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45 9( 0?BEZ w P[C:/ 1 |fu1jz‘g|y _ -$y -vf V w -;# E- $1.07 #!1PNY (1,2,3,···,N)y8*# \ _r# Y9 EhˆP=1P(1)2P(2)3P(3)······NP(N)!.(1)Iv P(k)rPNYkY94‘xyY x: ?Y9ˆP=1421354253(2)Z w 1P(1)=4,P(2)=1,P(3)=5,P(4)=2.P(5)=3 41TY9ybYz4N Y1T y Y AN! ~ yY9TxdM#e ˆP=1122······kj······jk······NN!.(3)I[Y9_r 9 Eh(k,j) dMRtj=k+1I 9(k,k+1)_r# 9= 9 } Y9ˆP16ˆP2ya? +rˆP1ˆP2=1P1(1)2P1(2)······NP1(N)!1P2(1)2P2(2)······NP2(N)!=1P3(1)2P3(2)······NP3(N)!≡ˆP3.(4)a!R# Y9 x:14253241531321354254=1224334551.(5)1 Y9a?y +R4 w l’P 8/# Y9 l’Æ‘ 9ya? x:13223441=(3,4)(2,3)(3,4)(1,2)(3,4)(2,3)(3,4)(2,3)(1,2)(2,3).(6)1a_r# Y9y 9a? Y # I# YRRq#y sR 8# Yy# R2 y ,i #Y9l’ Y‘#Y 9ya? w _r#Y9 A_r Y9 # dMUdyPKR +# Y9ˆPy# r(−1)ˆP AP} 8*} Y 94‘xyY9y# r(−1)ˆP+1 f4 w # Y yvf P;Y 1,2,3,···,NG-y+ vm oZ N,}# oZ1n1 + } oZ1n2 + }m oZ1nm +(a4 w ,n1+n2+···+nm=N) v#1 B[Rjy nRK=N!n1!n2!······nm!.(7)$1.1Æ 8*# ~‘{ |k ?d yFhO~ bSchr¨odinger by1TOR)#BMY d # n,Yd dMR? 0 z‘RLy 5h/R4 qg bSchr¨odingeryBM,Yd{ψn(x)} l’/s_\| Æz‘RL 5h/U=Iy z‘b r :%w ?Bk~‘{ obZ vy1’qz‘byQL 5w si} x‘ 1.1:j # O 1‘ DF\rL t# z‘?1 = I bSchr¨odinger br−¯h22m∂2∂x2+∂2∂y2+∂2∂z2!ϕ(r)=Eϕ(r).(8)I byiYrϕ(r)=Ckexp(ik·r).(9)r 2 k)#6[YCk w IStyFhO [/yFhOr‘u℄! FhO A,ϕ(r)?}} y1E. jyT x:?x y} 1E w ,ϕL2,y,z=ϕ−L2,y,z.(10)20iw xvexpikxL2+ikyy+ikzz=exp−ikxL2+ikyy+ikzz,(11)=Rexp(ikxL)=1.(12)Ia,kx=2πLnx,nx=0,±1,±2,······(13)ju w xvkykzl.y Tr ky=2πLny,kz=2πLnz,ny,nz=0,±1,±2,······(14),i I g|qz‘by1g0~N dϕ(nx,ny,nz)(x,y,z)=Cnx,ny,nzexpi2πL(nxx+nyy+nzz).(15)f4 w 0~Nf )#6[YCnx,ny,nzZL/2−L/2ZL/2−L/2ZL/2−L/2dxdydz|ϕ(n1,n2,n3)(x,y,z)|2=ZL/2−L/2ZL/2−L/2ZL/2−L/2dxdydz|Cnx,ny,nz|2=|Cnx,ny,nz|2L3=1.