?122122??x??x??2543253322??xe(?2?x?9?x?3?)?e(?8?x?18?x)?? 3?????(?x?7?)422422?3?e122??x2(2?x?3?x)33
?(?x?7?)?(x) 把
ddx22?(x)代入①式左边，得
左边???2d?(x)dx2222?2?12?2??x?(x)?x?(x)??24222 ? 7 ??2??(x)??21222???x?(x)1222 ? 7 ?????2??(x)?2?（???）x?(x)?4??x?(x)
227112222 ? ???(x)???x?(x)???x?(x)2227 ? ???(x)2右边?E?(x)72 当E???时，左边 = 右边。 n = 3
?(x)?为
72??。
?d3?dxe122??x2是线性谐振子的波函数，其对应的能量(2?x?3?x)，
33第三章 量子力学中的力学量
3.1 一维谐振子处在基态?(x)? (1)势能的平均值U? (2)动能的平均值T?12p22??e??x222i??t2，求：
??x；
22?；
(3)动量的几率分布函数。 解：(1) U?12??2x2?1212????2????????xe12??x22dx
12 ? ?
2?2?22?2????212?2?14??2????
14??
21
??nn?1)?0x2e?ax2dx?1?3?5???(22n?1ana (2) T?p2?*2??12???(x)p?2???(x)dx ??1?12?12?2x2(?2?2x2dx
?2???e???2ddx2)e ???22222?2??2????(1??x)e??xdx
???2?2[????2x22??2x2?2???edx??2????xedx] ???22?2] ?2??[?????2?3 ???2?2?2?2?2??2?2?4???4?????
?14?? 或 T?E?U?12???14???14??
(3) c(p)???*p(x)?(x)dx ?12??????1222?xPx??? ee?i?dx
?1???12?2x2i2?e?Px?dx
????? e21???1 ??2?dx
???2?2(x?ip2p?? e?2)??2?2221?p?12?2(x?ip2 ??2?2?2??2??e????? e2)dx
1??p22?p2 ?2?2?222???e???1?e2?2?
??
动量几率分布函数为 2 ?(p)?c(p)2?p?1e?2?2?
??#
3.2.氢原子处在基态?(r,?,?)?1e?r/a0，求：
?a30 (1)r的平均值； (2)势能?e2r的平均值；
22
(3)最可几半径； (4)动能的平均值；
(5)动量的几率分布函数。 1?2?? 解：(1)r??r?(r,?,?)2d???a3?re?2r/a0r2sin? drd? d?
00?0?0 ?4a3??/a0
00r3a?2rdr ??nax0xe?dx?n!an?1
?43!a34?30?2a0
?2????a?0?2)U?(?e2(e2r)???a3??2??1?2r/a00?0?0r2sin? drd? d?0re??e2?2??2r/a0?a3?0?0?0e?rsin? drd? d?0
4e2??
a3???2r/a00er dr04e2??1e2a3?2??0?2?a0???a?0?
(3)电子出现在r+dr球壳内出现的几率为
?(r)dr???2?224?2r/a020?0[?(r,?,?)]rsin? drd? d??a3erdr
0 ?(r)?4a3e?2r/a0r2 0 d?(r)4dr?a3(2?20ar)re?2r/a0
0 令
d?(r)dr?0， ? r1?0, r2??, r3?a0
当 r1?0, r2??时，?(r)?0为几率最小位置 2
d?(r)2?2r/a0dr2?4a3(2?80ar?42r)e
0a02
d?(r)?8?2dr2?r?aa3e?0
00 ∴ r?a0是最可几半径。 2 (4)T??12?p?2???22?? ? 2 ? 1 ??2?1?r2 ???r(r?r)?sin???(sin????)?1??sin2???2?? 23
2 T????2??1/a02?r/a02??)r2sin0?0?0?a3e?r?(e? drd? d?
0???2?2??1?r/a01d2?r/a022??0?0?0?a3e0r2dr[rddr(e)]rsin? drd? d?
2 ??4?1r22?a3(?0a(2r??r/a0 dr0??0a)e
0 4?222 ?(2a0?22?a404?a04)?2?a2
0 (5) c(p)????*p(r?)?(r,?,?)d? i c(p)?11?r/a2?0(2??)3/2??0r2dr?a3e??e??prcos?0sin? d??0d?
0 ?2??r2e?r/a?i0dr???prco?s(2??)3/2?a3?00e d(?cos?)
0? ?2?r2e?r/a?iprco0dr???s(2??)3/2?a3??00ipre
0 ?2???ipri?pr(2??)3/2re?r/a0(e??e?)dr
?a30ip?0??n?ax0xedx?n!an?1 ?2??[11
(2??)3/2??a30ip(1a?i(1i]0?p)2a?0?p)2 ?14ip
2a3?3ip?p20a20?(1a2?2)0?444 ?a0?2a3?3?a2222
00(a0p??) ?(2a3/20?)??
(a2p2??2)20 动量几率分布函数
35 ?(p)?c(p)2?8a0??2(a22)4
0p??#
3.3 证明氢原子中电子运动所产生的电流密度在球极坐标中的分量是 Jer?Je??0 Je? m2e??? rsin??n?m
证：电子的电流密度为
24
??i? Je??eJ??e(?2?n?m??*n?m??*n?m??n?m)
?在球极坐标中为
???1??1? ??er ?e??e??rr??rsin???式中er、e?、e?为单位矢量
??i?Je??eJ??e[?2????1??1?(e?e?e)?n?mr???rr??rsin???*n?m???
???1??1?* ? ? n?m(er?e??e?)??rr??rsin?????ie??[er(?2?*n?m
n?m]?n?m
?r?*n?m??*n?m??r???)?en?m?(??n?m1?n?mr??1?*n?m ??1?r????n?m)?e?(1rsin????*n?m?rsin??*n?m???
?n?m)] ??中的r和?部分是实数。 ?ie? ∴ Je??(?im?n?m2?rsin?n?m2?im?2n?m?)e? ??e?m?rsin??2n?m?e?
可见，Jer?Je??0 Je???e?m?rsin??2n?m
#
3.4 由上题可知，氢原子中的电流可以看作是由许多圆周电流组成的。 (1)求一圆周电流的磁矩。 (2)证明氢原子磁矩为
??????????me?2?me?2?c (SI)M?Mz
(CGS) 原子磁矩与角动量之比为
MLze?? )?2? ( SI? ????e ( C GS)??2?c
z这个比值称为回转磁比率。
解：(1) 一圆周电流的磁矩为 dM?iA?Je?dS?A （i为圆周电流，A为圆周所围面积）
e?m ????rsin? ??e?m2n?m2dS??(rsin?)
2??rsin??n?mdS
25