Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 81
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Second,we may consider the semiclassical approximation to the path integral as an exactpath integral associated with the lowest quadratic approximation to the action in(4.92), (4.93):Aqu [x, ẋ] = A[xcl ] +ZtbtaMdt(δ ẋ)2 + Ω2 (t)(δx)2 ,2(4.110)with Ω2 (t) = V 00 (xcl (t))/M of (4.94). Then, since the classical orbit running fromxa to xb satisfies the equation of motion (4.106), also a slightly different orbit (xcl +δxcl )(t) from x0a = xa + δxa to x0b = xa + δxb satisfies (4.106).
Although the smallchange of the classical orbit gives rise to a slightly different frequency Ω2 (t) =V 00 ((xcl + δxcl )(t))/M, this contributes only to second order in δxa and δxb . As aconsequence, the derivative Da (t) = −∂ ẋb (t)/∂xa satisfies Eq. (4.106) as well. Alsothe boundary conditions of Da (t) are the same as those of Da (t) in Eqs. (2.221).Hence the quantity Da (tb ) is the correct fluctuation determinant also for the generalaction in the semiclassical approximation under study.Another way to derive this formula makes use of the general relation (4.81), fromwhich we find∂M ∂E∂ ∂A(xb , xa ; tb − ta ) =p(xb ) =.(4.111)∂xb ∂xa∂xap(xb ) ∂xaOn the right-hand side we have suppressed the arguments of the functionE(xb , xa ; tb − ta ).
After rewriting∂E∂ ∂A∂ ∂A= −=−∂xa∂xa ∂tb∂tb ∂xa(4.112)H. Kleinert, PATH INTEGRALS4.3 Van Vleck-Pauli-Morette Determinant385M ∂E∂p(xa ) =,∂tbp(xa ) ∂tb=we see that∂ ∂∂E1A(xb , xa ; tb − ta ) =.∂xb ∂xaẋ(tb )ẋ(ta ) ∂tb(4.113)From (4.57) we calculate∂E∂tb∂tb∂E=="−Z!−1xbxa"= −M2dx 3pZxa#−1#−1M ∂pdx 2p ∂Exb"= −MZtbtadtp2(4.114)#−1"1= −MZtbtadt2ẋcl (t)#−1.Inserting this into (4.113), we obtain once more formula (4.109) for the fluctuationdeterminant.A relation following from (4.85):∂E=∂tb∂2S∂E 2!−1,(4.115)∂2S.∂E 2(4.116)leads to an alternative expressionDren = −M ẋcl (tb )ẋcl (ta )The fluctuation factor is therefore also here [recall the normalization fromEqs. (2.195), (2.197), and (2.205)]"1∂ ẋbF (xb , ta ; tb −ta ) = q2πih̄/M ∂xa#1/2=√1[−∂xb ∂xa A(xb , xa ; tb −ta )]1/2.
(4.117)2πih̄Its D-dimensional generalization of (4.117) isF (xb , xa ; tb − ta ) = √1D2πih̄no1/2detD [−∂xi ∂xja A(xb , xa ; tb − ta )]b,(4.118)and the semiclassical time evolution amplitude reads(xb tb |xa ta ) = √1D2πih̄no1/2detD [−∂xi ∂xja A(xb , xa ; tb − ta )]beiA(xb ,xa ;tb −ta )/h̄ .
(4.119)The D × D -determinant in the curly brackets is the so-called Van Vleck-PauliMorette determinant.4 It is the analog of the determinant in the right-hand part4J.H. Van Vleck, Proc. Nat. Acad. Sci. (USA) 14 , 178 (1928); W. Pauli, Selected Topics inField Quantization, MIT Press, Cambridge, Mass. (1973); C. DeWitt-Morette, Phys. Rev. 81 , 848(1951).3864 Semiclassical Time Evolution Amplitudeof Eq. (2.263). As discussed there, the result is initially valid only as long as thefluctuation determinant is regular.
