量子力学习题答案(4)

即 ?2??12???22ddp22C(p,t)?(E?p22?)C(p,t)?0

两边乘以，得

11ddp22 ?C(p,t)?(2E???p2???)C(p,t)?0

???令??1???p?? p, ??2E??1???

??

dd?22

2C(p,t)?(???)C(p,t)?0

跟课本P.39(2.7-4)式比较可知，线性谐振子的能量本征值和本征函数为

En?(n?1)??2C(p,t)?Nne?12?p22Hn(?p)e?i?Ent

式中Nn为归一化因子，即 Nn?(??1/22n!n)1/2

4.4.求线性谐振子哈密顿量在动量表象中的矩阵元。

1??122??1p?2???2x2?????x 解：H22?22??x2??(x)dx Hpp????*(x)Hpp22 ?12???2?e(?i?px(??2?222??x1??12i??x)e22?p?xdx

1i(p??p)x

??ii2??2p?)2?2??12??e?(p??p)xdx?12??2?2??i????xe?2dx

?p?2?p?2?(p??p)???2?2???i21???()e2?i?p??2?2(p??p)xdx

?

2??(p??p)?12??()16

2?22?p??1i?????e?(p??p)xdx

?p22?p2?(p??p)?1212???22?22?p?2?(p??p)

?

?4.5 设已知在L22??(p??p)????2?22?p?(p??p)

?和LZ?和L?的矩阵分别为 的共同表象中，算符Lxy1010??1? Ly?0???02???i2??0?i0i0???i? 0?? Lx?0????12??0 求它们的本征值和归一化的本征函数。最后将矩阵Lx和Ly对角化。 解：Lx的久期方程为

???2???20?2???0???????0

32

?20 ??1?0，?2??，?3???

?的本征值为0，?，?? ∴Lx?的本征方程 Lx?0?? ?12??0?a1? 其中???a2??a31010??a1??1??a2?0???a3??a1????????a2? ?????a3????的本征函数在L?2和L?共同表象中的矩阵 ?设为LZx?? 当?1?0时，有

?0?? ?12??01010??a1??1??a2?0???a3??0???????0? ??0???? 17

?a2??0??????a?a??0? ?a3??a1，a2?0 ?13?2???0??a2??? ∴ ?0 由归一化条件

?a1?????0? ??a?1?? 1??0??0?a1???**?(a1,0,?a1)?0??2a1??a?1??122

取 a1?

????的本征值?对应于Lx???? ?0?1??2??0?1???2?0 。

当?2??时，有

?0?? ?12??01010??a1??1??a2?0???a3??a1????????a2? ??a???3??1?a2???2??a2?2a1?a1??1?????(a1?a3)???a2???a2?2a3 ??2????a?aa3?31???1??a2???2? ∴ ???a1?????2a1? ???a1????a1???**2a1,a1)?2a1??4a1???a??1? 由归一化条件 1?(a1*,2

18

取 a1?12

????????????2?1???对应于Lx的本征值? 2?1??2?1 ∴归一化的?? 当?2???时，有

?0?? ?12??01010??a1??1??a2?0???a3??a1?????????a2? ?????a3??1???a1?2??a2??2a1??a1??1????? ?(a1?a3)????a2???a2??2a3

?2????a?a1?1???a3???3??a2???2??a1??????2a1? ???a??1? ∴ ??? 由归一化条件

?a1???*** 1?(a1,?2a1,a1)??2a1??4a1???a??1?2

取 a1?12

?1??2?1???2??1??2??????对应于Lx的本征值?? ???? ∴归一化的????表象的变换矩阵为 ?2和L?的共同表象变到L 由以上结果可知，从LZx 19

??111??222?? S???01?2?1?2?

?111?????222?? ∴对角化的矩阵为L??x?SLxS

??1??11?20?1??2? L???111???010??22x?2??222??101????01?111?????010???2??11?222?????22??????111??000??222?? ???11??2?2??01?1??21??22?

?1?1?111?????212??????222???000??000? ???2?020?????0?0?? ??00?2????00???? 按照与上同样的方法可得

L?y的本征值为0，?，?? L?y的归一化的本征函数为 ??1????1??2?2??? ?0????0? ?????i?? ?1?2????1??2?????2?? 从L?2和L??Z的共同表象变到Ly表象的变换矩阵为 20

1?2???1?2?

1??2????1???2????????i??2???1????2??