?(sinx?xx)?x(cosx?cosx?x)?x(x?x) ?sinx?2xcosx
?和B?都是厄米的，那么 4．说明：如果算符A?+B?)也是厄米的 (A??B??d???*B?)?d???*A??d? 证： ?*(A?12?12?12??)*d???(B ???2(A1?2??1)*d? ??B?)?]*d? ???2[(A1?+B?也是厄米的。 ∴ A
5．问下列算符是否是厄米算符：
1?p?x?p?xx?) ?p?x ②(x ①x2**?p?x)?2d????1?(p?x?2)d? (xx 解：①??1??1)*p?x?2d???(p?xx??1)*?2d? ??（x?xx???p?x 因为 p?p?x 不是厄米算符。 ∴ x11*1**?p?x?p?xx?)]?p?x)?2d????1?xx?)?2d? [(x?2d????1(x(p ②??122211?xx??1)*?2d???(x?p?x?1)*?2d? ??(p221?p?x?p?xx?))?1]*?2d? ??[(x21?xx??x?p?x)?1]*?2d? ??[(p21?p?x?p?xx?)是厄米算符。 ## ∴ (x2 6 （略）
?满足关系式???????、?????1，求证 7．如果算符??2???2?? ????2? ①??3???3??2 ????3? ②??2???2??2??2?????(1???)??? 证： ① ??2????????2???? ???2???(1???)???2???? ??? ?2??3???3?????2?????3?????(2??)?? ②??2???2?????3???? ?2??2???2(1?????3??)??? ?2??2 ?3??P??? 8．求 Lxx?PxLx?? ?P??? Lyx?PxLy?? ?P??P?L??? Lzxxz?P????????????? 解： Lxx?PxLx?(yPz?zPy)Px?Px(yPz?zPy)
?P??z?P??P?y??P?z?) ?P?P?P?P ?yzxyxxzxy?P??P????????? ?yzx?zPyPx?yPzPx?zPyPx)
= 0
?P????????????? Lyx?PxLy?(zPx?xPz)Px?Px(zPx?xPz)
?2?x?P??P?z??P?x?) ?P?P?P?P ?zxzxxzxz?2?x?P??z?2?P?x????P? ?zPPxzxxxPz) ??P?x? ?P?)P ??(xxxz? ??i?Pz?P????? Lzx?PxLz?(xPy??P??y?2?P??P?P ?xyxxx?)P??P?(x??y?) ?P?P?Pyxxxyx??P?y? ?P?Pxyxx?P??P??2?????2 ?xxy?yPx?PxxPy?yPx
??P?x? ?P?)P ?(xxxy? ?i?Py?x??? ??x?L 9． Lxx?x??? ??x?L Lyy?x??? Lz?xLz?? ?x??(y??z?)x??z?) ??x?L?P??x?(y?P?P?P 解： Lxxzyzy?x?x??x? ?P??z??x?y?P?z?P?P ?yzyzy?x?x?x?x?P??z??y?P??z??P?P ?yzyzy
= 0
?x??(z??x?)x??x?) ??x?L?P??x?(z?P?P?P Lyyxzxz?x?x??x? ??x?P??x?z?2P?P?P ?zxzxz?x?) ??x?P?(P ?zxx? ??i?z?x??(x??y?)x??y?) ??x?L?P?P??x?(x?P?P Lzzyxyx??y?x??y? ?2P?P??x?2P?x?P ?xyxyx??P?x?(x?P?) ?yxx? ??i?y
量子力学 第三章习题与解答
