量子力学讲义田光善QM10第十章力学量本征值的代数解法.pdf

?(?)?$10.1?Schr odinger?i ht(x)=h22md2(x)dx2+12m20 x2(x)(1)?En=?n+12?h0,n=0,1,2,(2)?n(x)=s2nn!exp?122x2?Hn(x).(3)?=qm0 h?Hn()?n?Hermite?H=p22m+12m20 x2=h0 p22m h0+m02 h x2!=h0 p2m h0+irm02 h x!p2m h0 irm02 h x!+12#.(4)?a p2m h0+irm02 h x,a p2m h0 irm02 h x,(5)?H=h0?a a+12?.(6)1?h a,ai=1.(7)?a a?|ni?a a?a a|ni=f(n)|ni.(8)?a?|ei a|ni.(9)?a a|ei=?a a?a|ni=a?a a?|ni+h a a,ai|ni=f(n)a|ni+h a a,ai|ni.(10)?h a a,ai=h a,ai a+a a,a=a.(11)?a a|ei=(f(n)1)a|ni=(f(n)1)|ei.(12)?|ei?a a?f(n)1?|e(m)i (a)m|ni=mz|a a a|ni(13)?a a?f(n)m g(n,m)?a a?h|a a|i=?,a a?=(a,a)0.(14)2?a a?f(n)m g(n,m)?m0?f(n)m0=0?a a?a a|0i=0(15)?0?h0|?h0|a a|0i=?0,a a0?=(a0,a0)=0.(16)?|0i?a|0i=0.(17)?|0i?a a?|emi=?a?m|0i.(18)?a a|emi=?a a?a?m|0i=?a?m?a a?|0i+h a a,?a?mi|0i=h a a,?a?mi|0i=ah a,?a?mi|0i=ah a,ai?a?m1|0i+?a?2h a,ai?a?m2|0i+?a?kh a,ai?a?mk|0i+?a?mh a,ai|0i=m?a?m|0i=m|emi.(19)?em?a a?m?hem|emi=D0?(a)m?a?m?0E=?0?(a)m1 a a?a?m1?0?=hem1|a a|em1i=hem1|a a|em1i+hem1|h a,ai|em1i=(m 1)hem1|em1i+hem1|em1i3=mhem1|em1i=m(m 1)hem2|em2i=m(m 1)2 1h0|0i=m!h0|0i.(20)?h0|0i=1?hem|emi=m!.(21)?|mi 1m!|emi=1m!?a?m|0i,(22)?|mi?n 6=m?|ni?|mi?hn|mi=nm.(23)?H?(22)?|mi?H|mi=?a a+12?h0|mi=?m+12?h0|mi.(24)?Em=(m+1/2)h0?|mi?a?a?a|ni=a 1n!?a?n|0i!=1n!?a?n+1|0i=n+1q(n+1)!?a?n+1|0i=n+1|n+1i(25)?a|ni=a 1n!?a?n|0i!=1n!a?a?n|0i=1n!?a?n a|0i+1n!h a,?a?ni|0i=1n!n?a?n1|0i=nq(n 1)!?a?n1|0i=n|n 1i.(26)4?|ni?Dirac?n(x)=hx|ni=*x?1n!?a?n?0+=Zdx*x?1n!?a?n?x+hx|0i(27)?hx|0i?0?a|0i=0.(28)?0=hx|a|0i=Zdex hx|a|exihex|0i=Zdex*x?p2m h0 irm02 h x?ex+0(ex)=Zdex*x?p2m h0?ex+0(ex)irm02 hZdexex(xex)0(ex)=12m h0 hixZdex(xex)0(ex)irm02 hx0(x)=12m h0 hix0(x)irm02 hx0(x).(29)?x0(x)+m0 hx0(x)=0.(30)?x=0?hx|0i=0(x)=exp?m02 hx2?.(31)?