Otherwise we must replace the determinant byits absolute value, and multiply the fluctuation factor by the phase factor e−iν/2with the Maslov-Morse index ν [see Eq. (2.264)]. Using the relation (4.81) in Ddimensions∂pib∂xi ∂xja A(xb , xa ; tb − ta ) = j ,(4.120)b∂xawe shall often write (4.119) as(xb tb |xa ta ) = √1D2πih̄"∂p− b∂xadetD!#1/2eiA(xb ,xa ;tb −ta )/h̄ ,(4.121)where the subscripts a and b can be interchanged in the determinant, if the sign ischanged [recall (2.245)].
This concludes the calculation of the semiclassical approximation to the time evolution amplitude.As a simple application, we use this formula to write down the semiclassicalamplitude for a free particle and a harmonic oscillator. The first has the classicalactionA(xb , xa ; tb − ta ) =M (xb − xa )2,2 tb − ta(4.122)and Eq. (4.109) givesDren = tb − ta ,(4.123)as it should. The harmonic-oscillator action isA(xb , xa ; tb − ta ) =hiMω(x2b + x2a ) cos ω(tb − ta ) − 2xb xa , (4.124)2 sin ω(tb − ta )and has the second derivative−∂xb ∂xa A =Mω,sin ω(tb − ta )(4.125)so that (4.117) coincides with fluctuation factor (2.209).4.4Fundamental Composition Law for SemiclassicalTime Evolution AmplitudeThe determinant ensures that the semiclassical approximation for the time evolutionamplitude satisfies the fundamental composition law (2.4) in D dimensions(xb tb |xa ta ) =N ZYn=1∞−∞Dd xn NY+1n=1(xn tn |xn−1 tn−1 ),(4.126)H.
Kleinert, PATH INTEGRALS4.4 Fundamental Composition Law for Semiclassical Time Evolution Amplitude387if the intermediate x-integrals are evaluated in the saddle point approximation. Toleading order in h̄, only those intermediate x-values contribute which lie on the classical trajectory determined by the endpoints of the combined amplitude. To nextorder in h̄, the quadratic correction to the intermediate integrals renders an inversesquare root of the fluctuation determinant. If two such amplitudes are connectedwith each other by an intermediate integration according to the composition law(4.126), the product of the two fluctuation factors turns into the correct fluctuation factor of the combined time interval.
This is seen after rewriting the matrix∂xi ∂xja A(xb , xa ; tb −ta ) with the help of (4.12) as ∂pb /∂xa . The intermediate integralbover x in the product of two amplitudes receives a contribution only from continuouspaths since, at the saddle point, the adjacent momenta have to be equal:∂∂A(xb , x; tb − t) +A(x, xa ; t − ta ) = −p0 (xb , x; tb − t) + p(x, xa ; t − ta ) = 0.∂x∂x(4.127)To obtain the combined amplitude, we obviously need the relationdetD − ∂pb detD −∂x xb 0∂p ∂x +xb∂p ∂x xa p0 =p∂pb ,= detD −∂xa xb−1detD !∂p −∂xa x(4.128)where we have indicated explicitly the variables kept fixed in p0 (xb , x; tb − t) andp(x, xa ; t − ta ) when forming the partial derivatives. To prove (4.128), we use theproduct rule for determinants ∂pb ∂pb ∂x −−det−1det=detDDD∂x xb∂xa xb∂xa xb(4.129)to rewrite (4.128) asdetD∂p ∂p0 ∂p = detD − +−∂xa x∂x xb∂x xa !p0 =p ∂x detD .∂xa xb(4.130)This equation is true due to the chain rule of differentiation applied to the momentum p0 (xb , x; tb − t)= p(x, xa ; t − ta ), after expressing p(x, xa ; t − ta ) explicitly interms of the variables xb and xa as p(x(xb , xa ; tb − ta ), xa ; t − ta ), to enable us tohold xb fixed in the second partial derivative:∂p0 ∂p ∂p(x(xb , xa ; tb − ta ), xa ; t − ta ) ∂p ∂xa ∂p = = = + .∂x xb∂x xb∂x∂x x ∂x xb∂x xaxb(4.131)3884 Semiclassical Time Evolution AmplitudeIt may be expected, and can indeed be proved, that it is possible to proceed inthe opposite direction and derive the semiclassical expressions (4.119) and (4.121)with the Van Vleck-Pauli-Morette determinant from the fundamental compositionlaw (4.126).5In the semiclassical approximation, the composition law (4.126) can also be written as a temporal integral (in D dimensions)(xb tb |xa ta ) =Zdt(xb tb |xcl (t)t) ẋcl (t) (xcl (t)t|xa ta )(4.132)over a classical orbit xcl (t), where the t-integration is done in the saddle pointapproximation, assuming that the fluctuation determinant does not happen to bedegenerate.Just as in the saddle point expansion of ordinary integrals, it is possible tocalculate higher corrections in h̄.
