Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 34

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For this we write the classicalaction (2.260) as a quadratic formMxbAcl =(2.270)(xb , xa ) Axa2with a matrix∂ ẋb∂xbA=The inverse of this matrix isA−1−∂ ẋa∂xb−∂xb ∂ ẋb= ∂xa∂ ẋb∂ ẋb∂xa .∂ ẋ (2.271)a∂xa−−∂xb∂ ẋa .∂x (2.272)a∂ ẋaThe partial derivatives of xb and xa are calculated from the solution of the homogeneous differentialequation (2.211) specified in terms of the final and initial velocities ẋb and ẋa :1Ḋa (tb )Ḋb (ta ) + 1nhihi o× Da (t) + Db (t)Ḋa (tb ) ẋa + −Db (t) + Da (t)Ḋb (ta ) ẋb ,x(ẋb , ẋa ; t) =which yieldsxa=xb=hi1Db (ta )Ḋa (ta )ẋb − Db (ta )ẋb ,Ḋa (tb )Ḋb (ta ) + 1hi1Da (tb )ẋa + Da (tb )Ḋb (ta )ẋb ,Ḋa (tb )Ḋb (ta ) + 1so thatA−1 =Da (tb )Ḋa (tb )Ḋb (ta ) + 1Ḋb (ta )−1−1−Ḋa (tb ).(2.273)(2.274)(2.275)(2.276)1322 Path Integrals — Elementary Properties and Simple SolutionsThe determinant of A is the Jacobiandet A = −∂(ẋb , ẋa )Ḋa (tb )Ḋb (ta ) + 1=−.∂(xb , xa )Da (tb )Db (ta )(2.277)We can now perform the Fourier transform of the time evolution amplitude and find, via a quadraticcompletion,ZZ−ipb xb /h̄dxa eipa xa /h̄ (xb tb |xa ta )(2.278)(pb tb |pa ta ) = dxb esr2πh̄Da (tb )=iM Ḋa (tb )Ḋb (ta ) + 1ihDa (tb )i 122× exp−Ḋb (ta )pb + Ḋa (tb )pa − 2pb pa .h̄ 2M Ḋa (tb )Ḋb (ta ) + 1Inserting here Da (tb ) = sin ω(tb − ta )/ω and Ḋa (tb ) = cos ω(tb − ta ), we recover the oscillator result(2.182).In D dimensions, the classical action has the same quadratic form as in (2.270)M T TxbAcl =xb , xa A(2.279)xa2with a matrix A generalizing (2.271) by having the partial derivatives replaced by the correspondingD × D-matrices.

The inverse is the 2D × 2D-version of (2.272), i.e.∂ ẋb∂xb∂xb∂ ẋb− ∂ ẋb ∂xb∂xa ∂ ẋa ,.A−1 = (2.280)A= ∂ ẋ∂ ẋa∂xa∂xa a−−−∂xb∂xa∂ ẋb∂ ẋaThe determinant of such a block matrixA=acbdis calculated after a triangular decomposition a ba 01a−1 b1A===c dc 10 d − ca−1 b0in two possible ways asadetcbd(2.281)bda − bd−1 cd−1 c01= det a · det (d − ca−1 b) = det (a − bd−1 c) · det d,depending whether det a or det b is nonzero. The inverse is in the first case −1 −1 −11 −a−1 bxa0a +a bxca−1 −a−1 bxA==, x ≡(d−ca−1b)−1.0x−ca−1 1−xca−1xThe resulting amplitude in momentum space isZZ(pb tb |pa ta ) = dxb e−ipb xb /h̄ dxa eipa xa /h̄ (xb tb |xa ta )1i 12πpb√pTb , pTa A−1exp.= √pah̄ 2M2πih̄M Dren det A(2.282)(2.283)(2.284)(2.285)Also in momentum space, the amplitude (2.285) reduces to the free-particle one in Eq.

