С.В. Петров - Лекции (1124220), страница 5
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Èòàê, äëèíà âîëíû ÷àñòèöû äîëæíàpìàëî èçìåíÿòüñÿ íà ðàññòîÿíèÿõ ïîðÿäêà å¼ ñàìîé. Òàêæå, èñïîëüçóÿ ñîîòíîøåíèådpdpm dV=,2m(E − V ) = −dxdxp dxìîæíî çàïèñàòü 1 dλ ~ dp m~ dV == 3 12π dx p2 dx p dx ýòî îçíà÷àåò, ÷òî ïðèáëèæåíèå ïðèìåíèìî â òåõ ñëó÷àÿõ, êîãäà èìïóëüñ ÷àñòèöû âåëèê,à ïîòåíöèàë èçìåíÿåòñÿ äîñòàòî÷íî ïëàâíî.~~~ ïåðâîì ïðèáëèæåíèè W = W0 + W1 ; W 0 = W00 + W10 , W 00 = W000 + W100 ; ïîäñòàâëÿÿiii2â óðàâíåíèå Øðåäèíãåðà è ïðåíåáðåãàÿ ÷ëåíàìè ïîðÿäêà ~ , ïîëó÷èì i~W000 − (W00 )2 −2~ 0 0W W + 2m(E − V ) = 0.
Íî 2m(E − V ) = (W00 )2 , ïîýòîìói 0 1p01W00000000=−⇒W=−ln |p| + C1 .W0 + 2W0 W1 = 0 ⇒ W1 = −12W002p2ãäå λ =Òàêèì îáðàçîì,i Ri R1· pdx− · pdxψ≈=pC1 e ~+ C2 e ~.|p|Ïîäîáíûé ïîäõîä ê ðåøåíèþ çàäà÷ êâàíòîâîé ìåõàíèêè íàçûâàåòñÿ êâàçèêëàññè÷åñêèìïðèáëèæåíèåì, ìåòîäîì ôàçîâûõ èíòåãðàëîâ èëè ïðèáëèæåíèåì Âåíòöåëÿ- ÊðàìåðñàÁðèëëþýíà (ÂÊÁ).ie ~ W0 +W1193.2.Ñòàöèîíàðíàÿ òåîðèÿ âîçìóùåíèé.Ïóñòü ãàìèëüòîíèàí ñèñòåìû ïðåäñòàâèì â âèäå H = H0 + H0 , ïðè÷¼ì âëèÿíèå H0äîñòàòî÷íî ìàëî, à ðåøåíèå çàäà÷è H0 ψ (0) = E (0) ψ (0) èçâåñòíî. Äëÿ óäîáñòâà çàïèøåìH = H0 +λ V, λ 1. Áóäåì èñêàòü k -îå ñîñòîÿíèå, òî åñòü ðåøåíèå çàäà÷è H ψk = Ek ψk ,ãäå Ek = Ek (λ), ψk = ψk (ri , λ).
Ðàçëîæèì ψk è Ek â ñòåïåííîé ðÿä ïî λ:(0)(1)(2)(0)(1)(2)ψk = ψk + λψk + λ2 ψk + . . . , Ek = Ek + λEk + λ2 Ek + . . . .(0)(0)Ôóíêöèè ψ1 , . . . ψn îáðàçóþò ïîëíóþ îðòîíîðìèðîâàííóþ ñèñòåìó, ïîýòîìó ∀ i ≥ 0P(i)(0)ψk = Cn ψn ; ïðè i = 0 Cn = δkn .nÍà÷í¼ì ñ ðàññìîòðåíèÿ ñëó÷àÿ íåâûðîæäåííîãî ñïåêòðà; ïîäñòàâèì ðàçëîæåíèÿ äëÿψk , Ek â óðàâíåíèå Øðåäèíãåðà: H ψk = Ek ψk ⇒(0)(1)(2)(0)(1)(2)(0)(1)(2)⇒ (H0 +λ V)(ψk +λψk +λ2 ψk +. .
