Лекция (2) (Лекции), страница 10

Íåòðóäíî ïîíÿòü, íå âû÷èñëÿÿÿâíî êîýôôèöåíòîâ Êëåáøà-Ãîðäîíà, êàêèå çíà÷åíèÿ ïðèíèìàþò ÷èñëà j, m ïðè ôèêñèðîâàííûõçíà÷åíèÿõ j1 , j2 . Ïðè ôèêñèðîâàííîì j ÷èñëà m ïðèíèìàþò 2j +1 çíà÷åíèå, ìàêñèìàëüíîåçíà÷åíèå j ðàâíî ìàêñèìàëüíî âîçìîæíîé ñóììå ïðîåêöèé m1 + m2 , ò.å. jmax = j1 + j2 .Ìèíèìàëüíîå çíà÷åíèå ïîëíîãî ìîìåíòà îïðåäåëÿåòñÿ ÷èñëîì jmin = |j1 − j2 |. Èíà÷åãîâîðÿ, ÷èñëî j ïðèíèìàåò çíà÷åíèÿj|j1 − j2 |,=|j1 − j2 | + 1,...,j1 + j2 .Íåòðóäíî ïðîâåðèòü, ÷òîjX1 +j2(2j + 1)=(2j1 + 1)(2j2 + 1).j=|j1 −j2 |5.

Ïîêàæèòå, ÷òî åñëè îïðåäåëèòü ðàäèàëüíûé èìïóëüñPròî1(r̂~p + p~r̂),2=11 ∂ 2 ∂(r) =((~rp~)2 − ih̄(~rp~)).2r ∂r∂rr26. Ïîêàæèòå, ÷òî Pr ìîæíî ïðåäñòàâèòü â ôîðìåPr 2−h̄2=Pr1~rp~r=à−h̄i ,rPr 2=1((~rp~)2 − ih̄(~rp~)).r2Ĥ=1 2p~2µ7. Ãàìèëüòîíèàí+V (r)ìîæíî ïðåäñòàâèòü â ôîðìåĤ1((~rp~)2 − ih̄(~rp~) + h̄2~l2 )2µr2=+V (r).8. Ñïðàâåäëèâû ïåðåñòàíîâî÷íûå ñîîòíîøåíèÿà)[~rp~, Ĥ]=−2ih̄Ĥih̄(2V (r) + xαá)[rs , Ĥ]=ih̄s s−2r (~rp~)µ+∂V),∂xαh̄2 s(s + 1) s−2r ,2µâ)[rs (~rp~), Ĥ]i=2ih̄rs Ĥh̄s s−2r (~rp~)2µih̄rs (2V (r) + xα−∂V)∂xα+h̄2 s(s + 1) s−2r (~rp~),2µ+2ã) åñëè ïîòåíöèàë ðàâåí V (r) = − Zer , òî[rs (~rp~), Ĥ]=2ih̄(s + 1)rs Ĥ50+ih̄Ze2 (2s + 1)rs−1−ih̄h̄2 s s−2~2r lµ−h̄2 s(s − 1) s−2r (~rp~).2µ9.