(27)?n(x)=Zdx*x?1n!?a?n?x+hx|0i=1n!Zdx*x?p2m h0+irm02 h x!n?x+0(x)=1n!hi2m h0 x+irm02 hx!nZdx(x x)0(x)=(i)nn!s h2m0 xrm02 hxnexp?m02 hx2?.(32)5?n(x)?$10.2?J2?Jz?hJx,Jyi=i hJz,hJy,Jzi=i hJx,hJz,Jxi=i hJy.(33)?Casimir?J2J2x+J2y+J2z.(34)?J+Jx+iJy,JJx iJy.(35)?hJ2,Jxi=hJ2,Jyi=hJ2,Jzi=0,(36)?hJz,Ji=hJ.(37)?hJz,J+i=hJz,Jx+iJyi=hJz,Jxi+ihJz,Jyi=i hJy+i(i hJx)=i hJy+hJx=hJ+.(38)?(37)?(36)?Jz?hJ2,Jzi=hJ2x+J2y+J2z,Jzi=hJ2x+J2y,Jzi6=?12(J+J+JJ+),Jz?=12hJ+J,Jzi+12hJJ+,Jzi=12hJ+,JziJ+12J+hJ,Jzi+12hJ,JziJ+12JhJ+,Jzi=12 hJ+J+12 hJ+J+12 hJJ+12 hJJ+=0.(39)?J2?Jz?Jx?Jy?hJ2,Jxi=hJ2,Jyi=0(40)?J2?Jz?|,mi?J2|,mi=h2|,mi,Jz|,mi=m h|,mi(41)?m?hJ2,J+i=0?D,m?hJ2,J+i?,mE=h,m|J2J+J+J2|,mi=0.(42)?h,m|J2J+J+J2|,mi=()h2h,m|J+|,mi.(43)?6=?h,m|J+|,mi=0.(44)?h,m|J+|,mi=,h,m|J+|,mi(45)?7?h,m|hJ+|,mi=D,m?hJz,J+i?,mE=h,m|JzJ+J+Jz|,mi=(m m)hh,m|J+|,mi,(46)?(m m 1)h,m|J+|,mi=0.(47)?m6=m+1?h,m|J+|,mi=0(48)?h,m|J+|mi=m,m+1h,m+1|J+|,mi.(49)?hJz,Ji=hJ?h,m|J|mi=m,m1h,m 1|J|,mi.(50)?h,m+1|J+|,mi?h,m1|J|,mi?12(J+J+JJ+)=J2x+J2y=J2J2z.(51)?12h,m|J+J+JJ+|,mi=12XXmh,m|J+|,mih,m|J|,mi+12XXmh,m|J|,mih,m|J+|,mi=12h,m|J+|,m 1ih,m 1|J|,mi+12h,m|J|,m+1ih,m+1|J+|,mi=h2 m2 h2=(m2)h2.(52)8?J+=?J?h,m 1|J|,mi=h,m|J+|,m 1i(53)?h,m|J|,m+1i=h,m+1|J+|,mi.(54)?12?h,m|J+|,m 1i?2+12?h,m+1|J+|,mi?2=(m2)h2.(55)?12(J+JJJ+)=hJz.(56)?12?h,m|J+|,m 1i?212?h,m+1|J+|,mi?2=m h2.(57)?(55)?(57)?h,m|J+|,m 1i?2=(m2+m)h2.(58)?m2 m=m(m 1).(59)?m?m?m?h,m+1|J+|,mi?2=0(60)?(m+1)2+(m+1)=m(m+1)=0.(61)?=m(m+1).(62)9?h,m 1|J|,mi?2=?h,m|J+|,m 1i?2=(m2+m)h2=(m(m+1)m2+m)h2=0.(63)?m=m,?m=m+1.(64)?m?m?m?1?m m=?.(65)?m=m?m m=2m=?.(66)?j=m?j?j=0,12,1,32,2,(67)?J2|,mi=h2|,mi=m(m+1)h2|,mi=j(j+1)h2|,mi.(68)?m?j=m=m m m=j.(69)?|j,mi?|,mi?(58)?hj,m|J+|j,m 1i?2=j(j+1)m(m 1)h2.(70)?hj,m|J+|j,m 1i=eiqj(j+1)m(m 1)h,(71)10?hj,m+1|J+|j,mi=eiqj(j+1)m(m+1)h.(72)?|j,mi?|j,m+1i?ei=1?hj,m+1|J+|j,mi=qj(j+1)m(m+1)h.