.) = (Ek +λEk +λ2 Ek +. . .)(ψk +λψk +λ2 ψk +. . .).Ïðèðàâíÿåì ÷ëåíû ïðè îäèíàêîâûõ ñòåïåíÿõ λ, òîãäà(0)(0)(0)(1)(1)(0)(H0 −Ek )ψk = 0, (H0 −Ek )ψk + (V −Ek )ψk = 0,(0)(2)(1)(1)(2)(0)(H0 −Ek )ψk + (V −Ek )ψk − Ek ψk = 0, . . .(0)(s)(1)(s−1)(H0 −Ek )ψk + (V −Ek )ψk(2)(s−2)− Ek ψk(s)(0)− . . . − Ek ψk = 0.(0)Äîìíîæèì âòîðîå óðàâíåíèå ñêàëÿðíî íà ψk ñëåâà:(0)(0)(1)(0)(1)(0)(ψk , (H0 −Ek )ψk ) + (ψk , (V −Ek )ψk ) = 0 ⇒(0)(1)(0)(0)(1)(0)(0)(1)(0)(0)(0)⇒ (H0 ψk , ψk ) − Ek (ψk , ψk ) + (ψk , V ψk ) − Ek = 0 ⇒ (H0 ψk = Ek ψk )(0)(0)(1)(0)(0)(1)⇒ (ψk , V ψk ) − Ek = 0 ⇒ (ψk , V ψk ) = Ek .(2)(0)(1)(s)(0)(s−1)Àíàëîãè÷íî Ek = (ψk , V ψk ), .
. . Ek = (ψk , V ψk).(0)Òåïåðü äîìíîæèì ýòî æå óðàâíåíèå íà ψm (m 6= k):(0)(1)(1)(0)(0)(0)(ψm, (H0 −Ek )ψk ) + (ψm, (V −Ek )ψk ) = 0 ⇒(1)⇒(1)ψk(0)n6=k(0)⇒ Cm = −(1)(0)(0)(0)(0)(0)⇒ Em(ψm, ψk ) − Ek (ψm, ψk ) + (ψm, V ψk ) = 0 ⇒!XXX(1)(0)(0)=Cn ψn(0) ⇒ (ψmCn (ψm, ψn(0) ) =Cn δmn = Cn ⇒, ψk ) =n6=k(0)(ψm , V ψk )(0)Em−(0)Ek(1)⇒ ψk = −n6=k(0)(0)X (ψm,Vψ )(0)m6=k Em−k(0)Ek(2)(0)· ψm, Ek = −(0)(0)X |(ψm, V ψ )|2m6=k(0)Em−k(0)Ek.(ïðè ñóììèðîâàíèè îïóùåí ÷ëåí Ck , êîòîðûé äîëæåí áûòü ðàâåí íóëþ ñîãëàñíî óñëîâèþ(0)(1)íîðìèðîâêè ψk = ψk + λψk â ïåðâîì ïðèáëèæåíèè ïî λ:(0)(0)(0)(1)(1)(0)(0)(1)(1)(0)1 = (ψk , ψk ) = (ψk , ψk ) + λ (ψk , ψk ) + (ψk , ψk ) = 1 + λ (ψk , ψk ) + (ψk , ψk ) ⇒(0)(1)⇒ Ck = (ψk , ψk ) = 0). Òàêèì ñïîñîáîì ìîæíî íàéòè âîëíîâóþ ôóíêöèþ è ýíåðãèþëþáîãî ñîñòîÿíèÿ. Óñëîâèåì ïîäîáíîãî ïðèáëèæåíèÿ áóäåò, î÷åâèäíî, ñõîäèìîñòü ðÿäîâäëÿ ýíåðãèè, òî åñòü(0)(0)(0)(0)|(ψm, V ψk )|2 |Em− Ek |.201, ãäå ĥ1 , ĥ2 îäíîýëåêòðîííûå ãàìèëüòîr121,íèàíû; e = 1 ðàáîòàåì â àòîìíîé ñèñòåìå åäèíèö.