Óñðåäíÿÿ ñîîòíîøåíèÿ 8á è 8ã â ñîñòîÿíèè ñ îïðåäåëåííûìè n è l ïîëó÷èì< rs−2 (~rp~) > =s+1< rs >n2ih̄s+1< rs−2 >,2− (2s + 1)r0 < rs−1 >s(2l + 1 + s)(2l + 1 − s) 2r0 < rs−2 > =40,r0+=h̄2.Ze2 µÝòî ðàâåíñòâî èçâåñòíî êàê ñîîòíîøåíèå Êðàìåðñà.×òîáû íàéòè ñ ïîìîùüþ ñîîòíîøåíèÿ Êðàìåðñà ñðåäíåå çíà÷åíèå ëþáîé ñòåïíè r,íåîáõîäèìî íåçàâèñèìî íàéòè äâà èç íèõ.10. Óñðåäíÿÿ ñîîòíîøåíèå 1 â ñîñòîÿíèè ñ îïðåäåëåííûì çíà÷åíèåì ýíåðãèè, ïîëó÷èì2 < T̂ > = < xα∂V>.∂xα ñëó÷àå ïîòåíöèàëîâ, îäíîðîäíûõ ïî êîîðäèíàòàì, ýòî ðàâåíñòâî ïðèâîäèò ê ñîîòíîøåíèÿììåæäó êèíåòè÷åñêîé, ïîëíîé è ïîòåíöèàëüíîé ýíåðãèÿì, èçâåñòíûì îáû÷íî êàê òåîðåìûâèðèàëà.11. Åñëè ãàìèëüòîíèàí, åãî íîðìèðîâàííûå íà åäèíèöó ñîáñòâåííûå âåêòîðû, ñîáñòâåííûåçíà÷åíèÿ çàâèñÿò îò ïàðàìåòðà λ, òî ñïðàâåäëèâà òåîðåìà Ôåéíìàííà-Ãåëüìàíà:<∂ Ĥ> =∂λ∂E(λ).∂λ12.

Äëÿ êóëîíîâà ïîòåíöèàëà ñîîòíîøåíèå (10) ïðîâîäèò ê ðàâåíñòâó<1> =r1.r0 n 213. Ðàññìàòðèâàÿ â ñðåäíåì çíà÷åíèè ãàìèëüòîíèàíà (7) âåëè÷èíó l êàê íåïðåðûâíóþïåðåìåííóþ è ïðèìåíÿÿ ðàâåíñòâî (11), ìîæíî ïîêàçàòü, ÷òî<1> =r2r01.+ 1/2)2 n3 (l14. Èç ñîîòíîøåíèÿ Êðàìåðñà ñëåäóåò, ÷òî<1> =r3r03 n3 l(l1,+ 1/2)(l + 1)1r0 (3n2 − l(l + 1)),2r0 2 n 2< r2 > =(5n2 + 1 − 3l(l + 1)).215. Íåîïðåäåëåííîñòü ðàäèóñà â ñîñòîÿíèè ñ çàäàííûìè ÷èñëàìè n è l ðàâíàr0 p 2 2∆r =n (n + 2) − l2 (l + 1)2 .2<r> =Âåêòîð Ëàïëàñà-Ðóíãå-Ëåíöà ýòîì ðàçäåëå îïåðàòîðû íå áóäóò ñíàáæàòüñÿ øëÿïêàìè.511. Ñïðàâåäëèâû ñîîòíîøåíèÿ(~l × p~)(~l × p~)~p = 0,−(~p × l)=p~(~p × ~l) = 0,+2i~p.p~(~l × p~) = 2i~p2 ,(~l × p~)~r = −h̄~l2 ,(~p × ~l)~p = 2i~p2 .~r(~p × ~l) = h̄~l2 ,~r(~l × p~) = −h̄~l2 + 2i(~rp~),(~p × ~l)~r = h̄~l2 + 2i(~p~r).Âåêòîð Ëàïëàñà-Ðóíãå-Ëåíöà îïðåäåëÿåòñÿ ôîðìóëîé~A~r1 ~((l × p~) − (~p × ~l)) + ,2p0r=p0 =h̄.r0~ ìîæíî ïðåäñòàâèòü â ôîðìå2.

