Лекция (2) (Лекции), страница 8

Óðàâíåíèå äëÿ ôóíêöèè R(r), êîòîðîå îïðåäåëÿåò çíà÷åíèå äìèñêðåòíûõ óðîâíåéýíåðãèè E , èíäåêñà m íå ñîäåðæèò. Òàêèì îáðàçîìêàæäîå ñîáñòâåííîå çíà÷åíèå ñôåðè÷åñêè ñèììåòðè÷íîãî ãàìèëüòîíèàíà âûðîæäåíî,ïî ìåíüøåé ìåðå, (2l + 1)-êðàòíî.Ýòî ñâîéñòâî óðîâíåé ÿâëÿåòñÿ ïðÿìûì ñëåäñòâèåì ñôåðè÷åñêîé ñèììåòðèè ãàìèëüòîíèàíà.8. Óðîâíè ýíåðãèè â öåíòðàëüíî-ñèììåòðè÷íîì ïîëå.Óðîâíè ýíåðãèè â öåíòðàëüíî ñèììåòðè÷íîì ïîëå îïðåäåëÿþòñÿ êâàäðàòè÷íî èíòåãðèðóåìûìèðåøåíèÿìè óðàâíåíèÿ−h̄2 1 d 2 drR(r)2m r2 drdr h̄2 1l(l + 1)2m r2++V (r) R(r)=ER(r),∞Z|R(r)|2 r2 dr∞.<0×òîáû óïðîñòèòü ñòðóêòóðó äèôôåðåíöèàëüíîãî îïåðàòîðà óðàâíåíèÿ, óäîáíî ïåðåéòè êôóíêöèè u(r):1Ψ(r) =u(r).rÓñëîâèå êâàäðàòè÷íîé èíòåãðèðóåìîñòè ïðèìåò âèäZ ∞|u(r)|2 dr < ∞,0à ñàìî óðàâíåíèå ñâåäåòñÿ ê−h̄2 d2u(r)2m dr2+ h̄2 1l(l + 1)2m r2+V (r) u(r)=Eu(r).×òîáû èçó÷èòü ïîâåäåíèå ðåøåíèÿ ïîëó÷åííîãî óðàâíåíèÿ ïðè àñèìïòîòè÷åñêè áîëüøèõçíà÷åíèÿõ ðàäèóñà îñòàâèì â óðàâíåíèè íàèáîëåå ñóùåñòâåííûå ïðè áîëüøèõ r ñëàãàåìûå.Åñëè ïîòåíöèàë óáûâàåò íà áîëüøèõ ðàññòîÿíèÿõ, òî óðàâíåíèå ñâîäèòñÿ ê−h̄2 d2u(r)2m dr2Îïðåäåëÿÿ âåëè÷èíó=√k=ïîëó÷èìu(r)=Eu(r).−2mE,h̄C1 ekr + C2 e−kr .Óáûâàþùåå íà áîëüøèõ ðàññòîÿíèÿõ ðåøåíèå âîçìîæíî ëèøü â òîì ñëó÷àå, åñëèE< 0,..

k> 0,C1=0.Çàìåòèì, ÷òî â êëàññè÷åñêèõ òåðìèíàõ óñëîâèå îòðèöàòåëüíîñòè ýíåðãèè âûäåëÿåò ôèíèòíûåäâèæåíèÿ. Ìîæíî ñêàçàòü, ÷òî38âîçìîæíûå äèñêðåòíûå óðîâíè ýíåðãèè ñîîòâåòñòâóþò êëàññè÷åñêèì ôèíèòíûìäâèæåíèÿì.Åñëè ïðè íåîãðàíè÷åííîì âîçðàñòàíèè ðàäèóñà ïîòåíöèàë ñòðåìèòñÿ ê íåêîòîðîìóïîëîæèòåëüíîìó ïðåäåëó,lim V (r) = V∞ > 0,r→∞òî ðåøåíèå óðàâíåíèÿ Øðåäèíãåðà íà áîëüøèõ ðàññòîÿíèÿõ ïîõîäèò íà ðåøåíèå óðàâíåíèÿ−h̄2 d2u(r)2m dr2+V∞ u(r)=Eu(r).Åñëè ýíåðãèÿ ìåíüøå ïðåäåëüíîãî çíà÷åíèÿ ïîòåíöèàëà, òîu(r)C1 ekr=C2 e−kr ,+k1p2m(V∞ − E).h̄=Êâàäðàòè÷íî èíòåãðèðóåìîå ðåøåíèå âûäåëÿåòñÿ óñëîâèåì C1 = 0. Åñëèlim r2 V (r)=r→00,òî ðåøåíèÿ óðàâíåíèÿ Øðåäèíãåðà âåäóò ñåáÿ íà ìàëûõ ðàññòîÿíèÿõ êàê ðåøåíèÿ óðàâíåíèÿ−d2u(r)dr21l(l + 1)u(r)r2+ò.å.