Îáîçíà÷àÿ H0 = ĥ1 + ĥ2 , V =r12ïðèõîäèì ê çàäà÷å òåîðèè âîçìóùåíèé.  ðåøåíèè çàäà÷è äëÿ H0 ïåðåìåííûå ðàçäåëÿþò(0)ñÿ, òî åñòü ψ0 = f0 ϕ0 (áóäåì èñêàòü òîëüêî ýíåðãèþ îñíîâíîãî ñîñòîÿíèÿ, ïîýòîìó íàñÏðèìåð (àòîì ãåëèÿ): H = ĥ1 + ĥ2 +(i)íå èíòåðåñóåò ψ0 (i > 0)). Èñïîëüçóÿ ôîðìóëó, ïîëó÷åííóþ äëÿ àòîìà âîäîðîäà (ñì. 2.5),√1 µe4 Z 2(0)− −2Eriíàõîäèì E = − ·=−2.f,ϕ≈e= e−2ri , i = 1, 2 ⇒ ψ0 = e−2(r1 +r2 ) .0022~11Çàïèñûâàÿ=÷åðåç ñôåðè÷åñêèå ôóíêöèè, ìîæíî ïðîèíòåãðèðîâàòür12|r1 − r2 |5(1)(0) 1(0)(0)(1)(1)E0 = ψ0 ,ψ0= . E0 = E0 + E0 = 2E + E0 = −2.75r124(0)(E0 = 2E, ïîñêîëüêó â àòîìå ãåëèÿ äâà ýëåêòðîíà).
Ýêñïåðèìåíòàëüíîå çíà÷åíèåE0 = −2.9037.Òåïåðü ðàññìîòðèì ñëó÷àé âûðîæäåííîãî ñïåêòðà, òî åñòü ðåøåíèå çàäà÷è äëÿ âûðîæ(0)äåííîãî ñîñòîÿíèÿ Ek , êîòîðîìó ñîîòâåòñòâóåò ñèñòåìà îðòîíîðìèðîâàííûõ âîëíîâûåôóíêöèé ϕ1 , . . . ϕr . Ê ýòîìó ñëó÷àþ ïðèìåíèìû âñå ïîëó÷åííûå ðàíåå ðåçóëüòàòû, îäíàêî(0)íåîáõîäèìî èìåòü â âèäó, ÷òî íå âñÿêèå ôóíêöèè ψk ìîæíî âûáðàòü â êà÷åñòâå ðåøåíèÿíåâîçìóù¼ííîé çàäà÷è äëÿ ïðîâåäåíèÿ äàëüíåéøåãî ïîñòðîåíèÿ. Îáû÷íî ïîëó÷àåòñÿ òàê,÷òî âîçìóùåíèå ÷àñòè÷íî èëè ïîëíîñòüþ ñíèìàåò âûðîæäåíèå, ïîýòîìó â êà÷åñòâå íóëåâî(0)ãî ïðèáëèæåíèÿ ïðèõîäèòñÿ èñïîëüçîâàòü ñîâñåì äðóãèå ôóíêöèè ψk ; ðàññìîòðèì ñïîñîá(0)(1)(1)(0)íàõîæäåíèÿ òàêèõ ôóíêöèé: êàê áûëî ïîëó÷åíî ðàíåå, (H0 −Ek )ψk + (V −Ek )ψk = 0.P(0)(0)Ñîñòîÿíèå ñ Ek âûðîæäåíî, ïîýòîìó ψk = Cm ϕm .