(16)0iw YxCnx,ny,nz=1√L3=1√V t(n1,n2,n3)6=(en1,en2,en3)I w 1ZL/2−L/2ZL/2−L/2ZL/2−L/2dxdydzϕ(n1,n2,n3)(x,y,z)∗ϕ(en1,en2,en3)(x,y,z)=1VZL/2−L/2ZL/2−L/2ZL/2−L/2dxdydzexp(i(n1−en1)x+i(n2−en2)y+i(n3−en3)z)=0.(17),i #eOR)#yqz‘b0~N dϕ(n1,n2,n3)(x,y,z)=1√Vexpi2πL(n1x+n2y+n3z).(18)3Iv ~‘Yn1,n2,n3=0,±1,±2,······ tz‘jI1a a0 I w lPg|yqz‘b Æreϕ(n1,n2,n3,↑)(x,y,z,↑)=1√Vexpi2πL(n1x+n2y+n3z)10!eϕ(n1,n2,n3,↓)(x,y,z,↓)=1√Vexpi2πL(n1x+n2y+n3z)01!.(19)~ w /E-k=(n1,n2,n3,σ)sMEBMby~‘Y /qsKEa0 (x,y,z,s,sz) tw q? I Rdq.uYr r,? Ta0 ,. w/Kronecker - w lPqz‘byOR)#hOÆ‘Zdqϕ∗k1(q)ϕk2(q)=δk1,k2.(20) 1.2:?#s K‘QV(x)=12mω2x2Z= yz‘y1T bBM,YlAÆh ψn(x)=mωπ¯h1/412n/2√n!e−mωx2/2¯hHnxrmω¯h,n=0,1,2,···(21)ZHn(x)r N a RSchr¨odinger b−¯h22md2dx2ψn(x)+12mω2x2ψn(x)=Enψn(x)(22)?x*;fr yOR)#Y ;G~‘{ yB9u Schr¨odinger b(22)y8/# Y l’;GI ,YhBh Aϕ(x,t)=Xnanψn(x)exp−iEn¯ht.(23)f4 w [C# ~ [/vy3zNXkϕk(q′)ϕ∗k(q)≡Xkhq′|ϕkihϕk|qi=δ(r′−r)δs′s≡δ(q′−q).(24)6 ;GDirac,Yy + 48/# qz‘Q,Yϕ(q,t) 3zNZdqϕ(q,t)δ(q′−q)=ϕ(q′,t)(25)4‘y 304ϕ(q,t)=Xkakϕk(q)exp−i¯hEkt,(26)w 1ZdqXk1ϕk1(q′)ϕ∗k1(q)Xk2ak2ϕk2(q)exp−i¯hEk2t=Xk1Xk2ϕk1(q′)ak2exp−i¯hEk2tZdqϕ∗k1(q)ϕk2(q)=Xk1Xk2ϕk1(q′)ak2exp−i¯hEk2tδk1,k2=Xk1ak1ϕk1(q′)exp−i¯hEk1t=ϕ(q′,t).(27)I w a P 3zN(24) $1.2$, 8 ’ ? w j g| 1N 1jz‘Iy)r 2a4 jqz‘Iy)r# N z‘y#Q,Y. l’;G#eOR)#,YBh Aψ(q1,q2,···,qN)=XlClψl(q1,q2,···,qN).(28)Iv {ψl}y . . ’~C hO(1)OR)# (2)g g|R0N qz‘e‘y (3) 1jz‘g|. ykD( 4} hO w g#~[ ~‘{ yB9u 1jz‘g|R0O?‘e‘y AQ,Y. ψ(q1,···,qi,···,qj,···,qN)=ψ(q1,···,qj,···,qi,···,qN).(29) 4 ‘g| A,ψ(q1,···,qi,···,qj,···,qN)=−ψ(q1,···,qj,···,qi,···,qN).(30)5,i w .t,(1) 4O?