êàê ôóíêöèèu(r)C1 r−l==0,C2 rl+1 .+Ôóíêöèÿ u(r) ðåãóëÿðíà â íà÷àëå êîîðäèíàò ïðè óñëîâèè C1 = 0. Òàêèì îáðàçîì, ðåãóëÿðíîåðåøåíèå óðàâíåíèÿ Øðåäèíãåðà âåäåò ñåáÿ íà ìàëûõ ðàññòîÿíèÿõ êàêΨ(r)rl u(r),∼u(0)6= 0.Åñëè ïîòåíöèàë ñòðåìèòñÿ ê íóëþ ïðè âîçðàñòàíèè ðàäèóñà, òî ðåøåíèå óðàâíåíèÿ Øðåäèíãåðàåñòåñòâåííî èñêàòü â ôîðìåu(r)rl eαr w(r),=α=1√−2mE,h̄E0.Ôóíêöèÿ w(r) äîëæíà óäîâëåòâîðÿòü óðàâíåíèþrd2 wdr2+(2(l + 1) − 2αr)dwdr−(2α(l + 1) −2mrV (r))w(r)h̄2=0. ñëó÷àå äîñòàòî÷íî ïðîñòûõ ïîòåíöèàëîâ ðåøåíèå óðàâíåíèå óäîáíî ïðåäñòàâèòü â ôîðìåñòåïåííîãî ðÿäà.Ðàññìîòðèì, íàïðèìåð, ñëó÷àé êóëîíîâà ïîòåíöèàëà ïðèòÿæåíèÿ, êîãäà−r=1x2α(c − x)dwdx−Ïîñëå çàìåíû ïåðåìåííîéóðàâíåíèå ïðèíèìàåò âèäxd2 wdx2a=+Ze2.r=V (r)ãäål+1−mZe2,αh̄239aw(x)c==0,2(2l + 1).Ýòî õîðîøî èçâåñòíîå âûðîæäåííîå ãèïåðãåîìåòðè÷åñêîå óðàâíåíèå, êîòîðîå äîïóñêàåòðåøåíèå â ôîðìå ðÿäà∞Xw(x) =bs xs .s=0Ïîäñòàíîâêà ðÿäà â óðàâíåíèå ïðèâîäèò ê ðåêóððåíòíûì ñîîòíîøåíèÿìbs+1s+abs .(s + 1)(s + c)= îáùåì ñëó÷àå ïðè áîëüøèõ s ñïðàâåäëèâû ñîîòíîøåíèÿ1bs ,s∼bs+1..