Äîìíîæèì ïîëó÷åííîå ðàâåíñòâîmñêàëÿðíî íà ϕ1 ñëåâà:!X(0)(0)(1)(1)Cm ϕm = 0 ⇒ H0 ϕj = Ek ϕk ∀ j = 1, r)(ϕ1 , (H0 −Ek )ψk ) + ϕ1 , (V −Ek ) ·m!(0)(1)(0)(1)⇒ (Ek ϕ1 , ψk ) − (ϕ1 , Ek ψk ) +(1)ϕ1 , (V −Ek ) ·XCm ϕm= 0.mÏðîâîäÿ àíàëîãè÷íûå îïåðàöèè ñî âñåìè ϕi , ïîëó÷èì ñèñòåìó r ëèíåéíûõ óðàâíåíèé íàCr : PP(1)(1)ϕ1 , (V −Ek ) · Cm ϕm = 0(ϕ1 , V ϕm )Cm = C1 EkmmPP(1) (ϕ2 , V ϕm )Cm = C2 E (1)ϕ2 , (V −Ek ) · Cm ϕm = 0kmm⇔ .. ....PP(1)(1) ϕr , (V −E ) · Cm ϕm = 0 (ϕr , V ϕm )Cm = Cr Ekmmk(1)Ïóñòü Vij = (ϕi , V ϕj ), òîãäà ñèñòåìà óðàâíåíèé çàïèøåòñÿ â âèäå V c = Ek c, ãäå V ìàòðèöà âûðîæäåíèÿ.(1)(0)Ðåøàÿ çàäà÷ó íà ñîáñòâåííûå çíà÷åíèÿ V, íàõîäèì Ek , c è ψk .21Äëÿ îïðåäåëåíèÿ ïåðâûõ ïîïðàâîê ê âîëíîâûì ôóíêöèÿì âíîâü èñïîëüçóåì ðàâåíñòâà,ñîîòâåòñòâóþùèå êîýôôèöèåíòàì ïðè λ è λ2 â óðàâíåíèè Øðåäèíãåðà:(0)(1)(1)(0)(H0 −Ek )ψk + (V −Ek )ψk = 0,(0)(2)(1)(1)(2)(0)(H0 −Ek )ψk + (V −Ek )ψk − Ek ψk = 0.(0)(0)(0)Ñêàëÿðíî äîìíîæèì îáà óðàâíåíèÿ íà ψl , ïîëàãàÿ El = Ek ; òîãäà, âûïîëíÿÿ ïðåîáðàçîâàíèÿ, àíàëîãè÷íûå íåâûðîæäåííîìó ñëó÷àþ, ïîëó÷èì(0)(1)Èìåÿ â âèäó ψk(0)(0)(1)(1)(0)(1)(ψl , V ψk ) = 0, (ψl , V ψk ) = Ek (ψl , ψk ).P(0)= Ci ψi , çàïèøåìi(1)(0)(0)Cl Ek = Cl (ψl , V ψl ) +X0(0)(0)Ci (ψl , V ψi ),i(0)ãäå øòðèõ ó çíàêà ñóììû îçíà÷àåò ó÷¼ò òîëüêî óðîâíåé ñ ýíåðãèåé, îòëè÷íîé îò Ek(0)(0)(Ei 6= Ek ).
ÎòñþäàX01(0)(0)·Ci (ψl , V ψi );Cl = (1)(1)Ek − Eliêîýôôèöèåíòû Ci â ñóììå îïðåäåëÿþòñÿ òàêæå êàê è â íåâûðîæäåííîì ñëó÷àå. Î÷åâèäíî, îïðåäåëåíèå ïîïðàâîê âòîðîãî ïîðÿäêà ê ýíåðãèè ïðè íàëè÷èè âûðîæäåíèÿ íè÷åì íåîòëè÷àåòñÿ îò îïèñàííîãî âûøå íåâûðîæäåííîãî ñëó÷àÿ.Ïðèìåð (àòîì âîäîðîäà â ýëåêòðè÷åñêîì ïîëå): ïóñòü ýëåêòðè÷åñêîå ïîëå îäíîðîäíî è íàïðàâëåíî âäîëü îñè z : ε = (0, 0, ε).