(73)?hj,m|J|j,m+1i=qj(j+1)m(m 1)h.(74)$10.3?Clebsch-Gordan?J1=(J1x,J1y,J1z)?J2=(J2x,J2y,J2z)?hJ1,J2i 0.(75)?J=J1+J2.(76)?hJ2,J21i=hJ2,J22i=0.(77)?hJ2,J21i=h(J1+J2)2,J21i=hJ21+J22+2J1J2,J21i=2hJ1J2,J21i=2hJ1xJ2x+J1yJ2y+J1zJ2z,J21i=2hJ1x,J21iJ2x+2hJ1y,J21iJ2y+2hJ1z,J21iJ2z=0.(78)11?hJx,J2i=hJy,J2i=hJz,J2i=0,(79)?J2,Jz,J21?J22?|j1,j2,j,mi?J21|j1,j2,j,mi=j1(j1+1)h2|j1,j2,j,mi,J22|j1,j2,j,mi=j2(j2+1)h2|j1,j2,j,mi,J2|j1,j2,j,mi=j(j+1)h2|j1,j2,j,mi,Jz|j1,j2,j,mi=m h|j1,j2,j,mi.(80)?J1?J2?|j1,m1i|j2,m2i(81)?j1?j2?|j1,j2,j,mi?|j,mi?|j,mi=j1Xm1=j1j2Xm2=j2Cjmj1m1,j2m2|j1,m1i|j2,m2i.(82)?Cjmj1m1,j2m2?Clebsch-Gordan?Cjmj1m1,j2m2=hj1,m1;j2,m2|j,mi(83)?Jz|j,mi=m h|j,mi=j1Xm1=j1j2Xm2=j2m h hj1,m1;j2,m2|j,mi|j1,m1i|j2,m2i=?J1z+J2z?|j,mi=j1Xm1=j1j2Xm2=j2(m1+m2)h hj1,m1;j2,m2|j,mi|j1,m1i|j2,m2i.(84)12?(m m1 m2)hj1,m1;j2,m2|j,mi=0.(85)?hj1,m1;j2,m2|j,mi=m1+m2,mhj1,m1;j2,m m1|j,mi.(86)?|j,mi?|j,mi=j1Xm1=j1hj1,m1;j2,m m1|j,mi|j1,m1i|j2,m m1i.(87)?j?|j1,m1i|j2,m2i?|j,mi?N1=(2j1+1)(2j2+1).(88)?|j,mi?m=m1+m2?m?mmax=m1,max+m2,max=j1+j2.(89)?10.2?mmax?j2?jmax?2jmax+1?|jmax,jmaxi,|jmax,(jmax 1)i,|jmax,(jmax 1)i,|jmax,jmaxi(90)?m=mmax 1?|j1,j11i|j2,j2i?|j1,j1i|j2,j21i?(87)?10.2?|jmax,jmax 1i?|j=j1+j2 1,j1+j2 1i?2jmax+113?j=j1+j2 1?J?2(j1+j2 1)+1=2j1+2j2 1?|j1+j2 1,j1+j2 1i,|j1+j2 1,j1+j2 2i,|j1+j2 1,(j1+j2 2)i,|j1+j2 1,(j1+j2 1)i.(91)?j=j1+j2 1?j=j1+j2k?2(j1+j2 k)+1?j?jmin?j=j1+j2?jmin?j?2j+1?j?|j,mi?N2=jmaxXj=jmin(2j+1)(92)?jmax=j1+j2?N2=j1+j2Xj=jmin(2j+1)=2j1+j2Xj=jminj+(j1+j2 jmin+1)=2(j1+j2 jmin+1)j1+j2+jmin2+(j1+j2 jmin+1)=(j1+j2 jmin+1)(j1+j2+jmin+1).(93)?|j1,m1i|j2,m2i?|j,mi?N1=N2?(j1+j2+1)2 j2min=(2j1+1)(2j2+1).(94)?j2min=(j1+j2+1)2(2j1+1)(2j2+1)=j21+j22 2j1j2=(j1 j2)2,(95)14?jmin=|j1 j2|.(96)?j?j=|j1 j2|,|j1 j2|+1,j1+j2.(97)15