bs∼1,s!ïîýòîìó îïðåäåëÿþùèé ôóíêöèþ w(x) ðÿä ñõîäèòñÿ ïðè âñåõ çíà÷åíèÿõ x. Îäíàêî, ýòîòðÿä îïðåäåëÿåò ýêñïîíåíöèàëüíî ðàñòóùóþ ïðè áîëüøèõ x ôóíêöèþ:eBx ,∼w(x)B> 0.Ôóíêöèÿ w(x) ìîæåò îêàçàòüñÿ êâàäðàòè÷íî èíòåãðèðóåìîé òîëüêî â îäíîì ñëó÷àå: îïðåäåëÿþùèéåå áåñêîíå÷íûé ðÿä ïðåâðàùàåòñÿ â êîíå÷íóþ ñóììó. Ýòî âîçìîæíî ëèøü â òåõ ñëó÷àÿõ,êîãäà ïàðàìåòð a ïðèíèìàåò öåëûå îòðèöàòåëüíûå çíà÷åíèÿ:a−nr ,=nr=0, 1, 2, ...ãäå nr ðàäèàëüíîå êâàíòîâîå ÷èñëî. Ïîäñòàâëÿÿ â ýòî ðàâåíñòâî ÿâíîå çíà÷åíèå a,ïîëó÷èìmZe2 1α =,n = nr + l + 1,h̄2 nn ãëàâíîå êâàíòîâîå ÷èñëî. Âñïîìèíàÿ, ÷òî α ðàâíîýíåðãèè ìîæíî ïåðå÷èñëèòü ôîðìóëîéEn−=Ýíåðãèÿ E0 ðàâíàE0=ãäå ïîñòîÿííàÿ ðàçìåðíîñòè äëèíû ra ðàâíàra√−2mE,h̄íàéäåì ÷òî óðîâíè1E0 .2n2Ze2,rah̄2.mZe2= ñëó÷àå Z = 1 âåëè÷èíà ra ïðåâðàùàåòñÿ â áîðîâñêèé ðàäèóñrB=h̄2me2=0.529117249 × 10−10 m,à ýíåðãèÿ E0 ïðèíèìàåò çíà÷åíèåE0=me4h̄2=27.21eV=4.36 × 10−11=2Ry.Çàìåòèì, ÷òî ïðèñòóïàÿ ê ïðîöåäóðå êâàíòîâàíèÿ ýíåðãèè ìû îæèäàëè, ÷òî óðîâíè ýíåðãèèìîæíî áóäåò ïåðå÷èñëèòü ïàðîé èíäåêñîâ nr , l, ïîýòîìó êàæäûé óðîâåíü â ñôåðè÷åñêèñèììåòðè÷íîì ïîëå áóäåò 2l + 1-êðàòíî âûðîæäåí.

 ñëó÷àå ïðîèçâîëüíîãî ñôåðè÷åñêîãîïîëÿ ýòî ïðàâèëüíî Îäíàêî, ïðèìåð êóëîíîâà ïîëÿ ïîêàçûâàåò, ÷òî äåëî äåëî îáñòîèò íåòàê ïðîñòî. Óðîâíè ýíåðãèè íóìåðóþòñÿ ÷èñëàìèn=nr + l + 1,40ïîýòîìó ïðè ôèêñèðîâàííîì çíà÷åíèè n ÷èñëà l ïðèíèìàþò n çíà÷åíèél0, 1, ...n − 1,=Êðàòíîñòü âûðîæäåíèÿ óðîâíÿ En ðàâíàd(n)n−1X=(2l + 1)=n2 .l=0Ïðè âûïîëíåíèè êîíêðåòíûõ ðàñ÷åòîâ ïðè÷èíó óâåëè÷åíèÿ êðàòíîñòè âûðîæäåíèÿ ïîíÿòüäîâîëüíî òðóäíî. Ïîýòîìó ñîîòâåòñâóþùåå âûðîæäåíèå èíîãäà íàçûâàþò ñëó÷àéíûì.Îäíàêî, ìû óæå çíàåì, ÷òî âñÿêîå âûðîæäåíèå ñâÿçàíî ñ íåêîòîðîé íåòðèâèàëüíîé ñèììåòðèåé.Êàæóùàÿñÿ ñëó÷àéíîñòü ñâÿçàíà âñåãî ëèøü ñ òåì, ÷òî äîïîëíèòåëüíàÿ ñèììåòðèÿ ãàìèëüòîíèàíàíå ñðàçó áðîñàåòñÿ â ãëàçà. Ìåæäó òåì, ïðè÷èíà è ñìûñë ýòîé ñèììåòðèè áûëè îòêðûòûåùå â XXVIII âåêå Ëàïëàñîì.Ñôåðè÷åñêèå ãàðìîíèêè×èñëà l â ýòîì ðàçäåëå ïðèíèìàþò çíà÷åíèÿ l = 0, 1, ... Âåêòîðû Φ(j, m) â êîîðäèíàòíîìïðîñòðàíñòâå áóäóò îáîçíà÷àòüñÿ Ylm (θ, φ).

