Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 18
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We remove the time evolution of the unperturbed Schrödingersolutions and define the states|ψI (t)i ≡ eiĤ0 t/h̄ |ψ(t)i.(1.286)Their time evolution comes entirely from the interaction potential V̂ . It is governedby the time evolution operatorÛI (tb , ta ) ≡ eiH0 tb /h̄ e−iHtb /h̄ eiHta /h̄ e−iH0 ta /h̄ ,(1.287)and reads|ψI (tb )i = ÛI (tb , ta )|ψI (ta )i.(1.288)H. Kleinert, PATH INTEGRALS431.10 Time Evolution AmplitudeIf V̂ = 0, the states |ψI (tb )i are time-independent and coincide with the Heisenbergstates (1.276) of the operator Ĥ0 .The operator ÛI (tb , ta ) satisfies the equation of motionih̄∂tb ÛI (tb , ta ) = VI (tb )ÛI (tb , ta ),(1.289)whereV̂I (t) ≡ eiH0 t/h̄ V̂ e−iH0 t/h̄(1.290)is the potential in the interaction picture.
This equation of motion can be turnedinto an integral equationÛI (tb , ta ) = 1 −ih̄ZtbdtVI (t)ÛI (t, ta ).ta(1.291)Inserting Eq. (1.290), this readsi Z tbÛI (tb , ta ) = 1 −dt eiĤ0 t/h̄ V e−iĤ0 t/h̄ ÛI (t, ta ).h̄ ta(1.292)This equation can be iterated to find a perturbation expansion for the operatorÛI (tb , ta ) in powers of the interaction potential:i tbdt eiĤ0 t/h̄ V e−iĤ0 t/h̄h̄ ta ZZ ti 2 tb00dt dt0 eiĤ0 t/h̄ V e−iĤ0 (t−t )/h̄ V e−iĤ0 t /h̄ + . . . .+ −h̄tataÛI (tb , ta ) = 1 −Z(1.293)Inserting on the left-hand side the operator (1.287), this can also be rewritten as−iH(tb −ta )/h̄ei+ −h̄−iH0 (tb −ta )/h̄=e2 ZtbtadtZttai−h̄Ztbtadt e−iĤ0 (tb −t)/h̄ V e−iĤ0 (t−ta )/h̄00dt0 e−iĤ0 (tb −t)/h̄ V e−iĤ0 (t−t )/h̄ V e−iĤ0 (t −ta )/h̄ + .
. . .(1.294)This expansion is seen to be the recursive solution of the integral equatione−iH(tb −ta )/h̄ = e−iH0 (tb −ta )/h̄ −ih̄Ztbtadt e−iĤ0 (tb −t)/h̄ V e−iĤ(t−ta )/h̄ .(1.295)Note that the lowest-order correction agrees with the previous formula (1.254)1.10Time Evolution AmplitudeIn the subsequent development, an important role will be played by the matrixelements of the time evolution operator in the localized basis states,(xb tb |xa ta ) ≡ hxb |Û(tb , ta )|xa i.(1.296)441 FundamentalsThey are referred to as time evolution amplitudes. The functional matrix (xb tb |xa ta )is also called the propagator of the system.
For a system with a time-independentHamiltonian operator where Û (tb , ta ) is given by (1.260), the propagator is simply(xb tb |xa ta ) = hxb | exp[−iĤ(tb − ta )/h̄]|xa i.(1.297)Due to the operator equations (1.266), the propagator satisfies the SchrödingerequationhiH(−ih̄∂xb , xb , tb ) − ih̄∂tb (xb tb |xa ta ) = 0.(1.298)In the quantum mechanics of nonrelativistic particles, only the propagators fromearlier to later times will be relevant. It is therefore customary to introduce theso-called causal time evolution operator or retarded time evolution operator :9RÛ (tb , ta ) ≡(Û (tb , ta ),0,tb ≥ ta ,tb < ta ,(1.299)and the associated causal time evolution amplitude or retarded time evolution amplitude(xb tb |xa ta )R ≡ hxb |Û R (tb , ta )|xa i.(1.300)Since this differs from (1.296) only for tb < ta , and since all formulas in the subsequent text will be used only for tb > ta , we shall often omit the subscript R.To abbreviate the case distinction in (1.299), it is convenient to use the Heavisidefunction defined by1 for t > 0,Θ(t) ≡(1.301)0 for t ≤ 0,and writeU R (tb , ta ) ≡ Θ(tb − ta )Û(tb , ta ),(xb tb |xa ta )R ≡ Θ(tb − ta )(xb tb |xa ta ).
