量子力学基础知识习题解答

101.1.,hEhpνλ==2.,,,xyzxphyphzphtEh∆∆≥∆∆≥∆∆≥∆∆≥3.1.1(,,,)xyztψ(,,)xyzψ*ψψψψ*ψψ*dψψτdτM.Bornψψψψ⋅=*22.2211221122ˆˆˆ()AcccAcAψψψψ+=+**2121ˆˆ()d()dAAψψτψψτ=∫∫123.3AˆAψaψˆAaψψ=ψAaaˆAψˆA4.412,,,nψψψψ1122nniiiccccψψψψψ=+++=∑A**ˆddAAψψτψψτ=∫∫5.5PauliSchrödinger1.Schrödinger222d2dEmxψψ−==2()sin()nnxxllπψ=2228nnhEml=1234531.4a10-10kg0.01·b0.1eVc300eV(1)34221016.62610Js6.62610m10kg0.01mshhpmvλ−−−−×⋅====××⋅()3412719-11(2)26.62610Js21.67510kg0.1eV1.60210JeV9.40310mhhpmTλ−−−−==×⋅=×××××⋅=×34311911(3)26.62610Js29.10910kg1.60210C300V7.0810mhhpmeVλ−−−−==×⋅=×××××=×1.6vc1234512hhEmvPmvvvνλ=====12mvmv=,/Ehphνλ==Planckpmv=λ/uλν=uv/uλν=Ehν=EE41.1222ddx3,sin,2cos,,sincosxexxxxx+22deedxxx=ex22ddx122dsin1sindxxx=−×,sinx22ddx−122d(2cos)(2cos)dxxx=−,2cosx22ddx−1232d6dxxx=3x22ddx22d(sincos)(sincos)dxxxxx+=−+(sincos)xx+22ddx−11.152()sinnnxxllπψ=1,2,3,n=lx(0)xl12222222222222222d2d2ˆ()(sin)(cos)8d8d2(sin)82sin8()8nnhnxhnnxHxmxllmxlllhnnnxmllllhnnxmlllnhxmlπππψπππππππππψ=−=−=−−==2228nhEml=2ˆ()()nnxxcxψψ≠,ˆx5**002000022ˆ()()dsinsind22112sindcosd2222dcosdllnnllllnxnxxxxxxxxllllnxnxxxxxllllnxxxxxllππψψπππ⎛⎞==⎜⎟⎜⎟⎝⎠⎛⎞==−⎜⎟⎝⎠⎡⎤=−⎢⎥⎣⎦∫∫∫∫∫∫00200022ddsin2222sinsind222llllllnxxxxlnlxlnxnxxxlnlllπππππ⎡⎤=−⎢⎥⎣⎦⎡⎤⎛⎞=−+⎢⎥⎜⎟⎝⎠⎢⎥⎣⎦=∫∫∫3ˆ()()nnpxcxψψ≠,ˆppx:*0020ˆ()()d2d2sinsind2dsincosd0lxnxnllpxpxxnxihnxxllxllinhnxnxxlllψψπππππ=⎛⎞=−⎜⎟⎝⎠=−=∫∫∫1.1822CHCHCHCHCHCHCHCH460nm4π8π4πn=4n=554EEE∆=−460nm22222222(1)(21)888hcnhnhhEhnmlmlmlνλ+∆===−=+1/23493181(21)8(241)6.62610Js46010m89.10910kg2.98810ms1120pmnhlmcλ−−−−+⎡⎤=⎢⎥⎣⎦⎡⎤×+××⋅××=⎢⎥××××⋅⎣⎦=1.19abc==5[h2/(8ma2)6]ma2222,,2()8xyznnnxyzhEnnnma=++h2/(8ma2)222,,xyznnnxyzEnnn=++1113E=1121212116EEE===1222122219EEE===11313131111EEE===22212E=1.20π1.3nml=222/8nEnhml=π510.0nmH3CNCCCCCCCNCH3CH3HHHHHHHCH310π5ππ5622222652226511888hchhhEEEmlmlmlλ∆==−=−=()22318193481189.109510kg2.997910ms1.310m116.626210Js506nmmclhλ−−−−=××××⋅××=××⋅=510.0nm-0.67%1.2122228nnhEmRπ=0,1,2,n=±±nR66ππ3691112E222E113=E131=E311E122=E212=E221E112=E121=E211E1117R=140pmnn=0|n|16πn=011−31.20014E∆↑↓↑↓↑↓1.2066π()222221222221388hhhcEEEmRmRππλ−∆=−===()()()()2223110813498389.1110kg1.4010m2.99810ms36.62610Js21310m=212nmmRchπλπ−−−−−=××××××⋅=××⋅=×Γα3184.0nm208.0nm263.0nm3π1.22222()2sin3inxxxaaaaππψ⎛⎞⎛⎞=−⎜⎟⎜⎟⎝⎠⎝⎠a12()sinxxaaπψ⎛⎞=⎜⎟⎝⎠222()sinxxaaπψ⎛⎞=⎜⎟⎝⎠()()()1223HxHxxψψψ∧∧=−⎡⎤⎣⎦()()1223HxHxψψ∧∧=−8()()22122242388hhxxmamaψψ=×−×≠()xψ×()xψˆH()xψ'()()xcxψψ=[]2222000221302'()d()d()d23d131aaaaxxcxxcxxcxcψψψψψ===−==∫∫∫∫113c=12123'()()131313xxψψψψ==−()xψ01212012112201222222222ˆ'()'()d2323ˆˆd131313132323dd131313