Èíòåãðàë ïî ñôåðå åäèíè÷íîãî ðàäèóñàZ2πZπdφsinθdθF (θ, φ)00áóäåò îáîçíà÷àòüñÿ ñèìâîëîìZdr̂F (r̂).4. Óðàâíåíèåˆl3 F (θ, φ)=lF (θ, φ)èìååò ðåøåíèåF (θ, φ) = Cexp(ilφ)g(θ).5. Ôóíêöèÿ F èç ïðåäûäóùåãî ðàçäåëà áóäåò ðåøåíèåì óðàâíåíèÿˆl+ F (θ, φ)=g(θ)sinl θ.åñëè=6. Âåêòîð Yll , íîðìèðîâàííûé óñëîâèåìZdr̂|Yll (r̂)|2ðàâåí=0,1,r(2l + 1)! 1exp(ilφ)sinl θ, |C| = 1.4π2l l!Âû÷èñëÿÿ íîðìèðîâî÷íûé èíòåãðàë, âîñïîëüçóéòåñü ôîðìóëîéYll = CπZ2dxsinµ−1 x=µ µ2µ−2 B( , ).2 207. Ïîêàæèòå ÷òîˆl− (eimφ f (θ))=ei(m−1)φ sin1−m θ41dsinm θf.dcosθ8. Ïîñëåäîâàòåëüíî äåéñòâóÿ íà âåêòîð Yll îïåðàòîðîì ˆl− ïîêàæèòå ÷òîYlml−m ilφB ˆl−(e sinl θ)=Beimφ sin−m θ=dl−msin2l θ.dcosθl−m9. Èñïîëüçóÿ ñîîòíîøåíèå1Yl,m−1 = ˆl− Ylm p(l + m)(l − m + 1)ïîêàæèòå, ÷òîsYlm=ˆll−m (Yll )−(l + m)!.(2l)!(l − m)!10.

Âåêòîð Ylm ðàâåí (|C| = 1):sdl−m2l + 1 (l + m)!1eimφYlm (θ, φ) = Csin2l θ.lm4π (l − m)! 2 l!sin θdcosθl−m11. Âåêòîð Yl,−l ïðîïîðöèîíàëåí e−ilφ sinl θ.12. Ñïðàâåäëèâî ðàâåíñòâîˆlm (e−ilφ sinl θ)+=(−1)m ei(m−l)φ sinm−l θdmsin2l θ.dcosθm13. Íà÷àâ ïîñòðîåíèå ñôåðè÷åñêèõ ãàðìîíèê ñ âåêòîðà Yl,−l , ïîëó÷èòå ýêâèâàëåíòíîå(10) âûðàæåíèå Ylm :s2l + 1 (l − m)! 1 imφ mdl+mesinθYlm (θ, φ) = (−1)m Csin2l θ.4π (l + m)! 2l l!dcosθl+mÏîëèíîìû Ëåæàíäðà îïðåäåëÿþòñÿ ôîðìóëîéPl (cosθ)=1dl(cosθ2 − 1)l .2l l! dcosθlÑïðàâåäëèâî ñîîòíîøåíèåZ1Pl0 (x)Pl (x)dx=δl 0 l2.2l + 1−1ßâíûé âèä ïåðâûõ ïÿòè ïîëèíîìîâ Ëåæàíäðà:P0 (x) = 1,P3 =P1 (x) = x,1(5x2 − 3x),2P2 (x) =P4 =1(3x2 − 1),21(35x4 − 30x2 + 3).8Ñïðàâåäëèâà ôîðìóëà1|~r − ~r0 |=∞01 X r< l( ) Pl (r̂r̂ ),r> 0 r>420r< = min(r, r ), r> = max(r, r0 ).Ïðèñåäèíåííûå ïîëèíîìû Ëåæàíäðà ðàâíûPl msinm θ=Pl −m (cosθ)Z1dmPl (cosθ), m ≥ 0,dcosθm(−1)m=mPlm0 (x)Pl (x)dx=(l − m)! mPl (cosθ).(l + m)!δl 0 l2 (l + m)!.2l + 1 (l − m)!−114.