(1.302)There exists also another Heaviside function Θ(tb − ta ) which differs from (1.303)only by the value at t = t0 :RΘ ≡1 for t ≥ 0,0 for t < 0.(1.303)Both Heaviside functions have the property that their derivative yields Dirac’sδ-function∂t Θ(t) = δ(t).(1.304)If it is not important which Θ-function is used we shall ignore the superscript.The retarded propagator satisfies the Schrödinger equationhiH(−ih̄∂xb , xb , tb )R − ih̄∂tb (xb tb |xa ta )R = −ih̄δ(tb − ta )δ (3) (xb − xa ).9(1.305)Compare this with the retarded Green functions to be introduced in Section 18.1H. Kleinert, PATH INTEGRALS451.10 Time Evolution AmplitudeThe nonzero right-hand side arises from the extra termhi−ih̄ ∂tb Θ(tb − ta ) hxb tb |xa ta i = −ih̄δ(tb − ta )hxb tb |xa ta i = −ih̄δ(tb − ta )hxb ta |xa ta i(1.306)and the initial condition hxb ta |xa ta i = hxb |xa i, due to (1.268).If the Hamiltonian does not depend on time, the propagator depends only on thetime difference t = tb − ta .
The retarded propagator vanishes for t < 0. Functionsf (t) with this property have a characteristic Fourier transform. The integralf˜(E) ≡Z∞0dt f (t)eiEt/h̄(1.307)is an analytic function in the upper half of the complex energy plane. This analyticityproperty is necessary and sufficient to produce a factor Θ(t) when inverting theFourier transform via the energy integralf (t) ≡Z∞−∞dE ˜f (E)e−iEt/h̄ .2πh̄(1.308)For t < 0, the contour of integration may be closed by an infinite semicircle in theupper half-plane at no extra cost.
Since the contour encloses no singularities, it canbe contracted to a point, yielding f (t) = 0.The Heaviside function Θ(t) itself is the simplest retarded function, with aFourier representation containing just a single pole just below the origin of thecomplex energy plane:Z ∞idEe−iEt ,(1.309)Θ(t) =−∞ 2π E + iηwhere η is an infinitesimally small positive number. The integral representation isundefined for t = 0 and there are, in fact, infinitely many possible definitions for theHeaviside function depending on the value assigned to the function at the origin. Aspecial role is played by the average of the Heaviside functions (1.303) and (1.301),which is equal to 1/2 at the origin:1for t > 0,Θ̄(t) ≡for t = 0,0 for t < 0.12(1.310)Usually, the difference in the value at the origin does not matter since the Heavisidefunction appears only in integrals accompanied by some smooth function f (t).
Thismakes the Heaviside function a distribution with respect to smooth test functionsf (t) as defined in Eq. (1.162). All three distributions Θr (t), Θl (t), and Θ̄(t) definethe same linear functional of the test functions by the integralΘ[f ] =Zdt Θ(t − t0 )f (t0 ),and this is an element in the linear space of all distributions.(1.311)461 FundamentalsAs announced after Eq.
(1.162), path integrals will specify, in addition, integralsover products of distribution and thus give rise to an important extension of thetheory of distributions in Chapter 10. In this, the Heaviside function Θ̄(t − t0 ) playsthe main role.While discussing the concept of distributions let us introduce, for later use, theclosely related distribution(t − t0 ) ≡ Θ(t − t0 ) − Θ(t0 − t) = Θ̄(t − t0 ) − Θ̄(t0 − t),(1.312)which is a step function jumping at the origin from −1 to 1 as follows:10(t − t ) = 0−11.11t > t0 ,t = t0 ,t < t0 .forforfor(1.313)Fixed-Energy AmplitudeThe Fourier-transform of the retarded time evolution amplitude (1.300)(xb |xa )E =Z∞−∞iE(tb −ta )/h̄dtb eR(xb tb |xb ta ) =Z∞tadtb eiE(tb −ta )/h̄ (xb tb |xb ta ) (1.314)is called the fixed-energy amplitudes.If the Hamiltonian does not depend on time, we insert here Eq.
