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îïåðàòîðà T̂a â èìïóëüñíîì ïðåäñòàâëåíèè: Z µ ¶ µ ¶ 0 1 i i 0 hp| T̂a |p i = 3 exp − pr T̂a exp p r d3 r = (2π}) } } µ ¶Z ½ ¾ µ ¶ 1 i 0 i 0 3 0 i 0 = exp pa exp r(p − p) d r = δ(p − p) exp pa . (2π})3 } } } | {z } (2π})3 δ(p0 −p) Ìíîæèòåëü, íàõîäÿùèéñÿ ïîñëå δ -ôóíêöèè, êàê ðàç è áóäåò îïåðàòîðîì ñäâèãà â èìïóëüñíîì ïðåäñòàâëåíèè: µ ¶ µ ¶ i i T̂a = exp pa = exp ap . (2.21) } } µ ¶ i  êîîðäèíàòíîì ïðåäñòàâëåíèè îïåðàòîð T̂a = exp ap̂ , ò.å. ïî ñòðóê- } òóðå ôîðìàëüíî ñîâïàäàåò ñ (2.21). Îäíàêî â ýòîì ñëó÷àå p̂ = −i}∇r , ò.å. îïåðàòîð, è T̂a âûðàæàåòñÿ ÷åðåç îïåðàòîðíóþ ýêñïîíåíòó, à â èì- ïóëüñíîì ïðåäñòàâëåíèè (2.21) T̂a îáû÷íûé ìíîæèòåëü, ÿâëÿþùèéñÿ ôóíêöèåé p. ¤ Ïðèìåð 2.7. Ôóíêöèè {Φn (r)} îáðàçóþò ïîëíóþ îðòîíîðìèðîâàííóþ ñèñòåìó. Âûðàçèòü hn0 | ÂB̂ |ni ÷åðåç hn0 |  |ni è hn0 | B̂ |ni. P ∗ Ðåøåíèå. Âñïîìèíàÿ óñëîâèå ïîëíîòû Φn (r)Φn (r0 ) = δ(r0 − r), ïîëó÷à- n åì: Z hn | ÂB̂ |ni = Φ∗n0 (r0 )ÂB̂Φn (r0 ) d3 r0 = 0 ZZ = Φ∗n0 (r0 ) δ(r0 − r)B̂Φn (r0 ) d3 r0 d3 r = X ZZ = Φ∗n0 (r0 )ÂΦn00 (r0 )Φ∗n00 (r)B̂Φn (r) d3 r0 d3 r = n00 XZ Z = Φ∗n0 (r0 )ÂΦn00 (r0 ) d3 r0 · Φ∗n00 (r)B̂Φn (r) d3 r = n00 X = hn0 |  |n00 i hn00 | B̂ |ni . n00 Èòàê, X hn0 | ÂB̂ |ni = hn0 |  |n00 i hn00 | B̂ |ni , (2.22) n00 28
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