Åñëè êîýôôèöèåíò Ñ âûáðàòü ðàâíûì il , òî ñôåðì÷åñêèå ãàðìîíèêè îïðåäåëÿòñÿôîðìóëîésm+|m|2l + 1 (l − |m|)! imφ |m|Ylm (θ, φ) = (−1) 2 ilePl (cosθ)4π (l + |m|)!15. Ñïðàâåäëèâû ñîîòíîøåíèÿrYl0 (θ, φ)l=Yl,−mi2l + 1Pl (cosθ).4π(−1)l−m Ylm + .=16. Ôîðìóëó ñëîæåíèÿ äëÿ ïîëèíîìîâ Ëåæàíäðà:Pl (cosγ)=0Pl (cosθ)Pl (cosθ )+2ãäåcosγ=cosθcosθlX00(l − m)! mPl (cosθ)Plm (cosθ )cos(m(φ − φ ),(l+m)!m=1000sinθsinθ cos(φ − φ ),+ìîæíî çàïèñàòü â ôîðìå0Pl (r̂r̂ )=lX4π0+Ylm (r̂)Ylm (r̂ ) .m=−l17. Ïðèìåíÿÿ ôîðìóëó ñëîæåíèÿ äëÿ ïîëèíîìîâ Ëåæàíäðà ïîêàæèòå, ÷òî ôîðìóëåeixcosφ∞X=(2l + 1)il jl (x)Pl (cosφ),l=0rjl (x)=πJ 1 (x).2x l+ 2ìîæíî ïðèäàòü âèä ðàçëîæåíèÿ ïëîñêîé âîëíû ïî ñôåðè÷åñêèì ãàðìîíèêàìrp~i~h̄e=4π∞Xl=0lpr X+i jl ( )Ylm (r̂)Ylm (p̂) .h̄lm=−l18.

Ôóíêöèè Ylm (θ, φ) = Ylm (r̂) îáðàçóþò îðòîíîðìèðîâàííûé áàçèñ â ïðîñòðàíñòâåêâàäðàòè÷íî èíòåãðèðóåìûõ ôóíêöèé, çàäàííûõ íà åäèíè÷íîé ñôåðå:Z+Ylm (r̂) Yl0 m0 (r̂)dr̂ = δll0 δmm0 ,X0+Ylm (r̂)Ylm (r̂ )lm43=0δ(r̂ − r̂ ),ZF (r̂)000δ(r̂ − r̂ )F (r̂ )dr̂ .=Cèììåòðèÿ êóëîíîâà ïîòåíöèàëàÍà ïðîøëîé ëåêöèè ìû ïîëó÷èëè ôîðìóëó äëÿ óðîâíåé ýíåðãèè â êóëîíîâîì ïîëåEn=−1 Ze2,2n2 r0r0=h̄2.mZe2Ìû íàøëè êðàòíîñòü âûðîæäåíèÿ óðîâíåé ýíåðãèè. Ñòàöèîíàðíûå ñîñòîÿíèÿ â öåíòðàëüíî-ñèììåòðè÷íîì ïîëå íóìåðóþòñÿ ÷èñëàìè nr , l, m, à óðîâíè ýíåðãèè - ÷èñëàìè nr , l:HΨ(nr , l, m)=Ψ(nr , l, m)E(nr , l).Ïîñêîëüêó ïðè çàäàííîì l ÷èñëî m ïðèíèìàåò 2l + 1 çíà÷åíèé, òî óðîâíè ýíåðãèè âñôåðè÷åñêè ñèììåòðè÷íîì ïîëå, âîîáùå ãîâîðÿ, âûðîæäåíû (2l+1)-êðàòíî. Îäíàêî, êóëîíîâîïîëå äàåò íàì ïðèìåð èñêëþ÷åíèÿ èç ïðàâèë. Ïîñêîëüêó ÷èñëî n ðàâíîn=nr + l + 1,òî ïðè çàäàííîé ýíåðãèè, ò.å. ïðè çàäàííîì ÷èñëå n, ÷èñëî l ìîæåò ïðèíèìàòü n çíà÷åíèé:l = 0, 1, ..., n − 1.