s = ; ìàãíèòíûì ñïèíîâûì2êâàíòîâûì ÷èñëîì íàçûâàåòñÿ âåëè÷èíà ïðîåêöèè ñïèíà íà ïðîèçâîëüíî âûáðàííóþ îñü,1äëÿ ýëåêòðîíà ms = sz = ± .2Îïðåäåëåíèå: ñïèíîâîé ôóíêöèåé íàçûâàåòñÿ âñÿêàÿ ôóíêöèÿ ñïèíà ÷àñòèöû, òî3 1åñòü, ïî ñóòè, ïðîèçâîëüíûé âåêòîð ïðîñòðàíñòâà.
Îáîçíà÷àÿ áàçèñíûå âåêòîðû ,è42 3 110 ,− 4 2 ÷åðåç 0 è 1 , çàïèøåì ñïèíîâóþ ôóíêöèþ χ â âèäåχ=a10+b01.Òåîðåìà: âñÿêîé ñïèíîâîé ôóíêöèè ñîîòâåòñòâóåò íàïðàâëåíèå â êîíôèãóðàöèîííîìïðîñòðàíñòâå, ïðîåêöèÿ ñïèíîâîé ôóíêöèè íà êîòîðîå ìàêñèìàëüíà, à êàæäîìó íàïðàâëåíèþ ñîîòâåòñòâóåò ñïèíîâàÿ ôóíêöèÿ, ïðîåêöèÿ êîòîðîé íà ñîîòâåòñòâóþùåå íàïðàâëåíèåìàêñèìàëüíà.4 Áóäåì ñ÷èòàòü ñïèíîâóþ ôóíêöèþ íîðìèðîâàííîé, òî åñòü |χ|2 = |a|2 + |b|2 = 1;h π iìîæíî çàïèñàòü a = eiα cos δ, b = eiβ sin δ α, β ∈ [0, π]; δ ∈ 0,. Òàêèì îáðàçîì,2~~~cos δχ = eiα; Sx = σx , Sy = σy , Sz = σz , ïîýòîìói(β−α)esin δ222 ~cos δ0 1i(α−β)+Sx = χ Sx χ = (cos δ, esin δ) ··=1 0ei(β−α) sin δ2~~= sin δ cos δ(ei(β−α) + ei(α−β) ) = sin 2δ cos(β − α);22 ~i~cos δ0 −1i(α−β)Sy = (cos δ, esin δ) ··= sin 2δ sin(β − α);i(β−α)10esinδ22 ~~cos δ1 0·= cos 2δ.Sz = (cos δ, ei(α−β) sin δ) ·i(β−α)0 −1esin δ22Ïóñòü n åäèíè÷íîå íàïðàâëåíèå, çàäàííîå â ñôåðè÷åñêîé ñèñòåìå êîîðäèíàò óãëàìèϕ, θ; òîãäà, î÷åâèäíî, nx = cos ϕ sin θ, ny = sin ϕ sin θ, nz = cos θ.
Ïðîåêöèÿ îïåðàòîðà Síà n (Sn ) ÿâëÿåòñÿ îïåðàòîðîì, ïðè÷¼ì Sn = Sx nx + Sy ny + Sz nz =~~cos θsin θ(cos ϕ − i sin ϕ)cos θ e−iϕ sin θ==.− cos θ2 sin θ(cos ϕ + i sin ϕ2 eiϕ sin θ − cos θ~. Íåñëîæíî óáå2äèòüñÿ â òîì, ÷òî ýòîìó ñîáñòâåííîìó çíà÷åíèþ ñîîòâåòñòâóåò ñîáñòâåííûé âåêòîð χ 1 =Ìàêñèìàëüíûì ñîáñòâåííûì çíà÷åíèåì Sz (à ïîòîìó è Sn ) ÿâëÿåòñÿ230θ cos 2 θ . Ñðàâíèâàÿ ýòîò âåêòîð ñ χ, âèäèì, ÷òî îíè ñîâïàäàþò ïðè θ = 2δ, ϕ = β −αeiϕ sin2(êîíñòàíòà eiα â äàííîì ñëó÷àå íå èìååò çíà÷åíèÿ, ïîñêîëüêó âñå âåêòîðû âèäà Cχ 1 òàêæå21ÿâëÿþòñÿ ñîáñòâåííûìè ñ ñîáñòâåííûì çíà÷åíèåì .
