1、33811hkThddc 2311cTcdd138hcc2hck1hkT231cTcd211hvkThvvckTT 2332111/221/c v Tcv dvcv dvcdTv dvec v Tc nE222(1)()2nsLnrvevrr向心力( )2(1)22222222222422*(2) :()()()(4)()(5)222(4)(5)2snnnssssnsneLnrvrrrnreeeepvErrreEn 又由 代入得:123,aaa22pE1111,22xxxnhhanpa,2233/2,/2yzpn hapn ha123,1,2,3n n n 222222223121231()(
2、)()()222222xyznnnphEpppaaa 222222222331212123123()()() ()()() 82nnnnnnhEaaaaaa()11( )ixtaf x e()22( )ixtbf x e1212,ff2222()()1212*()*()12122222*()*()1212()()221212212( )( )2,4( )( )*(2 )*( 2 )24( )( )4ixtixtixixixixixixaf ebf ea bf f eab f f eafxbfxa beab ef fabieefxfxf fiiifxfxf 代入1sin()fx2/20(1) (
3、2)00(3) ( )() 0 xxAxexAexxAx axxa,22222222222222()0 0200/21/41/412()211,xxyxyrrrxdxAedxA IIedxedyedxdyerdrdredeIAe 222201, ()xdxA x edx分 部 积 分22222220002222220002223303 / 23 / 22211221442,02,0,0 xxxxxxxxAAx edxx exedxAAxeedxedxAAexexAx 3222222300004425250002552( )()()()33222()3*43*43*4*5303030,aaaaa
4、aaxxdxA x a x dx Aa xx a x dxxxAaAa xdxxAAaa*0(1),(2),(3)010wJtEiEttiEiEttiwEEtttiiwJtJJ 定 态 :取 复 共 轭 : -定 态 几 率 密 度 分 布 不 随 时 间 变 化 , 即 :由 ( ),与 时 间 无 关 , 即为 与 t无 关 的 常 矢 量 。121211(1), ( 2 )ik rik reerr*111121()()2211sin()2()()2(2rrikrikrikrikrrikrikriJppeeerrreriJieeeeerrrrrrieerr 1111其 中 ,由 于=(r)
5、与 方 向 无 关 ,(r)2)()ikrikrikrikrrikeeeikeerrrr222121211111()()2,()rrrriikikkJeerrrrrrrJeJeJJ 平 行 于发 散 波 , 反 平 行 于, 收 敛 波取 上 号 , 取 下 号0( )0,0 xu xxaxa,uoa222( )(1)2du xEdx0,(0,)(2)xxa22( )( )0,(0)(3)Exxxa 2222222,0(4)cossin(5),:(0)(0)0,()sin(6)()()sin0,0,sin0,1, 2, 3,(7 )()sin, (0)(8):1sinEAxBxAxBxaaBa
6、BanannanxBxxaadxB 令则解 为由 波 函 数 的 连 续 性 要 求即归 一 化220021cos1222(9)aanxnaaxdxBdxBaBa 0 02( )sin 0,1, 2, (10)0 ( , )( ) niE txnxxxanaaxax tx e波 函 数其 中定 态 波 函 数2222222222222 (11)2,1, 2, (12)2( , )( ) (13)ni ntanEanEnax tx e 2由 (4)式 和 (7)式 :能 级1Aa 2222sin()121aandxAxa dxAaaAa00Eu00,( )0,uxau xxaoa-au02222
7、( )(1)22()( )0(1)du xEdxEux20,220 (2)2 ()0 (3)ExauExa 即22022,( ) (4)2 ()2,( ) (5)ikxikxxxxAeBexauEEkxaebexa 令,解为:(), 0 (6)(), 0 (7)( )( )()4 (8)sin ( )xxxxaaebxabeau xu xuxBABABAAkxx 根据波函数的有限性:时,0,有,否则又由于势能关于原点左右对称:波函数应具有确定的宇称即在( )式中,,可取,奇宇称波函数 9cos 10lnln cos,cot (12)sinsin,tan cosaaaaBkxxabeAkkaka
8、beAkakbeAkkakabeAkak( ),偶宇称波函数( )利用波函数连续性条件,在处:()=()(11)-对奇宇称解:-对偶宇称解: (13)220222( ),223 (14) (15)2xxaexaukkE 由于在处连续 不给出新的结果又由( )式和( )式,有而,knE010( )0 xuxau xuaxbxb,22221( ) (1)22 ()0 (2)0, (3)du xEdxE uEEuxbaou0-u122200222222112222 ()0 2 ()2 ()00(4)2 ()2 ()002 ()VEuuuEuEEuEukkE ,令,令224422 ()00VVVE ,
9、令442222344( )0 0 (5)( ) 0 (6)( )sin() (7)( ) xxVxxxA shxB chxxaxAkxaxbxA eB exb(由的有限性,连续性)解为 (8)22(0)(0)0,0,0(9)BA 中442234222220,(ln ) sin() (10) cos()cot() (11)sin()cos()sin(xVAx a x bAsh xAkxBech akkacth a kkash akakkbkb又根据波函数有限性要求:要求分别在和的边界上连续44cot() (12)kka22012222142242202422 () (13)42 (14)(11)
