2009级量子力学期中试题
一、计算下列对易关系
?x2????1、? 2、p,e??px???,r????
?0?二、若粒子的状态波函数为?(x)??Asinkx?Be????x?00?x?a x?a其中k和?是正常数。试求波函数的归一化常数，并给出在0?x?a的区间内发现粒子的几率。 三、设单粒子的归一化波函数?(x,y,z)、用?(x,y,z)表出下列概率： (1)、粒子坐标x介于x1与x2之间，即x1?x?x2。 (2)、x1?x?x2，同时粒子动量分量pz?0。
(3)、粒子动量分量px,py,pz同时分别为p1?px?p2、p3?py?p4和p5?pz?p6。 四、一个质量为m的粒子在无限深方势阱V(x)???? x?0,x?a中运动， 初始t?0时刻系统处于状态
?0 0?x?a?(x,t?0)?2??x??x，其中为常数。试求t时刻系统： 1?4cos??sin5a?a?a(1)、处于基态的几率， (2)、能量可能取值及相应几率， (3)、能量平均值， (4)、动量平均值、方均值。
五、在一维线性谐振子势中运动的粒子处于基态?0(x)和第一激发态?1(x)的叠加态?(x,t?0)上。其几率振幅分别为cos?和sin?，位相差为???0??1。试求下列各问题： 1、给出?(x,t)， 2、求t时刻的能量平均值H和能量方均值H；
23、求t时刻的位置平均值x和位置方均值x。
2?1?nn?1??n???n?1??n?1?提示：计算可能用到公式：x? ??22??
注：按顺序标明题号，将答直接写在答题纸上。
2
2009级量子力学期中试题答案及评分标准
de?x?x?x,e?一、(20分) [解] 1、??p???idx??i?e 5’
?x??,r???i2、利用?p???x??及?pr???x1??i得 3r?r???x?????2x??x??x???2???????????p,r?pp,r?p,rp??ip?p??ip,?p????????????????????rrrr????????x??2xx2x?1?1???1???x?????p???x????p?????i??3i?i?3???p??? 10’ ??i??prrrr?r????r?22?1?????1?r?p??r?i???rx???r??x??r?x2?x2?x2123??1?x3x???????????在球坐标系下?cos??x3r 3???2?xrsin?r???tan??xx??21????cos2????cos?x2???????2???1?tan???2?1????xxxrsin??1?1?????r?sin?cos????x1??r???sin?sin?、
?x?2??r?cos???x?3???cos?cos????sin???????xrrsin??1??x1???cos?sin????cos?和? ????xr?xrsin??2?2??????sin????0???xr?x?3?3??x12?x1??sin???x12?????2x??cos?cos????sin?cos?cos??cos?sin??1?????xr?rr??sin???r?r????????1?22?x2?x2??cos???x2???????2??x2???cos?sin????sin?cos?sin??sin?cos????
??sin????r?r????????x2r?rr?22??x3x3?x3????x3??sin???cos?sin???r?r?????x3r?rr?x?????2??p???ix????r??ir?r??p,r???2???x??r?x??r2??1???? 5’ ??rr?二、(20分) [解] 由归一化条件得
2
???22?2?x2??2?x?(x)dx?Asinkxdx?Bedx?Ax?sin2kx?e?????2?2k?02???0aa2a??11aB2??1?AB?2?a???A2a?sin2ka?e??12?2k??边界条件: Asinka?Be??a22 5’
5’
??A?2?a?1sin2ka?sin2ka????2k? ??B??e?asinka2???a?12ksin2ka?sin2ka?aa?1sin2kaa?1sin2kaW(0?x?a)???(x)2dx?2k?2k。 0a?1B21122ksin2ka??2?aA2?ea?2ksin2ka??sinka三、(20分) [解] (1)、粒子的位置分布几率为dW??(x,y,z)2dxdydz
x2????W(x1?x?x2)??dx?dy?dz?(x,y,z)2 5’
x1??????(2)、C(x,y,p)?1?ipzzz2?(x,y,z)edz
????粒子在位置x和y方向及动量z方向分布几率密度为
dW?C(x,y,p2z)dxdydpz???????1*(x,y,z?)eipzz?dz??2???????(x,y,z)e?ipzzdz??dxdydp?z ?????????1?dz?dz????*(x,y,z?)?(x,y,z)eipz(z??z)???2???dxdydpz??????????W??x1?x?x2,?1?z??z?x2???p??zz?0?2????dz?dz?dpeipz???0?dx?dy?*(x,y,z?)?(x,y,z) x1??(3)、C(p1??x,py,pz)?(2?)3?(r)e?ip?r2dr??? 2
5’
5’5’
动量空间粒子的几率分布为
dW?C(px,py,pz)dpxdpydpz???1??*?ip?r??ip?r)?????(r)edr?(r)edr?dpxdpydpz 5’ 3?????(2?)????1?????*ip?(r??r)??????drdr?(r)?(r)e?dpxdpydpz3?????(2?)?????2?p1?px?p2,???1W?p3?py?p4,??3(2?)?p?p?p?z6??5p2??????ip??r??r?*??dpdpdpdrdre?(r)?(r)? 5’ x?y?z????p1p3p5??????p4p6四、(20分) [解] 由题意知，能量本征值及对应的本征态分别为
n2?222n?x ?(x)?sin ?n?1,2, En?n2ma2aa初态按能量本征态展开，得：?(x,t?0)?故，任意t时刻体系的波函数为
?