Òàêèì îáðàçîì, óñòàíîâëåíî ñîîò2âåòñòâèå ìåæäó âèäîì ñïèíîâîé ôóíêöèè è ìàêñèìóìîì ïðîåêöèè, êîòîðîå è äîêàçûâàåòòåîðåìó. Ïðèìåð: âîñïîëüçóåìñÿ ðåçóëüòàòàìè òåîðåìû äëÿ ïðîñòåéøåãî ñëó÷àÿ ïðîåêöèè íàπîäíó èç êîîðäèíàòíûõ îñåé. Ïóñòü n = nx , òî åñòü θ = , ϕ = 0; ïîäñòàâëÿÿ â ôîðìóëû2π1äëÿ Sx , Sy , Sz δ = , β − α = 0, íàõîäèì Sx = , Sy = Sz = 0. Âåêòîð ñïèíà íàïðàâëåí42âäîëü îñè x, à ïîòîìó äâå äðóãèå åãî ïðîåêöèè îáðàùàþòñÿ â íîëü.Èòàê, íàëè÷èå ñïèíà ÿâëÿåòñÿ ôóíäàìåíòàëüíûì ñâîéñòâîì ÷àñòèöû, à å¼ ñîñòîÿíèåçàâèñèò îò âåëè÷èíû ñïèíà è åãî íàïðàâëåíèÿ â ïðîñòðàíñòâå; îäíàêî íèãäå ðàíåå çàâèñèìîñòü âîëíîâîé ôóíêöèè îò ñïèíà íå âîçíèêàëà, à, íàïðèìåð, â êîîðäèíàòíîì ïðåäñòàâëåíèè âîëíîâàÿ ôóíêöèÿ çàâèñåëà ëèøü îò êîîðäèíàò ÷àñòèöû.
Ýòî îçíà÷àåò, ÷òî ñóùåñòâîâàíèå ñïèíà, âîîáùå ãîâîðÿ, íå óêëàäûâàåòñÿ â ðàçâèòóþ òåîðèþ êâàíòîâîé ìåõàíèêè.Äèðàê óñòàíîâèë, ÷òî ñóùåñòâîâàíèå ñïèíà ÿâëÿåòñÿ ðåëÿòèâèñòñêèì ýôôåêòîì, à ïîòîìó, ðàçóìååòñÿ, íå ìîæåò âîçíèêíóòü â íåðåëÿòèâèñòñêîé êâàíòîâîé ìåõàíèêå, ðàññìàòðèâàåìîé ðàíåå. Ñîîòâåòñòâåííî, ñïèí âîîáùå íå ôèãóðèðóåò â óðàâíåíèè Øðåäèíãåðà,à âîçíèêàåò ëèøü â óðàâíåíèè Äèðàêà îñíîâíîì óðàâíåíèè ðåëÿòèâèñòñêîé êâàíòîâîéìåõàíèêè. Òåì íå ìåíåå, â áîëüøèíñòâå ñëó÷àåâ íåîáõîäèìî ðàññìàòðèâàòü ñèñòåìû ñ ó÷¼òîì íàëè÷èÿ ñïèíà, õîòÿ èõ ñêîðîñòè çíà÷èòåëüíî ìåíüøå ñêîðîñòè ñâåòà.
