喀兴林高等量子力学习题EX31

练习31.1证明)(ba与)'(ba的对易关系（31.4）和)(ba与)'(ba的对易关系（31.6）式。0)()'()'()(babababa（31.4）0)()'()'()(babababa（31.6）（解答：熊凯；校对：李泽超）证明：将)'()(baba和)()'(baba分别作用在n粒子基左矢bbbbn....;上bbbbbbnnnbbbbbbnnnbababbbbn....';2)2)(1(....';2)2)(1()'()(....;(1)bbbbbbnnnbababbbbn....';2)2)(1()'()(....;(2)由)2()1(得：0)()'()'()(babababa（2）将)'()(baba与)()'(baba分别作用在右矢bbbbn....;上bbbbbnbbbbbbnbbbbbbnbbbbbbnbbbbbbbnbanbbbbnbabavn....';)(........';)(....';)(....;)'(....';1)(1....;)'()(2（3）bbbbbnbbbbbbnbbbbbbnbbbbbnbbbbbnbbbbbnbbbbbnbbnbabbbbnbabavnvn....';)(........';)(....';)(]....;1)(........;1)(....;1)(....;1)([1)'(....;)()'(112（4）由)4()3(得：)'()()'()'()(bbbabababa□练习31.2计算下列对易关系：)]()'()'()(),()([babababababa)]()'()'()(),'()'([babababababa（解答：熊凯；校对：李泽超）解：（1）令)()()(bababN为处于b态的占有数算符由（31.10）、（31.11）两式可得：)'()()](),([bbbababN（31.10）)'()()](),([bbbababN（31.11）0)'()]()'()'()([)'()'()()'()()'()'()]'(),([)]'(),()['()]'()'(),([)]'(),([bbbababababbbababbbababababNbabNbabababNbNbN从上式可以看出当'bb时中括号为0，'bb时δ函数为0，所以上式为零因为：0)()]'(),()[()()]'()'(),()()[()()]'()'(1),()(1)[()()]'()'(),()()[()()]()()'()'()'()'()()()[()()()()'()'()()()'()'()()()()]()'()'()(),()([22babNbNbababababababababababababababababababababababababababababababababababababababababababababababa上式中第四步计算用到了（31.6）式∴0)]()'()'()(),()([babababababa(2))}'()'()()()'()'(){'()}'()'()()'()()'()'()'({)}'()'()'()()()'()'()'({)]}(),'()['()()()'()](),'({[)]}()'(),'()[()()'()](),'({[)]()'()(),'([)]()'()'()(),'([)]()'()'()()(),'([)]()'()'()()()(),'([)]())'()'(1)((),'([)]())'()'()''()((),'([)]()'()'()(),'()'([babNbababNbabbbabNbabbbabNbabbbbbabNbababNbbbababNbNbababNbabNbabNbNbababNbabNbabNbabNbababababNbababababNbNbababababababNbababababNbabababbbabNbabababababa从上式可以看出：当'bb时括号为0，'bb时δ函数为0，所以上式为0∴0)]()'()'()(),'()'([babababababa□练习31.3讨论全同粒子的自旋态，设自旋为1/2的粒子的单粒子zS的本征矢量为||和，相应的本征值为2/2/和；aaaa,,和分别是态和态的产生和消灭算符。现定义以下4个算符：（解答：陈玉辉核对；项朋）AABCDaaCaaBaaaaA2)(2求它们的对易关系：],[],,[],,[],,[],,[],,[DCDBDACBCABA解：aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaBA2222)(2}],[],[],[],[],[],[],[],[{2}],[],[],[],[{2],2[],2[]),(2[],[aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaCA2222}{2}],[],[],[],[],[],[],[],[{2}],[],[],[],[{2],2[],2[]),(2[],[)(}],[],[],[],[{}],[],[{],[],[],[2222aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaCB0],[],[],[],[],[222aaaaaaaaCBACABBCAAABCADA）（aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaABABAAABCBBABABCBBAABBCBAABCBDB333333333333222222222222222)())((2)(2)()(],[],[],[],[],[],[],[],[],[],[],[aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaACACAAACCBCAABCCDC3333333333333222222222222)(2)(2)(],[],[],[],[],[],[31.4练习31.5取单粒子算符为zSB,其两个本征态为2,2.aaaa,,,为相应的产生算符和消灭算符.现有对称的多粒子自旋态,,,求下列算符作用到这些态上的结果:aaaaA21212121aaaaB式中及取21两值,aaaa2121,.（做题人：杜花伟）解:(1)当aaaaA2121时,作用到多粒子自旋态上的结果:aaaaaaaaaaaaA2121212121210aaaaAaaaaaaaA51445142151aaa251aaaaaaaA665162151aaa(2)当aaaaA21212121时,作用到多粒子自旋态上的结果:aaaaaaaaaaaaA21212121212121210aaaaAaaaaaaaA5145142151aaa4251aaaaaaaA65162151aaa6aaaaB21212121212121212121212121212121aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaBaaaaaaaa551551aaaaaa2051202151aaa20aaaaaaaaBaaaaaaaa451451aaaaaaaaaaaa515142151122151aaaa42151aa4514511251aaa4412aaaaaaaaB