 ýòîì ñëó÷àå V = ε z, H = H0 +λ ε z.Íàéä¼ì ýíåðãèþ ñîñòîÿíèÿ ñîñòîÿíèÿ ñ n = 2 (÷åòûð¼õêðàòíîâûðîæäåííîãî); â íåâîçìóù¼ííîì ñëó÷àå (ñ òî÷íîñòüþ äî êîíñòàíò):n, l, mψr −r= 1−e 22n = 2, l = 0, m = 0R20 Y00n = 2, l = 1, m = 1R21 Y11 = re− 2 sin θeiϕn = 2, l = 1, m = 0R21 Y10 = re− 2 cos θrrrn = 2, l = 1, m = −1 R21 Y1−1 = re− 2 sin θe−iϕrÏåðåõîäÿ ê äåêàðòîâûì êîîðäèíàòàì, ìîæíî çàïèñàòü R21 Y10 = ze− 2 pz -îðáèòàëü,rr11(R21 Y11 + R21 Y1−1 ) = xe− 2 px -îðáèòàëü, (R21 Y11 − R21 Y1−1 ) = iye− 2 py -îðáèòàëü.22Î÷åâèäíî, ÷åòûðå ïîëó÷åííûå âîëíîâûå ôóíêöèè ïîïàðíî îðòîãîíàëüíû. Íàõîäÿ ýëåìåíòû ìàòðèöû âûðîæäåíèÿ, çàìåòèì, ÷òî ñêàëÿðíûå ïðîèçâåäåíèÿ (ϕi , zϕj ) îòëè÷íû îò íóëÿòîëüêî â äâóõ ñëó÷àÿõ (2s, z · pz ) = (pz , z · 2s) = a (â îñòàëüíûõ ñëó÷àÿõ ïî R èíòåãðèðóåòñÿíå÷¼òíàÿ ôóíêöèÿ, ïîýòîìó èíòåãðàëû ðàâíû íóëþ), òî åñòü 0 0 0 a 0 0 0 0 .V= 0 0 0 0 a 0 0 022(1)Ñîáñòâåííûå çíà÷åíèÿ E1 = x îïðåäåëÿþòñÿ óðàâíåíèåì (E åäèíè÷íàÿ ìàòðèöà)det(V − xE) = 0 ⇒ x2 (x2 − a2 ) = 0 ⇒ x = 0, x = ±a ñèñòåìà îáëàäàåò òðåìÿ óðîâ(1)íÿìè ýíåðãèè.
Óðîâíè ñ E1 = ±a ÿâëÿþòñÿ ëèíåéíîé êîìáèíàöèåé 2s− è pz -îðáèòàëåé, à(1)óðîâíè ñ E1 = 0 px - è py -îðáèòàëÿìè. Òàêîå ðàñùåïëåíèå ýíåðãåòè÷åñêèõ óðîâíåé ïîääåéñòâèåì âíåøíåãî ïîëÿ íàçûâàåòñÿ ëèíåéíûì ýôôåêòîì Øòàðêà.3.3.Íåñòàöèîíàðíàÿ òåîðèÿ âîçìóùåíèé.Ïóñòü ãàìèëüòîíèàí ñèñòåìû ïðåäñòàâèì â âèäå H = H0 +λ H0 , ïðè÷¼ì H0 íå çàâèñèò îòâðåìåíè ÿâíî. Ïðè îòñóòñòâèè âîçìóùåíèÿ èçâåñòíî ðåøåíèå íåñòàöèîíàðíîãî óðàâíåíèÿiPØðåäèíãåðà ψ(x, t) = Cm ψm (x)e− ~ Em t , ãäå ψm (x) ñîáñòâåííûå ôóíêöèè, à Em ñîám2ñòâåííûå çíà÷åíèÿ H0 . |Cm| âåðîÿòíîñòü òîãî, ÷òî ýíåðãèÿ ñèñòåìû ïðèíèìàåò çíà÷åíèåEm . Çàïèøåì ðåøåíèå ïðè íàëè÷èè âîçìóùåíèÿ â âèäå ðàçëîæåíèÿ â ðÿä Ôóðüå ïî ψm ,ïîëàãàÿ êîýôôèöèåíòû Cm çàâèñÿùèìè îò âðåìåíè:Xii∂ψ X .i− Em tψ=Cm (t) · ψm e ~⇒=Cm − Em ψm e− ~ Em t .∂t~mmÏîäñòàâëÿÿ ýòè âûðàæåíèÿ â íåñòàöèîíàðíîå óðàâíåíèå Øðåäèíãåðà, ïîëó÷èì:X .XiiiCm − Cm Em ψm e− ~ Em t =i~ ·(Cm Em + λCm H0 )ψm e− ~ Em t ⇔~mmXX .ii⇔ i~ Cm ψm e− ~ Em t = λ Cm H0 ψm e− ~ Em tmm(çäåñü ñ÷èòàåòñÿ, ÷òî îïåðàòîð H0 ìóëüòèïëèêàòèâåí ïî t áîëåå îáùèé ñëó÷àé íå ðàññìàòðèâàåì).