Äëÿ ðåøåíèÿýòîé çàäà÷è Äèðàê ðàñøèðèë ôîðìàëèçì íåðåëÿòèâèñòñêîé êâàíòîâîé ìåõàíèêè, çàìåíèâ1âåêòîðû ñîñòîÿíèÿ òàê íàçûâàåìûìè ñïèíîðàìè : â ïðîñòåéøåì ñëó÷àå ýëåêòðîíà s =2åãî ñîñòîÿíèå îïèñûâàåòñÿ ñòîëáöîì 1 ψ ms = 2 ,ψ=1 ψ ms = −2òî åñòü â âåêòîðå ñîñòîÿíèÿ ïðîñòî ó÷èòûâàþòñÿ äâà âîçìîæíûõ ñïèíîâûõ ñîñòîÿíèÿ. Äëÿ÷àñòèö ñ áîëüøèì ñïèíîì àíàëîãè÷íûì îáðàçîì ñòðîèòñÿ ñïèíîð 2s-ãî ðàíãà, ÿâëÿþùèéñÿ,ïî ñóòè, òåíçîðîì 2s-ãî ðàíãà.Ïîñòðîèì òåïåðü ãàìèëüòîíèàí, ñîîòâåòñòâóþùèé ñïèíîðó ïåðâîãî ðàíãà. Äëÿ ýòîãîíåîáõîäèìî îïðåäåëèòü ýíåðãèþ âçàèìîäåéñòâèÿ òîêà è ìàãíèòíîãî ïîëÿ; áóäåì ðàññìàòðèâàòü òîëüêî ñëó÷àé îäíîðîäíîãî ìàãíèòíîãî ïîëÿ. Çàäàäèì ïîëå H1 = const ÷åðåç âåêòîðíûé ïîòåíöèàë111A = [H1 r] (rot A = rot· [H1 r] = ((r ∇) H1 −(H1 ∇) r + H1 div r − r div H1 ) = H1 ,222ïîñêîëüêó div r = 3, H1 = const, div H1 = 0); äëÿ H2 âûïîëíÿåòñÿ óðàâíåíèå Ìàêñâåëëà4πerot H2 =j .
Ìàãíèòíàÿ ñîñòàâëÿþùàÿ ñèëû Ëîðåíöà Fm = · [v H1 ] åé ñîîòâåòñòâóåòcc31ïîòåíöèàëüíàÿ ýíåðãèÿ U =1A j, ïîñêîëüêócgrad(A j) = (j ∇) A +(A ∇) j +[j rot A] + [A rot j] = [j H1 ],X ∂ vαdrdj = e v, rot j = e rot= e rot r = 0, (A ∇) j = e(A ∇) v = e Aα= 0, rot A = H2 ,dtdt∂ααcc[∇ H2 ]∇ =[∇, ∇] H2 = 0, ïîñêîëüêó [∇, ∇] = 0. Òàêèì îáðàçîì, ýíåðãèÿ4π4πee1[H1 r] v =[r m v] H1 =âçàèìîäåéñòâèÿ òîêà è ìàãíèòíîãî ïîëÿ εm = A j =c2c2mcee→→l · H1 = −µ H1 , ãäå l êèíåòè÷åñêèé ìîìåíò, à −µ =l ìàãíèòíûé ìîìåíò2mc2mc÷àñòèöû.Äëÿ îäíîýëåêòðîííîãî àòîìà ñ l = 0 ãàìèëüòîíèàí çàïèøåòñÿ â âèäå H = H0 +gµB (S H),ãäå H0 ìåõàíè÷åñêàÿ ñîñòàâëÿþùàÿ ãàìèëüòîíèàíà (êèíåòè÷åñêàÿ ýíåðãèÿ ýëåêòðîíà èýíåðãèÿ ýëåêòðîñòàòè÷åñêîãî âçàèìîäåéñòâèÿ ñ ÿäðîì), H ïîñòîÿííîå ìàãíèòíîå ïîëå,e~µB = ìàãíåòîí Áîðà, à g ôàêòîð ñïåêòðîñêîïè÷åñêîãî ðàñùåïëåíèÿ (g-ôàêòîð ),2mcââîäèìûé äëÿ ñâÿçè ìàãíèòíîãî ìîìåíòà è îïåðàòîðà ñïèíà.
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