Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 60

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. . ,h̄2!h̄3!h̄(3.497)with the interactionAint ≡ −Ztbtadt V (x(t)).(3.498)The expectation values are defined by−1hn| . . . |niω ≡ ZQM,ω,nZdxb dxa ψn∗ (xb )Zx(tb )=xbx(ta )=xa!Dx . . . eiAω /h̄ ψn (xa ), (3.499)whereZQM,ω,n ≡ e−iω(n+1/2)(tb −ta )(3.500)is the projection of the quantum-mechanical partition function of the harmonicoscillator∞ZQM,ω =Xe−iω(n+1/2)(tb −ta )n=0[see (2.40)] onto the nth excited state.The expectation values are calculated as in (3.494), (3.495). To first order inV (x), one has−1hn|Aint |niω ≡ −ZQM,ω,nZtbtadt1Zdxb dxa dx1 ψn∗ (xb )(xb tb |x1 t1 )ω× V (x1 )(x1 t1 |xa ta )ω ψn (xa ).(3.501)The time evolution amplitude on the right-hand side describes the temporal development of the initial state ψn (xa ) from the time ta to the time t1 , where theH. Kleinert, PATH INTEGRALS3.18 Rayleigh-Schrödinger and Brillouin-Wigner Perturbation Expansion277interaction takes place with an amplitude −V (x1 ).

After that, the time evolutionamplitude on the left-hand side carries the state to ψn∗ (xb ).To second order in V (x), the expectation value is given by the double integralZ t2Z tbZ1−1dt2hn|A2int|niω ≡ ZQM,ω,ndt1 dxb dxa dx2 dx12tata∗×ψn (xb )(xb tb |x2 t2 )ω V (x2 )(x2 t2 |x1 t1 )ω V (x1 )(x1 t1 |xa ta )ω ψn (xa ).(3.502)As in (3.495), the integral over t1 ends at t2 .By analogy with (3.481), we resum the corrections in (3.497) to bring them intothe exponent:ii1231 + hn|Aint |niω −(3.503)2 hn|Aint |niω −3 hn|Aint |niω + . . .h̄2!h̄3!h̄1ii23= exphn|Aint|niω −hn|A|ni−hn|A|ni+....intω,cintω,ch̄2!h̄23!h̄3The cumulants in the exponent arehn|A2int |niω,c ≡ hn|A2int|niω − hn|Aint |ni2ωhn|A3int |niω,c= hn|[Aint − hn|Aint |niω ]2 |niω ,(3.504)323≡ hn|Aint|niω − 3hn|Aint|niω hn|Aint|niω + 2hn|Aint |niω= hn|[Aint − hn|Aint |niω ]3 |niω ,....(3.505)From (3.503), we obtain the energy shift of the nth oscillator energyih̄∆En =limtb −ta →∞ t − tbai1hn|Aint |niω −hn|A2int |niω,ch̄2!h̄2i3− 3 hn|Aint |niω,c + .

. . ,3!h̄(3.506)which is a generalization of formula (3.485) which was valid only for the ground stateenergy. At n = 0, the new formula goes over into (3.485), after the usual analyticcontinuation of the time variable.The cumulants can be evaluated further with the help of the real-time versionof the spectral expansion (3.496):(xb tb |xa ta )ω =∞Xψn (xb )ψn∗ (xa )e−iEn (tb −ta )/h̄ .(3.507)n=0To first order in V (x), it leads tohn|Aint |niω ≡ −ZtbtadtZdxψn∗ (x)V (x)ψn (x) ≡ −(tb − ta )Vnn .(3.508)2783 External Sources, Correlations, and Perturbation TheoryTo second order in V (x), it yieldstbt21−1hn|A2int |niω ≡ ZQM,ω,ndt2dt12tataX −iE (t −t )/h̄−iE (t −t )/h̄−iE (t −t )/h̄n 1a1k 2e n b 2Vnk Vkn .×ZZ(3.509)kThe right-hand side can also be written asZtbtadt2Zt2tadt1Xei(En −Ek )t2 /h̄+i(Ek −En )t1 /h̄ Vnk Vkn(3.510)kand becomes, after the time integrations,−Xkhih̄2Vnk Vknih̄(tb − ta )−ei(En −Ek )(tb −ta )/h̄ −1 .Ek − EnEn − Ek()(3.511)As it stands, the sum makes sense only for the Ek 6= En -terms.