(2.133)in the limit of infinitesimally short time tb − ta : For the time-independent harmonic oscillator, thiswas shown in Eq. (2.184), and the time-dependence of Ω(t) becomes irrelevant in the limit of smalltb − ta → 0.H. Kleinert, PATH INTEGRALS1332.6 Free-Particle and Oscillator Wave Functions2.6Free-Particle and Oscillator Wave FunctionsIn Eq. (1.330) we have expressed the time evolution amplitude of the free particle(2.71) as a Fourier integral(xb tb |xa ta ) =Zdp ip(x−x0 )/h̄ −ip2 (tb −ta )/2M h̄ee.(2πh̄)(2.286)This expression contains the information on all stationary states of the system.

Tofind these states we have to perform a spectral analysis of the amplitude. Recallthat according to Section 1.7, the amplitude of an arbitrary time-independent systempossesses a spectral representation of the form(xb tb |xa ta ) =∞Xψn (xb )ψn∗ (xa )e−iEn (tb −ta )/h̄ ,(2.287)n=0where En are the eigenvalues and ψn (x) the wave functions of the stationary states.In the free-particle case the spectrum is continuous and the spectral sum is anintegral.

Comparing (2.287) with (2.286) we see that the Fourier decompositionitself happens to be the spectral representation. If the sum over n is written as anintegral over the momenta, we can identify the wave functions asψp (x) = √1eipx .2πh̄(2.288)For the time evolution amplitude of the harmonic oscillator1(xb tb |xa ta ) = q2πih̄ sin [ω(tb − ta )] /Mω(2.289))(hiiMω(x2b + x2a ) cos ω(tb − ta ) − 2xb xa ,× exp2h̄ sin [ω(tb − ta )]the procedure is not as straight-forward.

Here we must make use of a summationformula for Hermite polynomials (see Appendix 2C) Hn (x) due to Mehler:141()120220qexp −2 [(x + x )(1 + a ) − 4xx a]22(1−a)1−a∞Xan0202= exp(−x /2 − x /2)n Hn (x)Hn (x ),n=0 2 n!(2.290)withH0 (x) = 1, H1 (x) = 2x, H2 (x) = 4x2 − 2, . . . , Hn (x) = (−1)n ex142dn −x2e .dxn(2.291)See P.M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York,Vol.

I, p. 781 (1953).1342 Path Integrals — Elementary Properties and Simple SolutionsIdentifyingx≡so thatqx0 ≡Mω/h̄ xb ,qMω/h̄ xa ,a ≡ e−iω(tb −ta ) ,(2.292)1 + a21 + e−2iω(tb −ta )cos [ω(tb − ta )]=2 =−2iω(tb −ta )i sin [ω(tb − ta )]1−a1−e1a,2 =2i sin [ω(tb − ta )]1−awe arrive at the spectral representation(xb tb |xa ta ) =∞Xψn (xb )ψn (xa )e−i(n+1/2)ω(tb −ta ) .(2.293)n=0From this we deduce that the harmonic oscillator has the energy eigenvaluesEn = h̄ω(n + 1/2)(2.294)and the wave functionsψn (x) = Nn λω−1/2 e−x2 /2λ2ωHn (x/λω ).(2.295)Here, λω is the natural length scale of the oscillatorλω ≡sh̄,Mω(2.296)and Nn the normalization constant√Nn = (1/2n n! π)1/2 .(2.297)It is easy to check that the wave functions satisfy the orthonormality relationZ∞−∞dx ψn (x)ψn0 (x)∗ = δnn0 ,(2.298)using the well-known orthogonality relation of Hermite polynomials15Z ∞21√dx e−x Hn (x)Hn0 (x) = δn,n0 .n2 n! π −∞2.7(2.299)General Time-Dependent Harmonic ActionA simple generalization of the harmonic oscillator with time-dependent frequency allows also fora time-dependent mass, so that the action (2.300) becomesZ tbMg(t)ẋ2 (t) − Ω2 (t)x2 (t) ,(2.300)A[x] =dt2ta15I.S.

