量子力学第七章习题

1、4010110201102)0 1 (2222121xxss4)(2222xxxsss 001002)0 1 (2121iissyy 401002002)0 1 (2222121iiiissyy 4)(2222yyysss16)()(422yxss16)()(422yxss1111201102baba 111111 abbaab1),(11*1*1aaaa222222 abbaabi12121 i12121cos10012cos002cos01102iisncoscoscoscoscoscos2iisn其相应的久期方程为 0cos2)cos(cos2)cos(cos2cos2ii即 0)cos

2、(cos4cos42222220422 ) 1coscoscos(222利用 2babaii2coscoscoscoscoscos2bbiacos)cos(cos cos1coscosib由归一化条件，得22*),(12121bababa 1cos1coscos222aia 1cos122a)cos1 (2coscos1cos1)(21isn)cos1 (2coscos1cos1)(21isn2121)cos1 (2coscos2cos110)cos1 (2coscos012cos1)(21iisncos22cos122cos12zs)cos1 (2coscos2cos1)(21isncos2

3、zs 解：可改写成 10),()(2301),()(2110211121yrryrr )(),()(23)(),()(21211021211121zzsyrrsyrr4zl44324122ziizscs )4(422eeselemzzz bme4142 7.6 一体系由三个全同的玻色子组成，玻色子之间无相互作用。玻色子只有两个可能的单粒子态。问体系可能的状态有几个？它们的波函数怎样用单粒子波函数构成？)()()(3211qqqiii )()()(3212qqqjjj)()()()()()()()()(311322313213qqqqqqqqqjiijiijii )()()()()()()()(

4、)(311322313214qqqqqqqqqijjijjijj 解： )()()()(22/112/122/112/1)1()1(zzzzssssss )s()s()s()s(z22/1z12/1z12/1z22/1 )s()s(z22/1z22/1 = 1)()()()(22/112/122/112/1)2()1(zzzzssssss )s()s()s()s(z22/1z12/1z12/1z22/1 = 0)()()()()()(2122/112/122/112/122/112/1)3()1(zzzzzzssssssss )()()()( )()()()(2122/112/112/122/

5、122/112/112/122/1zzzzzzzzssssssss 0)()(2122/122/1zzss = 0同理可证其它的正交归一关系。)()()()()()()()(2122/112/122/112/122/112/122/112/1)3()3(zzzzzzzzssssssssss )s()s()s()s(21z12/1z22/1z22/1z12/1 )s()s()s()s(21z12/1z12/1z12/1z22/1 )s()s()s()s(21z12/1z22/1z12/1z22/1 1210021)()()()(2122/112/122/112/1zzzzssss 解：电子波函数

6、的空间部分满足定态s-方程 )()()()(22rerrur)()(21)()(2222222222rerrrzyx)()(21)()(2222222222rerrrzyx)z(z)y(y)x(x) r ( exyzxyzzyxxyzzyx)(21)(222222222222ezxzzyxyyxxxx)2112()2112()2112(222222222222222xexxxx)2112(22222)()(2221xhenxxnxnn)()(2221yhenyymymm)()(2221zhenzzz)()()()(2221zhyhxhennnrmnrmnnmyeyxyy)2112(22222zezxzz)2112(22222zyxeeee)()()()(2221zhyhxhennnrmnrmnnm)mn(e23nm22r212/30000e)() r (22214/32/5100122)(rxer 两电子的空间波函数能够组成一个对称波函数和一个反对称波函数，其形式为 )()()(21),(2011211021rrrrrrs )(211)(2122/342221222212rrrrexex )(21122/3422212)(rrexx22r212/30000e)() r ()r ()r ()r ()r (21)r ,r (1120211021a )r