2??x2?x?14sin?2sin????2 1??5a?aa?55i?t2i?t?1?iE1t4?iE2t2??2ma2?x2?x?ma2?e? 5’ ?(x,t)?e?1?e?2?sin?2esin?555a?aa??1(1)、P， 2’ ?15222?2214,; P(E)?,P(E)?(2)、E?E1,E2?； 3’ 122ma2ma2551?2242?2217?22???(3)、H?E?P(E1)E1?P(E2)E2?? 2’ 22252ma5ma10maa?22(4)、p?*?x?(x,t)dx ??(x,t)p02i?t2i?t?i?2t?d2??i?2t?2?x2?x?x2?x?22mama2mama2?e?2?e?dx??sin?2esinsin?2esin????5a0?aa?dx?aa?? 2222i?2ti?t2i?ta?i?t222i?x2?x????2ma2?x8??ma22?x?2ma2ma2?e???2e?dx??sin?2esinsin?2ecos????5a?aaaaaa0????2a2222
2
3i?t3i?t2i??2?x4?4?x4??2ma2?x2?x2?2ma22?x?x??sin?dx???sin?esincos?esincos??5a?2aaaaaaaaaa0??aaa3i?2t3i?2ta???2i?12?x4?x2?2ma2?3?x?x??2ma2?3?x?x??????cos???esin?sin?esin?sindx??cos?????5a?4a0a00?aaaaaa?????????? aa??3i?2t?3i?2t2i?2ma2?23?x?x?13?x?x?2??2ma??cos?2cos?e?cos?cos?e?????5a?aa?3aa???3?00??22?3?2t?32??sin? 5?2?15a2ma??d2?n??*利用公式：2?n(x)?????n(x)及??n(x)?n?(x)dx??nn?得
dx?a?0a2?xp???*(x,t)p?(x,t)dx022a????22?1iE1t*?4iE2t*?d2?1?iE1t4?iE2te?(x)?e?(x)e?(x)?e?(x)dx???12122???5???550??dx?5?a22iEt??1iE1t*4iE2t*?????1?iE1t?2??4?2e?(x)?e?(x)e?(x)?e?(x)????dx??1212????5?5?a?5?0?????a?5?a222?1???2?4?2??2?2??i(E1?E2)t*2???i(E2?E1)t*22?1(x)?2(x)???e?2(x)?1(x)?dx????1(x)????2(x)???e?5?a?5?a?5?a??5?a??0??a2222??17?21?42?????2?????????25a5a5a???????? 5?i?i?五、(20分) [解] 由题意，设?(x,t?0)?cos?e0?0(x)?sin?e1?1(x)，则 1、?(x,t)?cos?ei(?0??t2)?0(x)?sin?ei???3?t2??1(x) 5’
1cos2?3sin2?1??????2?cos2?? 2’ 2、H?E0P(E0)?E1P(E1)?222cos2?9sin2?1H?EP(E0)?EP(E1)??????5?4cos2???； 3’
44422024?1?nn?1??n???n?1??n?1?3、利用公式x?得 ??22?? 2
物理学相关 09级期中题
