84北京大学物理学院-量子力学chap3

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0,$/)$3.1+.#Æ!nAXufgHeisenberg(x}HA;i$Mj}℄|fT~x}j℄GBFjbCHx3t}Schr¨odinger~i$JLBFoˆp=¯hi∇iug}axXujA?;H:Mjez7%i5CZp:j“!”+j-jo~JE04mR&oLandau$Lifschitz℄&\j“QuantumMechanics”NjnAXi~7|j‘L}eZp!j-g:Mj}aje;uOoW~\Lj(Operation)jHigARjej^_jwr}jRNEs0|Y%mYHi~n\M}L*!W.sjwRdH;-ij!xijein}\M}n:\M}!iW.sj/|HFijwREYM}uOonqM}‘yRjM}RP3Eqn}qM}?4Li~n\M}L*!W.sjwR;-ij!xijein}\M}n:\M}!5$6L6j;Wi.sjwRn}YM}:Ooi6Mj(Predictable)}axX‘7oZjj-EYM}RWjwR:oAR}aje[X^_jx}/a-Tn}YM}IHx}jM}7x3_jYx}jM}z|\Wi.sRNu\OE_ox}j;H_|x+jwR?Oo^j;HNU*jkRXuJ$;Qj’(rS“x}”Zj(Q0x3x}jM}ui$*87|jRB_Eψnox}u7}j;H_D$^[x+j;HNoEnu*8Y(Operator)ˆHoBFx3};H_ψnW3“}”Ex}M}jLj(Operation)Æ_;Q0x}M}js(u0ˆHψn=Enψn.(1)1~/EB_H0.s’(jirui$s(kAt}2HMt}kx}[x+jY9[x+j;HN$;H_dHuOWjHkAH7bMjx}x^j|\zM}iWFijwREH3;Qx}M}s(jnjul#qojHgoPuVVxx}M}j;H_s(~YjLjo~o0jt-ERB_uJs(A$nTL}u℄’gx3}ajejx}ˆQM}[igwRj,e{qn}+H&X~[0jiN/a-T+j;H_e{φn}}ajej[0ij^_PjlDX+Hm9j\HT3%}ajeji^_Φui$P{φn}jf&:u0Φ=∞Xn=1Cnφn.(2)x3E_us(ˆQΦ=ˆQ∞Xn=1Cnφn=∞Xn=1CnˆQφn=∞Xn=1Cnqnφn.(3)Ni#qojHEs(4x3_ΦWM}~?j“ER\Z”Rv(RHv-RB_EQj)*8Es(jRXOf8?H}ax_+dj_rA8r|wr!?H4x}^_Φ|jdN0j/a-Tey}ajeijVDgH|:V9WoL*2x^[mWix}ˆQjM}8?;1igFijwRdHEwRjdN-.E(3)bs|o3Eou1|Wo{jSIE(3)jZ3^Jx}YˆQjs(℄;H_8&{φn}kg~}ajej[0ij^_PjlDXj/Eou\0it-R#Z~oj0v&plD$9IvElDXjYj,HrEHuEAXKQj6$3.2’*’*2s(YLu*8lDj^H8&^j7WLDi$s(A“+”ii$s(^j7W$RjRQELjLDd[sjlDXj7Wy;Oo}3.1:*lDXj)5Ω(u7plDXui$*Ω=(0,a)2HΩ=(−∞,∞))us(L2(Ω)o[0)5Ω;ji5Rj8&:9f∈L2(Ω)?ZΩ|f(r)|2dr∞(4)Pv?ui$JL2(Ω)HlD23’*|L2(Ω)XjRf(r)$g(r)5Cs(^dgHZΩ|f(r)|2dr∞,ZΩ|g(r)|2dr∞.(5)uJ^j3%f&G(r)=af(r)+bg(r)(6)Hi5j)[HL2(Ω)Xj}A?;u0ZΩ|G(r)|2dr=ZΩ|af(r)+bg(r)|2dr=|a|2ZΩ|f(r)|2dr+|b|2ZΩ|g(r)|2dr+¯abZΩf(r)g(r)dr+a¯bZΩf(r)g(r)dr≤|a|2ZΩ|f(r)|2dr+|b|2ZΩ|g(r)|2dr+2|ab|sZΩ|f(r)|2drsZΩ|g(r)|2dr≤∞.(7)Esu-g~Cauchy-SchwartzFkEZΩf(r)g(r)dr≤sZΩ|f(r)|2drsZΩ|g(r)|2dr.(8)diJlDV;9/ˆA:V→VdzˆA(c1v1+c2v2)=C1ˆAv1+c2ˆAv2,(9)3Esv1$v2HVj3’|}|c1$c2H|3’jNR?^:OoYu7ˆpx=−i¯h∂/∂x:HlDL2(−∞,∞);jYY‘07|jW(i)ˆA=ˆBPv’q‘x33%VXj}ψuu0ˆAψ=ˆBψ(ii)YˆIOor79ˆIψ=ψx3VX[0j}ψPv(iii)YL$s(jˆA+ˆBψ=ˆAψ+ˆBψ(iv)YL5s(jˆAˆBψ=ˆAˆBψ7|ˆAˆB6=ˆBˆA0plDXui$Bgfvj}f℄|JYP℄GdHvpjlDu7L2(Ω)XuFsigEobCHYˆx$ˆpxFiP℄GjÆEBsERFPv0ux3kEˆxˆpx−ˆpxˆx=i¯hˆI(10)j|?C(6u0Tr(ˆxˆpx−ˆpxˆx)=Tr(ˆxˆpx)−Tr(ˆpxˆx)=0.