In these, the secondterm in the curly brackets can be neglected in the limit of large time differencestb − ta . The term with Ek = En must be treated separately by doing the integraldirectly in (3.510). This yieldsVnn Vnn(tb − ta )2,2(3.512)so thatX Vnm Vmn1(t − ta )2ih̄(tb − ta ) + Vnn Vnn bhn|A2int |niω = −.22m6=n Em − En(3.513)The same result could have been obtained without the special treatment of the Ek =En -term by introducing artificially an infinitesimal energy difference Ek − En = in (3.511), and by expanding the curly brackets in powers of tb − ta .When going over to the cumulants 21 hn|A2int |niω,c according to (3.504), the k = n term is eliminated and we obtainX Vnk Vkn1ih̄(tb − ta ).hn|A2int |niω,c = −2k6=n Ek − En(3.514)For the energy shifts up to second order in V (x), we thus arrive at the simple formula∆1 En + ∆2 En = Vnn −Vnk Vkn.E − Enk6=n kX(3.515)The higher expansion coefficients become rapidly complicated.

The correction ofthird order in V (x), for example, is∆3 En =XVnk VknVnk Vkl Vln− Vnn2.(E−E)(E−E)(E−E)knlnknk6=nk6=n l6=nXX(3.516)H. Kleinert, PATH INTEGRALS3.18 Rayleigh-Schrödinger and Brillouin-Wigner Perturbation Expansion279For comparison, we recall the well-known formula of Brillouin-Wigner equation 14∆En = R̄nn (En + ∆En ),(3.517)ˆ (E)|ni of the level shift operatorwhere R̄nn (E) are the diagonal matrix elements hn|R̄ˆ (E) which solves the integral equationR̄ˆ (E).ˆ (E) = V̂ + V̂ 1 − P̂n R̄R̄E − Ĥω(3.518)ˆ (E) = V̂ + V̂ 1 − P̂n V̂ + V̂ 1 − P̂n V̂ 1 − P̂n V̂ + . .

. .R̄E − ĤωE − Ĥω E − Ĥω(3.519)The operator P̂n ≡ |nihn| is the projection operator onto the state |ni. The factors1 − P̂n ensure that the sums over the intermediate states exclude the quantumnumber n of the state under consideration. The integral equation is solved by theseries expansion in powers of V̂ :Up to the third order in V̂ , Eq. (3.517) leads to the Brillouin-Wigner perturbationexpansionXXX Vnk VknVnk Vkl Vln+ .

. . , (3.520)+E − En = Rnn (E) = Vnn +k6=n l6=n (E − Ek )(E − El )k6=n E − Ekwhich is an implicit equation for ∆En = E − En . The Brillouin-Wigner equation(3.517) may be converted into an explicit equation for the level shift ∆En :00200∆En = Rnn (En ) + Rnn (En )Rnn(En )+[Rnn (En )Rnn(En )2 + 21 Rnn(En )Rnn(En )]0320003000+[Rnn (En )Rnn (En ) + 32 Rnn (En )Rnn (En )Rnn (En )+ 16 Rnn (En )Rnn (En )]+ . . . .(3.521)Inserting (3.520) on the right-hand side, we recover the standard RayleighSchrödinger perturbation expansion of quantum mechanics,which coincides preciselywith the above perturbation expansion of the path integral whose first three termswere given in (3.515) and (3.516).

Note that starting from the third order, theexplicit solution (3.521) for the level shift introduces more and more extra disconnected terms with respect to the simple systematics in the Brillouin-Wignerexpansion (3.520).For arbitrary potentials, the calculation of the matrix elements Vnk can becomequite tedious. A simple technique to find them is presented in Appendix 3A.The calculation of the energy shifts for the particular interaction V (x) = gx4 /4is described in Appendix 3B. Up to order g 3 , the result ish̄ωg∆En =(2n + 1) + 3(2n2 + 2n + 1)a424 21g2(34n3 + 51n2 + 59n + 21)a8(3.522)−4h̄ω 3g14 · 3(125n4 + 250n3 + 472n2 + 347n + 111)a12 2 2 .+4h̄ ω14L.