Äîìíîæèì ðàâåíñòâî ñêàëÿðíî ñëåâà íà ψn (n 6= m, (ψn , ψm ) = δnm ), òîãäà.ii~Cn e− ~ En t = λXiCm (ψn , H0 ψm )e− ~ Em t ⇔m.⇔ i~C n = λXCm · H0nm eiωnm t , H0nm = (ψn , H0 ψm ), ωnm =m(0)En − Em.~(1)(2)Ðàçëîæèì Cn â ñòåïåííîé ðÿä ïî λ: Cn = Cn + λCn + λ2 Cn + . . . , ïðè÷¼ì(0)(0)(i)(i)Cn 6= Cn (t), Cn = Cn (t) ∀ i > 0. Ïîäñòàâëÿÿ ðàçëîæåíèÿ â íåñòàöèîíàðíîå óðàâíåíèå Øðåäèíãåðà, ïîëó÷èì..i~(λCn(1) + λ2 Cn(2) + . . .) =X(0)(1)(λCm+ λ2 Cm) · H0nm eiωnm t .mÏðèðàâíÿåì ÷ëåíû ïðè îäèíàêîâûõ ñòåïåíÿõ λ, òîãäà.(1)i~ · Cn=P (0) 0 iωnm tCm Hnm e. Êîýôôèm(0)öèåíòû Cm(0)â âèäå Cmîïðåäåëÿþòñÿ íà÷àëüíûìè óñëîâèÿìè. Çàäàäèì íà÷àëüíûå óñëîâèÿ ïðè t = 0= δkm (ýòî îçíà÷àåò, ÷òî ïðè t = 0 ñèñòåìà íàõîäèòñÿ â k -îì ñòàöèîíàðíîìñîñòîÿíèè). Î÷åâèäíî, â ýòîì ñëó÷àå t2ZZt.11 (1)0iωnk t(1)0iωnk τ(1) 20iωnk τ⇒ Cn (t) =· Hnk (τ )ei~ · Cn (t) = Hnk edτ, |Cn | = 2 · Hnk (τ )edτ .i~~ 00 âåðîÿòíîñòü òîãî, ÷òî âîçìóù¼ííàÿ ñèñòåìà íàõîäèòñÿ íà n-ì ýíåðãåòè÷åñêîì óðîâíå.23Ïðèìåð (ñëó÷àé ãàðìîíè÷åñêîãî âîçìóùåíèÿ): ïóñòü ñèñòåìà íàõîäèòñÿ âî âíåøíåìïîëå (íàïðèìåð, ýëåêòðè÷åñêîì), òàê ÷òî âêëàä ýòîãî ïîëÿ â ãàìèëüòîíèàí ñîñòàâëÿåòH0 = F e−iωt +G eiωt ; H0 òàêæå ýðìèòîâ, ïîýòîìó F+ eiωt +G+ e−iωt = F e−iωt +G eiωt ⇒ F = G+ ; t2Z 1 i(ωnk +ω)ti(ωnk −ω)t0−iωtiωt(1) 2Gnk e+ Fnk edt =Hnk = Fnk e+ Gnk e ⇒ |Cn | = 2 · ~ 021 iFnk i(ωnk −ω)tiGnkiFnk iGnk i(ωnk +ω)t= 2 · −e−e++.~ωnk + ωωnk − ωωnk + ω ωnk − ω Òåì íå ìåíåå, ïîäîáíîå çíà÷åíèå âåðîÿòíîñòè íàõîæäåíèÿ íà n-ì ýíåðãåòè÷åñêîì óðîâíåëèøåíî ôèçè÷åñêîãî ñìûñëà ïðè ωnk − ω = ε → 0: çíàìåíàòåëü äðîáåé ìàë, ÷òî ïðèäà¼òèì äîñòàòî÷íî áîëüøèå (è àáñóðäíûå äëÿ âîëíîâûõ ôóíêöèé, íîðìèðîâàííûõ íà åäèíèöó)çíà÷åíèÿ.
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