10、,(12),(13),(14),442(14) (15)uukukku又由( )式:式中, ,为待求量,个方程 个未知数,此四式为决定束缚态能级的方程。或将换为:00( )00Vxau xxxa,或E-V0oax221T=11sin4kkkakk(-)022EuEkk(),02EVk ()20R2sin0k a ,1,2,3k ann 222022nnEVa 2sin1k a 1() ,0,1,22k ann 222021()22nnEVa1112202nnnnnHHnHHnH,111111( )( )( ) (1)221( )( )( ) (2)22nnnnnnnnxxxxdnnxxxdxx
11、22222212112211( )()2!1( )()(2)2xnnnnnxxnnnnnnxN eHxNnxxN exHxN enHH,其中22221211121111111122!2!2(1)1222(1)!2(1)!11( )( )22xnnnnxnnnnnnenHHnnnneHHnnnnxx222211222( )()()xxnnnndxNxeHxeHxdx222222211112211211111111( )( )( )22()2()2!22()2( )2(1)!211( )( 2)( )( )2222nnnxxnnnnxnnnnnnnnnnxxxxN eHxenHxnnenHxnxn
12、dnnnnxnxxdx 222221()222yxppHxy22222222222222222222( )( )( )21()()22122122xyxyu rrErxyExyHHHHdHxdxdHydy 本题即:显然可以分成两项之和:,其中yHxHyH( ,)x y12( , )( )( )x yxyxH22112222222121221112222()212121( )(),()21( )(),()2( , )()()(1),0,1,2xnnynnxynnnnxN eHx EnaxN eHy Enaax yN N eHx HyEnnn n其中 为常数121,NfNNnn2222222222
13、22221111222211122221122221111( )()*2( )*44 228(1)010,0,01xxxxxxnxN eHxN exwxN exwNx exx exxN exxxwxwx 第一激发态,几率密度令或,由的表达式可见,当时,取极小值几率密度最大处20s ine reen lmJJemJr *()2iJ qe *()2ei eJ 11sinreeerrr ( , , )( )( , )( 1)(cos )( )nlmnllmmmimimlmlm llmnlrRr YYN PefeYRr 而氢原子波函数其中中 的函数为实函数, 的函数为复函数,而也为 的实函数*2*20
14、,011()()( , , )2sin2sin1sinrrereenlmnlmJJi ei eJimimrrrm er 即2zmeMM2zzMeL zyxordssinreJzedMdIJdsdI2222sinsinzeedMJrdsJrrdrd2220 0220 02220 001sinsin2sin2sin2()2zznlmnlmnlmnlmmeMdMrrdrdrmerdrdmerdrddme 利 用正 交 归 一2zzzMMeLm 2110211 113( , ,)( )( , )( )( , ),22rRr YRr Y 21021 113( ,)22r 4228seE 2222213*
15、22EEEE21*(1 1)221*(1 1)22222(1/ 43/ 4)*22LL平均值0*1*133*0*()444zzLL 平均值22221()1()d rRddRrrdrr drdr2222221( )1111( )2111()22d rRddrRR rRRR rRR rRr drrdrrdrrrddRrrR r RR rRr drdrrr得 证222( )xxe2212ux22pT*( )( )Fx Fx dx222222221/21/20210221/22212222222(21)!22(21)!11,2 ()21112224xxnaxnnxx edxx edxnx edxaan
16、axux由积分公式代入上式中:,有22pT222222222222222*2222222222222323222220( )( )()(1)2(1)(1)xxxxxxxdPx Px dxeiedxdxdeedxxedxdxxedxxedx 2012axedxa2210(21)!2naxnnnxedxaa3222222222222221!122212444PPT P2*22*22*2211112222021( )( )() ()11()222222nnnnnnnnnnnPx Px dxPPPdx abnnnnPPdxdxdx( ) x( )px( )( )( )pxc px dp22222222
17、22222222222*21/21/22()221/21/21/21/2()()22221/21/21/21/21( )( ) ( )(2)11(2)(2)11(2)(2)1(ixpxpiixxppxi pppi pxxc pxx dxeedxedxedxedxeedx222222221/21/221/21/212)/2()ppee222222222221/21/21/21/21/211( )1()ppppc peeee( )c p0( )p2( )c p2PT2222*22222222112212444ppppPcpc dpcp dpep dpPT0/3 1/201( , , )()r ar