‘1ψl(q1,···,qi,···,qj,···,qN)=ψl(q1,···,qj,···,qi,···,qN);(31)(2) 4 ‘1ψl(q1,···,qi,···,qj,···,qN)=−ψl(q1,···,qj,···,qi,···,qN)(32)‘y ?l’Pz‘RLy 5h/hrp6fuy)r~ # gg|yBMQ,Yψl(q1,···,qi,···,qj,···,qN)l’;Gqz‘Q,Yya?gBh F(1) 4O?‘ w l’ +ψl(q1,···,qi,···,qj,···,qN)=ψk1,······,kN(q1,···,qi,···,qj,···,qN)=DXˆPˆP(ϕk1(q1)······ϕkN(qN))=fDX{ˆP}ϕk1qP(1)······ϕkNqP(N).(33)Iv E-ΣˆP.Σ{ˆP} MpL 41TyY9. dRjy yY9ty,. .| D.fD Mr1TBhN.0Njt 4yBhNy)#6[Y (2) 4 ‘ w l’ +ψl(q1,···,qi,···,qj,···,qN)=ψk1,······,kN(q1,···,qi,···,qj,···,qN)=DXˆP(−1)ˆPˆP(ϕk1(q1)······ϕkN(qN))=DXˆP(−1)ˆPϕk1qP(1)······ϕkNqP(N).(34)?iQ,YZ w ,Æ# bUK\d #j ,i BhNZy }}Rj )#6|Y 1D6fDy-M ?[ I +.,y0u R& 5w i# dgyx‘ 6 1.3:JD1 1jO?‘ \1} bk1.k2 w j ~ } l ybψ2k1,k2(q1,q2,q3).ψ2k1,k2(q1,q2,q3) ;G + w 1ψ2k1,k2(q1,q2,q3)=D1[ϕk1(q1)ϕk1(q2)ϕk2(q3)+ϕk1(q2)ϕk1(q3)ϕk2(q1)+ϕk1(q3)ϕk1(q1)ϕk2(q2)+ϕk1(q2)ϕk1(q1)ϕk2(q3)+ϕk1(q1)ϕk1(q3)ϕk2(q2)+ϕk1(q3)ϕk1(q2)ϕk2(q1)]=fD1[ϕk1(q1)ϕk1(q2)ϕk2(q3)+ϕk1(q2)ϕk1(q3)ϕk2(q1)+ϕk1(q3)ϕk1(q1)ϕk2(q2)].(35)ju w 1ψk1,2k2(q1,q2,q3)=fD2[ϕk1(q1)ϕk2(q2)ϕk2(q3)+ϕk1(q2)ϕk2(q3)ϕk2(q1)+ϕk1(q3)ϕk2(q1)ϕk2(q2)].(36) 4 4I} ,Y 8* 9Z# iK ,YTNR I j 04(2k1,k2)6=(k1,2k2) w l’P I} ,YOR KG w 1Zdq1dq2dq3ψ∗2k1,k2(q1,q2,q3)ψk1,2k2(q1,q2,q3)=Zdq1dq2dq3fD∗1fD2×hϕ∗k1(q1)ϕ∗k1(q2)ϕ∗k2(q3)+ϕ∗k1(q2)ϕ∗k1(q3)ϕ∗k2(q1)+ϕ∗k1(q3)ϕ∗k1(q1)ϕ∗k2(q2)i×[ϕk1(q1)ϕk2(q2)ϕk2(q3)+ϕk1(q2)ϕk2(q3)ϕk2(q1)+ϕk1(q3)ϕk2(q1)ϕk2(q2)].(37)ah4 #19 w .Z}# si 1Zdq1dq2dq3ϕ∗k1(q1)ϕ∗k1(q2)ϕ∗k2(q3)ϕk1(q1)ϕk2(q2)ϕk2(q3)=Zdq1ϕ∗k1(q1)ϕk1(q1)Zdq2ϕ∗k1(q2)ϕk2(q2)Zdq3ϕ∗k2(q3)ϕk2(q3).(38)04} ,‘r uXN a?r ju w lP 5y !r IR04 t} byqz‘b~‘YR jI bl’Fv# iKqi Mxϕ∗α(qi)6ϕβ(qi)d1Rjy~‘Yα.β ,i ? TZϕ∗α(qi)ϕβ(qi)dqi=0.(39)7 ? w D_)#6[YfD1.fD2 0)#6hO1=Zdq1dq2dq3ψ∗2k1,k2(q1,q2,q3)ψ2k1,k2(q1,q2,q3