由
?2aabm?n?asinx?sinxdx??mn
aa2
?A?
??2(x)? ?k2?2n?sinxaa2mE?2
n2 (n?1,2,3,?)可见E是量子化的。
?En??2?22ma2对应于En的归一化的定态波函数为
i?2n???Entsinxe, 0?x?a? ?n(x,t)??a a? 0, x?a, x?a?3 求一维谐振子处在激发态时几率最大的位置。
??? 解：?(x)??2?xe22?2122x
?1(x)??1(x)?4?2?
??x2e??2?22x2? ? ?x2e??3
22x?22d?1(x)2?3 ?[2x?2?2x3]e??x
dx? 令
d?1(x) ?0，得 dx1x?? x??? x?0 ? x???时，?1(x)?0。 由?1(x)的表达式可知，x?0 ，显然不是最大几率的位置。
d2?1(x)2?322223??2x2而 ?[(2?6?x)?2?x(2x?2?x)]edx2? 3224??[(1?5?2x2?2?4x4)]e??x?d2?1(x)4?31 ??2?0 2dx1?ex??2 可见x??1??????是所求几率最大的位置。
4 一维谐振子处在基态?(x)??e???2x2i2??t2，求：
(1)势能的平均值U?1??2x2； 2p2 (2)动能的平均值T?；
2? (3)动量的几率分布函数。 解：(1) U?11???2x2???222?????x2e??2x2dx
?1?1?111?2 ??2?222???2????2224??2??2??1?? 4 ???0x2ne?axdx?21?3?5???(2n?1)? n?1n2aap21?*2? (2) T???(x)p?(x)dx
2?2?????1??2? ?e????2?122xd2?2?2x2(??)edx 2dx2221??22? ???(1??2x2)e??xdx
???2????22???x ??[?edx??2?x2e??xdx]
?????2?2222??22?? ??[??2?3]
?2??2???22??22?2?? ? ?????2?2?4?4???1?? 4111 或 T?E?U?????????
244 ??(x)dx (3) c(p)???*p(x) ??2??12??1??????1 e2??1??2x2222xei?Px?dx
?1??????? eei?Px?dx
?2??12??12?????? e1ipp2??2(x?2)2?222??2??dx
??e??e???p22??22?2??? e1ip??2(x?2)22??dx
?p22?2?2p22?2?2 ????1???e
动量几率分布函数为 ?(p)?c(p)?21???e?p2?2?2
5 氢原子处在基态?(r,?,?)?13?a0e?r/a0，求：
(1)r的平均值；
e2 (2)势能?的平均值；
r (3)最可几半径； (4)动能的平均值；
(5)动量的几率分布函数。 解：(1)r??r?(r,?,?)d??4 ?3a0213?a0???00?2??0re?2r/a0r2sin? drd? d?
??0r3a?2r/a0dr
?
?0xne?axdx?
n! an?143!3?3?a0 42a0?2???a???0?e2e2(2)U?(?)??3r?a0e2??3?a04e2??3a0???00?2??01?2r/a02ersin? drd? d?r???00?2??0e?2r/a0rsin? drd? d?
??0e?2r/a0r dr4e21e2??3??2a0?2?a0??a???0?
(3)电子出现在r+dr球壳内出现的几率为 ?(r)dr???0?2?0[?(r,?,?)]2r2sin? drd? d??4?2r/a02er 3a04?2r/a02erdr 3a0 ?(r)? 令
d?(r)42?3(2?r)re?2r/a0 dra0a0d?(r)?0， ? r1?0, r2??, r3?a0 dr 当 r1?0, r2??时，?(r)?0为几率最小位置
d2?(r)4842?2r/a0 ?(2?r?r)e32a0dr2a0a0d2?(r)
dr2??r?a08?2e?0 3a0 ∴ r?a0是最可几半径。
1???1??1??12?22?2?2?(r2)?(sin?)?????? (4)T?p? r ? ? r?rsin?????sin2???2?2?2??2?2??1?r/a02?r/a02 T??e?(e)rsin? dr?d d? 3???0002??a0?2?2??1?r/a01d2d?r/a02??e[r(e)]rsin? drd? d? 32???0002?dr?a0rdr4?21 ??(?3a02?a0??0r2?r/a0(2r?)e dr
a022a0a04?2?2 ? (2?)?42442?a02?a0??(r)?(r,?,?)d? (5) c(p)???*p1 c(p)?(2??)3/2??10?a30e?r/a0rdr?e02?i?prcos??sin? d??d?
0i?prco?s?2? ?2?(2??)3/2?a30??0re2?r/a0dr?e0? d(?co?s)
?2?(2??)3/2?a30??0r2e?r/a0dri?eipr?i?prco?s?0i
?2?(2??)3/2?pr???r/a0?pr?re(e?e)dr ?3ip0?a0??0xne?axdx?n! an?1 ?2?(2??)3/2?11[?] 3ip1i1i?a0(?p)2(?p)2a0?a0? ?4ip 2332a0?ip?a?(1?p)202a0?24303244a0?2221 ?2a??a0(a0p??)
?(2a0?)3/2??(a0p??)2222
动量几率分布函数