134913134192138138513aannnaijijExHxxHxHEEExEEhhmamahmaψψψψψψψψψψψψψψτδ=⎡⎤⎡⎤=−−=⎢⎥⎢⎥⎣⎦⎣⎦⎡⎤⎡⎤=−−=⎢⎥⎢⎥⎣⎦⎣⎦=+=+=∫∫∫∫1()xψ2()xψ9()()()()()()()()*0*012120121201211220121201222ˆddˆ2323dˆ2323d2323dd2323d4949513aaannnaaijijaHxExHxHExEExxEEhmaψψψψψψψψψψψψψψψψψψψψτδψψψψ=−−==−−−−==−−+=+=∫∫∫∫∫∫∫1.1λ=670.8nmLi(1s)2(2p)1(1s)2(2s)11kJmol−⋅8114172.99810ms4.46910s670.810mcνλ−−−×⋅===××417111.49110cm670.810cmνλ−−===××34114123116.62610Js4.46910s6.0310mol174.4kJmolAEhNν−−−−−==×⋅××××=⋅1.2λ/nm312.5365.0404.7546.1Ek/10-19J3.412.561.950.75-Plank(h)(W)(0ν)ν,kEλ/nm312.5365.0404.7546.1v/10141s−9.598.217.415.49Ek/10-19J3.412.561.950.751.2104567891001234Ek/10-19Jν/1014g-11.2~kEν0khhEνν=+kk0EEhννν∆==−∆PlanckkEv−kEvh19341141(2.71.05)10J6.6010Js(8.506.00)10sh−−−−×==×⋅−×v0v14104.3610sv−=×3411411906.6010Js4.3610s2.8810JWhv−−−==×⋅××=×1.35.46300nm2012hhmvνν=+08134141931512()2.99810ms26.2610Js(5.46410s)30010m9.10910kg8.1210mshvmνν−−−−−−−=×⋅××⋅−××=×=×⋅1.5200kVdeBroglie1134311951226.62610Js29.10910kg1.60210C210V2.74210mhhhpmvmeVλ−−−===×⋅=××××××=×1.70.01kg1000·10-9kg10·10-13kg1·s-11000·s-110%343416.62610Js6.6310m0.01kg100010%mshxmv−−−×⋅∆===×∆××⋅3425916.62610Js6.6310m10kg1010%mshxmv−−−−×⋅∆===×∆××⋅34201316.62610Js6.6310m10kg110%mshxmv−−−−×⋅∆===×∆××⋅3463116.62610Js7.2710m9.1.0910kg100010%mshxmv−−−−×⋅∆===×∆××⋅1.81000Vv∆v10%3431193102/10%102106.62610Js29.10910kg1.60210C10V3.8810mhhxmvmeVmhmeV−−−−∆==∆×=××⋅=×××××=×1.9610m−10000V12991111.22610m2/2/21=1.22610m10000=1.22610mhhhhhxpmvmTmmeVmeVmV−−−∆======×∆××510−510−610m−3428166.62610Js6.62610Jsm10mxhpx−−−−×⋅∆===×⋅⋅∆104V311941229.10910kg1.60210C10V5.402JsmxxpmvmeV−−−===×××××=⋅⋅xp∆xp5oarcsinarcsin100xxppθ−∆==≈1.101minv∆(a)2Hb2HHcCd2He5m2O1ma119.110kg−×b271.710kg−×cH271.710kg−×d2H273.410kg−×e2O265.310kg−×x∆2H10310m−×C15110m−×9110m−×5m5mxxph∆∆=minxpmv∆=∆minv∆()min/vhmx∆=⋅∆minv∆13(e)(d)(b)(a)(c)(2)(a)2H()()()3461min11106.610Js/2.410ms9.110kg310mvhmx−−−−×⋅∆=⋅∆==×⋅××e5m2O()()()3491min266.610Js/2.510ms5.310kg5mvhmx−−−−×⋅∆=⋅∆==×⋅×1.112axxeψ−=2222d4daxx⎛⎞−⎜⎟⎝⎠III222222222222222222222222232323dd44ddd()4()dd(2)4d244466axaxaxaxaxaxaxaxaxaxaxaxaxxexxxeaxxexeaxeaxexaxeaxeaxeaxeaxeaψψ−−−−−−−−−−−⎛⎞⎛⎞−=−⎜⎟⎜⎟⎝⎠⎝⎠=−=−−=−−+−=−=−6a−1.13imeφcosmφddiφddimimieimeφφφ=imeφddiφimdcossincosdimimmcmφφφφ=−≠cosmφddiφ1.14l2()sinnnxxllπψ=01xn=123n'n14'0000022'()()dsinsind2'11sinsindsinsincos()cos()2221''coscos)d21'1'coscos0'''llnnlllnxnxxxxxllllnxnxxlllnnnnxxxllllnnlnnxxnnlnnllnnlππψψππαβαβαβππππππ===−++−+−=++−⎧⎫=−+=≠⎨⎬+−⎩⎭∫∫∫∫()nxψ'()nxψ1.161ψ2ψ0.49~0.51ll1.3.2ba12()sinxxllπψ=2212()sinxxllπ