Brillouin and E.P. Wigner, J. Phys. Radium 4, 1 (1933); M.L. Goldberger and K.M. Watson,Collision Theory, John Wiley & Sons, New York, 1964, pp. 425–430.2803 External Sources, Correlations, and Perturbation TheoryThe perturbation series for this as well as arbitrary polynomial potentials canbe carried out to high orders via recursion relations for the expansion coefficients.This is done in Appendix 3C.3.19Level-Shifts and Perturbed Wave Functionsfrom Schrödinger EquationIt is instructive to rederive the perturbation expansion from ordinary operator Schrödinger theory.This derivation provides us also with the perturbed eigenstates to any desired order.The Hamiltonian operator Ĥ is split into a free and an interacting partĤ = Ĥ0 + V̂ .(3.523)Let |ni be the eigenstates of Ĥ0 and |ψ (n) i those of Ĥ:(n)Ĥ0 |ni = E0 |ni,Ĥ|ψ (n) i = E (n) |ψ (n) i.(3.524)We shall assume that the two sets of states |ni and |ψ (n) i are orthogonal sets, the first with unitnorm, the latter normalized by scalar products(n)a(n)i = 1.n ≡ hn|ψ(3.525)Due to the completeness of the states |ni, the states |ψ (n) i can be expanded asX|ψ (n) i = |ni +wherem6=na(n)m |mi,(n)a(n)im ≡ hm|ψ(3.526)(3.527)are the components of the interacting states in the free basis.

Projecting the right-hand Schrödingerequation in (3.524) onto hm| and using (3.527), we obtain(m) (n)amE0+ hm|V̂ |ψ (n) i = E (n) a(n)m .(3.528)Inserting here (3.526), this becomes(m) (n)amE0+ hm|V̂ |ni +X(n)k6=nak hm|V̂ |ki = E (n) a(n)m ,and for m = n, due to the special normalization (3.525),X (n)(n)E0 + hn|V̂ |ni +ak hn|V̂ |ki = E (n) .(3.529)(3.530)k6=n(n)Multiplying this equation with am and subtracting it from (3.529), we eliminate the unknown(n)exact energy E (n) , and obtain a set of coupled algebraic equations for am :X (n)1hm − a(n)a(n)(3.531)ak hm − a(n)m =m n|V̂ |ni +m n|V̂ |ki ,(n)(m)E0 − E0k6=n(n)(n)where we have introduced the notation hm − am n| for the combination of states hm| − am hn| ,for brevity.H. Kleinert, PATH INTEGRALS2813.20 Calculation of Perturbation Series via Feynman DiagramsThis equation can now easily be solved perturbatively order by order in powers of the inter(n)action strength.

To count these, we replace V̂ by g V̂ and expand am as well as the energies E (n)in powers of g as:a(n)m (g)=∞X(n)am,l (−g)ll=1(m 6= n),(3.532)(n)(3.533)and(n)E (n) = E0−∞X(−g)l El .l=1Inserting these expansions into (3.530), and equating the coefficients of g, we immediately find theperturbation expansion of the energy of the nth level(n)E1(n)El= hn|V̂ |ni,X (n)=ak,l−1 hn|V̂ |ki(3.534)l > 1.(3.535)k6=n(n)The expansion coefficients am,l are now determined by inserting the ansatz (3.532) into (3.531).This yields(n)am,1 =and for l > 1:(n)am,l =1−a(n) hn|V̂m,l−1(m)(n)E0 −E0hm|V̂ |ni(m)E0(n)− E0,(3.536)l−2X (n)XX (n)(n)ak,l−1 hm|V̂ |ki−am,l0ak,l−1−l0 hn|V̂ |ki.|ni+l0 =1k6=nk6=n(3.537)Using (3.534) and (3.535), this can be simplified tol−1XX1(n)(n)(n)(n)am,l = (m)ak,l−1 hm|V̂ |ki −am,l0 El−l0  .(n)E0 −E0k6=nl0 =1(3.538)Together with (3.534), (3.535), and (3.536), this is a set of recursion relations for the coefficients(n)(n)am,l and El .The recursion relations allow us to recover the perturbation expansions (3.515) and (3.516) forthe energy shift.

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