The result is a saddle point expansion of the pathintegral which is again a semiclassical expansion. The counting of the h̄-powers isthe same as for the integral. The lowest approximation is of the exponential formeiAcl /h̄ . Thus, in the exponent, the leading term is of order 1/h̄.6 The fluctuationfactor F contributes to this an additive term log F , which is of order h̄0 . To firstorder in h̄, one finds expressions containing the third and fourth functional derivativeof the action in the expansion (4.92), corresponding to the expressions (4.42) and(4.43) in the integral. Unfortunately, the functional case offers little opportunity forfurther analytic corrections, so we shall not dwell on this more academic possibility.4.5Semiclassical Fixed-Energy AmplitudeAs pointed out at the end of Subsection 4.2.1, we have observed that the semiclassical approximation allows for a simple evaluation of Fourier integrals.
As anapplication of the rules presented there, let us evaluate the Fourier transform of thetime evolution amplitude, the fixed-energy amplitude introduced in (1.308). It isgiven by the temporal integral(xb |xa )E = √12πih̄Z∞tadtb [−∂xb ∂xa A(xb , xa ; tb − ta )]1/2× ei[A(xb ,xa ;tb −ta )+(tb −ta )E]/h̄ ,(4.133)which may be evaluated in the same saddle point approximation as the path integral.The extremum lies at∂A(xb , xa ; tb − ta ) = −E.(4.134)∂t5H. Kleinert and B. Van den Bossche, Berlin preprint 2000 (http://www.physik.fu-berlin.de/~kleinert/301).6Since h̄ has the dimension of an action, the dimensionless number h̄/Acl should really beused as an appropriate dimensionless expansion parameter, but it has become customary to countdirectly the orders in h̄.H.
Kleinert, PATH INTEGRALS4.5 Semiclassical Fixed-Energy Amplitude389Because of (4.86), the left-hand side is the function −E(xb , xa ; tb − ta ). At theextremum, the time interval tb − ta is some function of the endpoints and the energyE:tb − ta = t(xb , xa ; E).(4.135)The exponent is equal to the eikonal function S(xb , xa ; E) of Eq. (4.80), whosederivative with respect to the energy gives [recalling (4.85)]∂S(xb , xa ; E) = t(xb , xa ; E).∂E(4.136)The expansion of the exponent around the extremum has the quadratic termi ∂ 2 A(xb , xa ; tb − ta )[tb − ta − t(xb , xa ; E)]2 .2h̄∂tb(4.137)The time integral over tb yields a factor√∂ 2 A(xb , xa ; tb − ta )2πih̄∂t2b"#−1/2.(4.138)With this, the fixed-energy amplitude has precisely the form (4.48):h(xb |xa )E = −∂xb ∂xa A(xb , xa ; t)/∂t2 A(xb , xa ; t)i1/2eiS(xb ,xa ;E)/h̄ .(4.139)Since the fluctuation factor has to be evaluated at a fixed energy E, it is advantageous to express it in terms of S(xb , xa ; E).
For ∂t2 A, the evaluation is simplesince∂2A∂t∂E=−2 = −∂t∂E∂t!−1∂2S=−∂E 2!−1.(4.140)For ∂xb ∂xa A, we observe that the spatial derivatives of the action must be performed at a fixed time, so that a variation of xb implies also a change of the energyE(xb , xa ; t).
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