( ) () we define the interaction representation by the unitary transformation () = ()


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1 Hgher Order Perurbaon Theory Mchael Fowler 3/7/6 The neracon Represenaon Recall ha n he frs par of hs course sequence, we dscussed he chrödnger and Hesenberg represenaons of quanum mechancs here n he chrödnger represenaon, he operaors are mendependen (excep for explcly medependen poenals) he kes represenng he quanum saes develop n me n he Hesenberg represenaon, he kes say he same, he me dependence s n he operaors These dfferng represenaons descrbe he same physcs marx elemens of operaors beween kes mus be he same n boh The mos naural o use depends on he problem a hand n he classcal lm, for example, he Hesenberg operaors have he me dependence of he correspondng classcal operaors n fac, for perurbaon heory problems wh a medependen poenal, an nermedae represenaon, he neracon represenaon, s very convenen Usng a subscrp o denoe he chrödnger represenaon, d () H () H V () d = = + ( ) (), we defne he neracon represenaon by he unary ransformaon () e H / () = so he neracon represenaon kes and he chrödnger represenaon kes concde a =, and f he neracon were zero, he neracon represenaon kes would be consan n me, lke hose n he Hesenberg represenaon For nonzero V( ), hen, he me developmen of he neracon represenaon kes s enrely due o V( ), and s easly found by dfferenang boh sdes of he equaon: d H / d () = H () + e () d d = H + e H + V e H () () H () () H ( ) () / / H / / () = e V e = V (), where we have nroduced he neracon represenaon operaor V (), defned by H () = () / H / V e V e
2 Operaors n hs represenaon mus have hs me dependence relave o he chrödnger operaors o ensure ha marx elemens, he only quanes of physcal sgnfcance, are he same n he wo represenaons Tha s o say, we mus have f O = f O, he wo represenaons mus predc he same probably amplude for any ranson negrang boh sdes of he dfferenal equaon, () = ( ) dv ( ) ( ) Ths s no a soluon we ve jus gone from a dfferenal equaon o an negral equaon Ths s only worh dong f V s small, n whch case he negral equaon can be solved eravely The zeroh approxmaon s hen ( ) = ( ) Pung hs value no he small erm on he rgh hand sde of he negral equaon gves he frs order soluon, = () ( ) dv ( ) ( ) The second order soluon s now gven by pung he frs order soluon no he negral on he rgh: () = ( ) dv ( ) ( ) d V ( ) ( ) Ths can be wren: () = 1 dv ( ) + d d V ( ) V ( ) ( ) The complee perurbaon seres s generaed by repeang he eraon o all orders can be expressed as a meordered produc:
3 exp () = T dv ( ) ( ) The T symbol means ha on expandng ou he exponenal, he operaors a dfferen mes are arranged n order of me, he laes on he lef, whou worryng abou commuaors f we jus blndly expand he exponenal, we wll ge, for example, a hrdorder erm 1 T dv d V d V 3! ( ) ( ) ( ) The T operaor ells us o rearrange he V () s n chronologcal order nce here are hree of hem, hey clearly appear n all possble orders before T operaes, ha s o say, here are 3! dfferen ordered erms ha T makes he same Ths jus ncely cancels he 3! n he exponenal expanson, o gve us he expresson we found by eraon Ths meordered exponenal s herefore he neracon represenaon propagaor: () = U ( ) ( ) U ( ) = T dv ( ),,, exp Gong Back o he chrödnger Represenaon s nsrucve o recas hs resul n he chrödnger represenaon (followng hankar) Frs, noe ha pung he above equaon for U ogeher wh he orgnal defnon of neracon represenaon kes gves () e H / () = () H / () H / e e U ( ) ( ) e H / = = = U ( ), (,) o he chrödnger represenaon propagaor s relaed o he neracon represenaon propagaor by: ( ) H / = ( ) U, e U, Now le us see how o pu our perurbaon expanson for he propagaor back from he neracon represenaon no he chrödnger represenaon nsead of ryng o handle he
4 whole nfne seres a once, we concenrae on he secondorder erm We wll dscover a paern ha works for all he hgher order erms as well o, ransformng he operaors n he secondorder erm of he neracon propagaor back o he chrödnger form, usng we fnd H () = () / H / V e V e ( ) ( ) / ( ) / H H H / H / d d V V = d d e V e e V ( ) e H / e 1 1 Recall also ha he chrödnger propagaor has he exra erm mulplyng he neracon represenaon propagaor Pung hs n, and combnng some of he exponenals, we fnd he secondorder conrbuon o he chrödnger propagaor o be: whch can also be wren: 1 H ( )/ H ( )/ H / d d e V ( ) e V ( ) e 1 (, ) ( ) (, ) ( ) ( ) d d U V U V U, To fnd he probably amplude correspondng o hs secondorder process, we mus sandwch beween nal and fnal saes We ake as our bass se he egensaes of H f we nser he un operaor = n n beween he wo V s, he exponenaed H s become smply numbers snce hey are now acng on egensaes, and he expresson becomes 1 n ( ) ( ) E ( )/ f En ( )/ E / d d e f V n e n V e The nerpreaon s now clear: he nal sae evolves from = o under H, ha s o say, only s phase changes n he sandard fashon A, he neracon V ( ) kcks no anoher egensae n of H, and only he phase changes unl, when V ( ) sends o he fnal sae f Ths process mus be summed over all mes, beween and, and over
5 all possble nermedae saes The n h order erm has precsely he same srucure, wh V comng no play n mes
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