(1.297) and findthat the fixed-energy amplitudes are matrix elements(xb |xa )E = hxb |R̂(E)|xa i(1.315)of the so-called of the so-called resolvent operatorR̂(E) =ih̄,E − Ĥ + iη(1.316)which is the Fourier transform of the retarded time evolution operator (1.299):R̂(E) =Z∞−∞iE(tb −ta )/h̄dtb eRÛ (tb , ta ) =Z∞tadtb eiE(tb −ta )/h̄ Û (tb , ta ).(1.317)Let us suppose that the time-independent Schrödinger equation is completelysolved, i.e., that one knows all solutions |ψn i of the equationĤ|ψn i = En |ψn i.(1.318)These satisfy the completeness relationXn|ψn ihψn | = 1,(1.319)H. Kleinert, PATH INTEGRALS471.11 Fixed-Energy Amplitudewhich can be inserted on the right-hand side of (1.297) between the Dirac bracketsleading to the spectral representation(xb tb |xa ta ) =Xnψn (xb )ψn∗ (xa ) exp [−iEn (tb − ta )/h̄] ,(1.320)withψn (x) = hx|ψn i(1.321)being the wave functions associated with the eigenstates |ψn i.
Applying the Fouriertransform (1.314), we obtain(xb |xa )E =Xnψn (xb )ψn∗ (xa )Rn (E) =Xnψn (xb )ψn∗ (xa )ih̄.E − En + iη(1.322)The fixed-energy amplitude (1.314) contains as much information on the systemas the time evolution amplitude, which is recovered from it by the inverse FouriertransformationZ ∞dE −iE(tb −ta )/h̄e(xb |xa )E .(1.323)(xb ta |xa ta ) =−∞ 2πh̄The small iη-shift in the energy E in (1.322) may be thought of as being attachedto each of the energies En , which are thus placed by an infinitesimal piece below thereal energy axis. Then the exponential behavior of the wave functions is slightlydamped, going to zero at infinite time:e−i(En −iη)t/h̄ → 0.(1.324)This so-called ensures the causality of the Fourier representation (1.323).
When doing the Fourier integral (1.323), the exponential eiE(tb −ta )/h̄ makes it always possibleto close the integration contour along the energy axis by an infinite semicircle inthe complex energy plane, which lies in the upper half-plane for tb < ta and in thelower half-plane for tb > ta . The iη-prescription guarantees that for tb < ta , there isno pole inside the closed contour making the propagator vanish. For tb > ta , on theother hand, the poles in the lower half-plane give, via Cauchy’s residue theorem, thespectral representation (1.320) of the propagator. An iη-prescription will appear inanother context in Section 2.3.If the eigenstates are nondegenerate, the residues at the poles of (1.322) renderdirectly the products of eigenfunctions (barring degeneracies which must be discussed separately). For a system with a continuum of energy eigenvalues, there isa cut in the complex energy plane which may be thought of as a closely spaced sequence of poles.
In general, the wave functions are recovered from the discontinuityof the amplitudes (xb |xa )E across the cut, using the formuladiscih̄E − En!≡ih̄ih̄−= 2πh̄δ(E − En ).E − En + iη E − En − iη(1.325)481 FundamentalsHere we have used the general relation to be used in integrals over E:P1∓ iπδ(E − En ),=E − En ± iηE − En(1.326)where P indicates that the principal value of the integral.The energy integral over the discontinuity of the fixed-energy amplitude (1.322)(xb |xa )E reproduces the completeness relation (1.319) taken between the local stateshxb | and |xa i,Z∞−∞XdEdisc (xb |xa )E =ψn (xb )ψn∗ (xa ) = hxb |xa i = δ (D) (xb − xa ).2πh̄n(1.327)The completeness relation reflects the following property of the resolvent operator:Z∞−∞dEdisc R̂(E) = 1̂.2πh̄(1.328)In general, the system possesses also a continuous spectrum, in which case thecompleteness relation contains a spectral integral and (1.319) has the formXn|ψn ihψn | +Zdν |ψν ihψν | = 1.(1.329)The continuum causes a branch cut along in the complex energy plane, and (1.327)includes an integral over the discontinuity along the cut.
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