(11)dHx3;E1?j(6~/FonjSIHE|YvP℄GjÆE)[(60’(u|1fgXˆx$ˆpxFP℄GjÆE.^f&|PjYˆLx,ˆLy$ˆLz-i$P℄GYjn\Æs(iˆAn=ˆA·ˆA···ˆA.(12)eYHxjui$s(^jˆA−1D$0ˆA·ˆA−1=ˆA−1·ˆA=ˆI.(13)t-TaylorejB_EF(x)=∞Xn=01n!dndxnF(x) x=0xn,(14)4ui$7|s(YjRˆF(ˆA)=∞Xn=01n!dndxnF(x) x=0ˆAn.(15)Es(u|u1-gu*8[s“5”js(uJ4|}u$v0jSR(u,v)OoIvlDV;j59^d$|gH(i)x33%}u∈V,u0(u,u)≥0(ii)(u,v)=(v,u)(iii)(u,C1v1+C2v2)=C1(u,v1)+C2(u,v2)(iv)(C1u1+C2u2,v)=C1(u1,v)+C2(u2,v)u.((u,u)=0e$Veu=0Pvjouaui$*0plDVu3*^jfv3{ei}?VXj3%}vui$Ef3jeui$s(|}u$vj5o(u,v)=NXn=1unvn.(16)|x3i5RlDL2(Ω)ui$s(Xj5o(ϕ,ψ)=ZΩdrϕ(r)ψ(r).(17)0~5js(ui$s(sYˆAjlYˆA†9x3lDVj3’|}u$vuu0u,ˆAv=ˆBu,v,(18)?uOYˆBoYˆAjlYDJ^=jˆA†bCHeˆA†=ˆAuJˆAOozY3.2:$0plDVou[ui$*^jfIL3l9ˆAHzYui$s(^j℄G7oAij=(ϕi,ˆAϕj).(19)5t-5jWuigAij=(ϕi,ˆAϕj)=(ˆA†ϕi,ϕj)=(ϕj,ˆA†ϕi)=ˆA†ji.(20)2HA†ij=A∗ji.(21)9ˆAozY?uWG0A†ij=A∗ji=Aij.(22)d#qojHv&plDX;j℄Gz7|H0’(j[u℄Eϕ,ˆAψ=ˆAϕ,ψ(23)W~os(bl(z)Y3.3:ˆpx=−i¯h∂/∂xlDL2(−∞,∞)X^Hzj5Cs(u0(ϕ,ˆpxψ)=Z∞−∞ϕ(x)−i¯h∂∂xψ(x)!dx=(−i¯h)ϕ(x)ψ(x) ∞−∞+i¯hZ∞−∞∂∂xϕ(x)!ψ(x)dx=i¯hZ∞−∞∂∂xϕ(x)!ψ(x)dx=Z∞−∞−i¯h∂∂xϕ(x)!ψ(x)dx=(ˆpxϕ,ψ).(24)4s((ϕ,ˆpxψ)=ˆp†xϕ,ψ(25)Ouigˆp†x=ˆpx)[lDL2(−∞,∞)XˆpxHzY5CzYjs(FJ(1)9ˆA$ˆBHzY?ˆC=ˆA+ˆB&HzY6(2)9ˆA$ˆBHzYD$ˆAˆB=ˆBˆA?ˆD=ˆAˆB&HzYuL[$JzYvfWo[H.307|jGA3.1:YˆAHzje$Vex33%ERψdNψ,ˆAψuHANj2L}5CzYjs(u0ψ,ˆAψ=ˆA†ψ,ψ=ˆAψ,ψ=ψ,ˆAψ.(26))[ψ,ˆAψoARL9x33%Rψuu0ψ,ˆAψ=ψ,ˆAψ=ˆAψ,ψ,(27)?ui$3*|Rψ1$ψ2D$s(ϕ1=ψ1+ψ2,ϕ2=ψ1+iψ2.(28)u0ϕ1,ˆAϕ1=ψ1+ψ2,ˆA(ψ1+ψ2)=ϕ1,ˆAϕ1=ˆA(ψ1+ψ2),ψ1+ψ2,ϕ2,ˆAϕ2=ψ1+iψ2,ˆA(ψ1+iψ2)=ϕ2,ˆAϕ2=ˆA(ψ1+iψ2),ψ1+iψ2.(29)J^e*u0ψ2,ˆAψ1+ψ1,ˆAψ2=ˆAψ2,ψ1+ˆAψ1,ψ2,−iψ2,ˆAψ1+iψ1,ˆAψ2=−iˆAψ2,ψ1+iˆAψ1,ψ2.(30)JnEQ$i4n}EA*uig2iψ1,ˆAψ2=2iˆAψ1,ψ2,(31)2Hψ1,ˆAψ2=ˆAψ1,ψ2.(32).3ERψ1$ψ2j3’uigRYˆAHzj7℄bRuKhx33%℄Gui$Bg}vCiˆAv=λv(33)PvEsλHROoˆAj;HN|v?Oo+j;H}i$JERi$kgj-3v&plDXjYt-;[Jjsr$zYjs(ui$’6%mJ3.2:zYj;HNoAR25CzYjs(dNψ,ˆOψoAR*ψoYˆOj;HRψn?u20ψn,ˆOψn=(ψn,λnψn)=λn(ψn,ψn).(34))[λn=ψn,ˆOψn(ψn,ψn)(35)oAR.3A;i$Mj}uHARDirac([0x+3x}jYu+HzYu.03.3:zYjP3Fi

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