Gradshteyn and I.M. Ryzhik, op. cit., Formula 7.374.1.H. Kleinert, PATH INTEGRALS1352.7 General Time-Dependent Harmonic Actionwith some dimensionless time-dependent factor g(t). This factor changes the measure of pathintegration so that the time evolution amplitude is no longer given by (2.304). To find the correctmeasure we must return to the canonical path integral (2.29) which now reads(xb tb |xa ta ) =Zx(tb )=xbx(ta )=xa0DxZDp iA[p,x]/h̄e,2πh̄(2.301)with the canonical actionA[p, x] =Ztbtap222dt pẋ −− M Ω (t)x (t) .2M g(t)(2.302)Integrating the momentum variables out in the sliced form of this path integral as in Eqs.

(2.49)–(2.51) yields#"ZN∞Y1i Ndxnp(xb tb |xa ta ) ≈ pexp.(2.303)Ah̄2πh̄i/M g(tN +1) n=1 −∞ 2πh̄i/M g(tn )The continuum limit of this path integral is written asZ√iA[x] ,(xb tb |xa ta ) =Dx g exph̄with the action (2.300).The classical orbits solve the equation of motion−∂t g(t)∂t − Ω2 (t) x(t) = 0,(2.304)(2.305)which, by the transformationpx̃(t) = g(t)x(t),1 ġ 2 (t) 1 g̈(t)12Ω (t) +,−Ω̃ (t) =g(t)4 g(t)2 g(t)2can be reduced to the previous formhipg(t) −∂t2 − Ω̃2 (t) x̃(t) = 0.(2.306)(2.307)The result of the path integration is thereforeZ√(xb tb |xa ta ) = Dx g eiA[x]/h̄ = F (xb , tb ; xa , ta )eiAcl /h̄ ,(2.308)with a fluctuation factor [compare (2.256)]11p,F (xb , tb ; xa , ta ) = p2πih̄/M Da (tb )(2.309)where Da (tb ) is found from a generalization of the formulas (2.257)–(2.262).

The classical actionisAcl =M(gb xb ẋb − ga xa ẋa ),2(2.310)where gb ≡ g(tb ), ga ≡ g(ta ). The solutions of the equation of motion can be expressed in termsof modified Gelfand-Yaglom functions (2.221) and (2.222) with the properties[∂t g(t)∂t + Ω2 (t)]Da (t) = 0 ;2[∂t g(t)∂t + Ω (t)]Db (t) = 0 ;Da (ta ) = 0,Ḋa (ta ) = 1/ga ,(2.311)Db (tb ) = 0,Ḋb (tb ) = −1/gb,(2.312)1362 Path Integrals — Elementary Properties and Simple Solutionsas in (2.241):x(xb , xa ; t) =Db (t)Da (t)xa +xb .Db (ta )Da (tb )(2.313)This allows us to write the classical action (2.310) in the formAcl =ihMgb x2b Ḋa (tb ) − ga x2a Ḋb (ta ) − 2xb xa .2Da (tb )(2.314)From this we find, as in (2.262),Dren = Da (tb ) = Db (ta ) = −M∂ 2 Acl∂xb ∂xa−1,(2.315)so that the fluctuation factor becomesF (xb , tb ; xa , ta ) = √12πih̄s−∂ 2 Acl.∂xb ∂xa(2.316)As an example take a free particle with a time-dependent mass term, whereDa (t) =Zt0dt g−10(t ), Db (t) =Ztb0dt g−10(t ), Dren = Da (tb ) = Db (ta ) =tbdt0 g −1 (t0 ), (2.317)tattaZand the classical action readsAcl =M (xb − xa )2.2 Da (tb )The result can easily be generalized to an arbitrary harmonic actionZ tbMg(t)ẋ2 + 2b(t)xẋ − Ω2 (t)x2 ,dtA=2tawhich is extremized by the Euler-Lagrange equation [recall (1.8)]hi∂t g(t)∂t + ḃ(t) + Ω2 (t) x = 0.(2.318)(2.319)(2.320)The solution of the path integral (2.308) is again given by (2.308), with the fluctuation factor(2.316), where Acl is the action (2.319) along the classical path connecting the endpoints.A further generalization to D dimensions is obvious by adapting the procedure in Subsection 2.4.6, which makes Eqs.

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