18、ea 2ser*( )( )Fx Fx dx01222 /*3200 0 003 1/2400( , , )( , , )()sin43!3()(2/)2r arrrrdaer rdrd daaa 002222 /2300 002222 /332000001sin441(2 /)r assr assseeerdrd drareeeredraaaa 00002222 /222100300 02 /2 /2 /221010330004( )( , , )sin4442( )(2)0r anlnlmr ar ar aw r drrrdrdr drer dradww rerrereadraa 几率密度
19、,令02 /002 (1)0,0,r arrerrraa即有或10( )wr0,r 10( )wr00000000/2*2210010030222/232322000022222 /2 /2322320000001()112()()1242()sin()r ar ar ar ar ar ar ar aPPdeiedarreredeedarrraraarrrrerdrd dedraraaaaa 2232232000000421!12!(2/)(2/)aaaaaa222022PTa0000100*1003320cos23320cos2320 02023020( , ,)()11(2)1sin(2)
20、2(cos)(2)2()(2)2ppprip rappirprairprariiprprarcp dcdeedaerdrd daer drdaeeer dripra 0011()()3020()(2)iiprpraaeerdripa 223002034032222200211()()(2)8(2) ()ipipaaipaaaap22683522002322242224000( )464324(2) ()()pw p dpcp dpaapp dpdpaa pa p( )()xAx ax2( )1xdx222222340055()(2)13030,( )()aaxaxA dxAa xaxxdxAx
21、x axaa1( )( )Ennxcx2( )sinnn xxaa*5002302230000223300230( ) ( )sin()2 15sin()2 15coscoscos2 cos4 154 15()sin() sin(aEnaaaaaaan xcxx dxx a x dxaaan xax x dxaaan xan xan xan xaxdx xxdxananananaan xan xaxdxananaan33304 15) cos1 ( 1) ann xan 330 2 (0,1, 2,)8 15 21Enkcknkn2660 264*15 21Enkcnkn22*52022550
22、0223255030()() ()23030()()( 2)()2305()23aaaadEHdxx axx ax dxadxx axdxx ax dxaaaxxaa22222662212122422442222196024801480596EnEnnnknknknEcEcEnaanaa22LHI22zzzLLL HLiI,22222( )02dIEEI d ,即2( )2/iiAeBeIE 解为,(0)(2):0,1,2,nmm ,212( )2imIEem ,2zL1( )0, 1,2imem ,2,2zLHHI( )2220, 1, 2,22znLmEmII,, ( , )LL 2L(
23、, )lmY 22LHIH( , )lmY 2(1)0,1,2,2nl lElI,21( )sincos2xAkxkx22222011( )sincos()()22222( )4ikxikxikxikxikxikxikxikxnpneeeexAkxkxAiAeeeecx1/21/211( )(2)(2)pixikxppxeek,k0, 2kkk ( ) x21/ 221/ 21/ 21/ 2022111( )(2)(2)(2)(2)244inkxnnxceAAAccccc ,2221nnc 22222(2*2*)141616AAAA1/22222222222222222222()111112
24、*0*()* 2*(2 )041616161611152* *0*()*(44)416164nnnnnnnnApkpcpAkkkkpcpkkkkk,又=222528pkT*(1)( )( )mnmnxx xx dx,mn*(2)( )( )mnmxnpx px dx111111( )( )( )221( )( )( )22nnnnnnnnxxxxdnnxxxdx*11,1,111( )( )( )221122mnmnnm nm nnnxxxxdxnn*11,1,11( )()( )( )22122mnmnnm nm nnnpxixx dxnni,mnmnxpxxp, , , xxyylyi z
25、lzi ylxi zlzi x , , , , , , ( , , )00 *()0 , , , , , , ( , , )*()0 00 xzyzyzzyyxzyzyzzyylyy pz pyy pyz pyy pyy y pz pyz y pzii zlzy pz pzy p zz pzy p zy z pz pzz z pyii y , , , , , , ( , , )*()0 00 , , , , , , ( , , )00 *()0yxzxzxxzzyxzxzxxzzlxz px p xz pxx p xz pxz x px p xx x pzii zlzz px p zz pzx p zz pzz z px p zx z pxii x 2222222222222222200()2sssseepHrrepEprprrlppppppperrrrprp rpErrr ,又,取对氢原子基态:,球对称,而,222032224442min14221022ssssseEraErrreeeeEE 令,则有,代入 中,得