Âåêòîð A~A1 ~i~r(l × p~) − p~ + .p0p0r=~A~A−=−=i~r[~p, ~l2 ] + .2p0ri~r1(~rp~2 − (~rp~)~p) − p~ + .h̄p0p0r3. Ìîìåíò èìïóëüñà è âåêòîð Ëàïëàñà-Ðóíãå-Ëåíöà îðòîãîíàëüíû:~lA~~~lA==0.4. Ñïðàâåäëèâû ïåðåñòàíîâî÷íûå ñîîòíîøåíèÿ[xα p~2 , xβ p~2 ][xα p~2 , (~rp~)pβ ]−2ih̄2 αβγ lγ p~2 .=−ih̄(xα pβ − δαβ p~2 ),=~r11[xα p~2 − (~rp~)pα , ] =xα pβ r2 − xα xβ (~rp~) − ixα xβ .rh̄h̄5.

Âåêòîð Ëàïëàñà-Ðóíãå-Ëåíöà óäîâëåòâîðÿåò ïåðåñòàíîâî÷íûì ñîîòíîøåíèÿì[lα , Aβ ][H, Aα ][Aα , Aβ ]===0,−iiαβγ Aγ ,H=1 2 Ze2p~ −.2µr2Hαβγ lγ , E0E0=Ze2.r06. Ñïðàâåäëèâà ôîðìóëà~2A=2H ~2(l + 1) + 1.E0 ïîäïðîñòðàíñòâå ñîñòîÿíèé, ïðèíàäëåæàùèõ äèñêðåòíûì óðîâíÿì ýíåðãèè îïåðàòîð− 2HE0 ïîëîæèòåëüíî îïðåäåëåí, ïîýòîìó ìîæíî îïðåäåëèòü îïåðàòîðûkα = BAα=Aα B,12H=−.2BE07. Ñïðàâåäëèâû ñîîòíîøåíèÿ[lα , lβ ] = iαβγ lγ , [lα , kβ ] = iαβγ kγ , [kα , kβ ] = iαβγ lγ .52~k 2 + ~l2 + 1=−E0.2H~l~k = ~k~l = 0.8. ÎïåðàòîðûJ~1=óäîâëåòâîðÿþò ñîîòíîøåíèÿì1 ~ ~(l + k),2[Jaα , Jbβ ]J~2==1 ~ ~(l − k)2iδab αβγ Jaγ ,1J~12 = J~22 = (~l2 + ~k 2 ).49.

Îïåðàòîðû Jaα ìîæíî ñ÷èòàòü äèíàìè÷åñêèìè ïåðåìåííûìè, îïèñûâàþùèìè ïîâåäåíèå÷àñòèöû â êóëîíîâîì ïîëå.  êà÷åñòâå ïîëíîãî íàáîðà íàáëþäàåìûõ ìîæíî âçÿòü âåëè÷èíû(Ja 2 , Ja3 ). Ýòî ñîîòâåòñâóåò âûáîðó áàçèñà â ïðîñòðàíñòâå ñîñòîÿíèé â ôîðìå ïðÿìîãîïðîèçâåäåíèÿ Φ(j1 , m1 )Φ(j2 , m2 ). Ïîñêîëüêó J1 = J2 , òî âåêòîðû áàçèñà íóìåðóþòñÿ òðåìÿ÷èñëàìè - (n = 2j1 + 1, m1 , m2 ).Êðàòíîñòü âûðîæäåíèÿ óðîâíåé ðàâíàd(n)=(2j1 + 1)(2j2 + 1)=n2 .Ïîñêîëüêó ìîìåíò èìïóëüñà ðàâåí~l=J~1 + J~2 ,òî âîçìîæíûå çíà÷åíèÿ l íàõîäÿòñÿ ïî ïðàâèëàì ñëîæåíèÿ ìîìåíòîâ.  íàøåì ñëó÷àå ln−1n−1n−1èçìåíÿåòñÿ îò | n−12 − 2 | = 0 äî 2 + 2 = n − 1.Áàçèñ Ψ(n, l, m) ïî ïðàâèëàì ñëîæåíèÿ ìîìåíòîâ ïðåäñòàâëÿåòñÿ êàê ëèíåéíàÿ êîìáèíàöèÿâåêòîðîâ áàçèñà Φ(n, m1 , m2 ).53.