Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 78
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Weiss, Quantum Dissipative Systems, World Scientific, Singapore, 1993.See also Notes and References in Chapter 18.For alternative approaches to the damped oscillator seeF. Haake and R. Reibold, Phys. Rev. A 32, 2462 (1985),A. Hanke and W. Zwerger, Phys. Rev. E 52, 6875 (1995);S. Kehrein and A. Mielke, Ann. Phys.
(Leipzig) 6, 90 (1997) (cond-mat/9701123).X.L. Li, G.W. Ford, and R.F. O’Connell, Phys. Rev. A 42, 4519 (1990).The effective potential (5.259) was derived in D dimensions byH. Kleinert and B. Van den Bossche, Nucl. Phys. B 632, 51 (2002) (http://arxiv.org/abs/cond-mat/0104102">cond-mat/0104102).By inserting D = 1 and changing the notation appropriately, one finds (5.259).The finite-temperature expressions (3.760)–(3.763) are taken from S.F. Brandt, Beyond EffectivePotential via Variational Perturbation Theory, M.S. thesis, FU-Berlin 2004 (http://hbar.wustl.edu/~sbrandt/diplomarbeit.pdf).
See alsoS.F. Brandt, H. Kleinert, and A. Pelster, J. Math. Phys. 46, 032101 (2005) (quant-ph/0406206) .H. Kleinert, PATH INTEGRALSNovember 20, 2006 ( /home/kleinert/kleinert/books/pathis3/pthic4.tex)Take the gentle path.George Herbert (1593-1633), Discipline4Semiclassical Time Evolution AmplitudeThe path integral approach renders a clear intuitive understanding of quantummechanical effects in terms of quantum fluctuations, exhibiting precisely how thelaws of classical mechanics are modified by these fluctuations. In some limitingsituations, the modifications may be small, for instance, if an electron in an atomis highly excited. Its wave packet encircles the nucleus in almost the same wayas a point particle in classical mechanics.
Then it is relatively easy to calculatequite accurate quantum-mechanical amplitudes by expanding them around classicalexpressions in powers of the fluctuation width.4.1Wentzel-Kramers-Brillouin (WKB)ApproximationIn Schrödinger’s theory, an important step towards understanding the relation between classical and quantum mechanics consists in proving that the center of aSchrödinger wave packet moves like a classical particle. The approach to the classicallimit is described by the so-called eikonal approximation, or the Wentzel-KramersBrillouin approximation (short: WKB approximation), which proceeds as follows:First, one rewrites the time-independent Schrödinger equation of a point particle(in the formh̄2 2∇ − [E − V (x)] ψ(x) = 0−2M)hi−h̄2 ∇2 − p2 (x) ψ(x) = 0,wherep(x) ≡q2M[E − V (x)](4.1)(4.2)(4.3)is the local classical momentum of the particle.In a second step, one re-expresses the wave function as an exponentialψ(x) = eiS(x)/h̄ .368(4.4)4.1 Wentzel-Kramers-Brillouin (WKB) Approximation369For the exponent S(x), called the eikonal , the Schrödinger equation amounts to thea differential equation :−ih̄∇2 S(x) + [∇S(x)]2 − p2 (x) = 0.(4.5)To solve this equation approximately, one assumes that the function p(x) showslittle relative change over the de Broglie wavelengthλ≡2πh̄2π≡,k(x)p(x)i.e.,2πh̄ ∇p(x) ε≡ 1.p(x) p(x) (4.6)(4.7)This condition is called the WKB condition.
In the extreme case of p(x) being aconstant the condition is certainly fulfilled and the Riccati equation is solved by thetrivial eikonalS = px + const ,(4.8)which makes (4.4) a plain wave. For slow variations, the first term in the Riccatiequation is much smaller than the others and can be treated systematically in a“smoothness expansion”. Since the small ratio (4.7) carries a prefactor h̄, the Planckconstant may be used to count the powers of the smallness parameter ε, i.e., it mayformally be considered as a small expansion parameter.The limit h̄ → 0 of the equation determines the lowest-order approximation tothe eikonal, S0 (x), by[∇S0 (x)]2 − p2 (x) = 0.(4.9)Being independent of h̄, this is a classical equation.
Indeed, it is equivalent to theHamilton-Jacobi differential equation of classical mechanics: For time-independentsystems, the action can be written asA(x, t) = S0 (x) − tE,where S0 (x) is defined byS0 (x) ≡Zdt p(t)ẋ(t),(4.10)(4.11)and E is the constant energy of the orbit under consideration. The action solves theHamilton-Jacobi equation (1.64). In three dimensions, it is a function A(x, t) of theorbital endpoints. According to Eq.
(1.62), the derivative of A(x, t) is equal to themomentum p. Since E is a constant, the same thing holds for S0 (x, t). Hencep ≡ ∇A(x) = ∇S0 (x).(4.12)In terms of the action A, the Hamilton-Jacobi equation reads1(∇A)2 + V (x) = −∂t A.2M(4.13)3704 Semiclassical Time Evolution AmplitudeBy inserting Eqs. (4.10) and (4.12), we recover Eq. (4.9). This is why S0 (x) is alsocalled the classical eikonal .The corrections to the classical eikonal are calculated most systematically byimagining h̄ 6= 0 to be a small quantity and expanding the eikonal around S0 (x) ina power series in h̄:S = S0 − ih̄S1 + (−ih̄)2 S2 + (−ih̄)3 S3 + .
. . .(4.14)This is called the semiclassical expansion of the eikonal. Inserting it into the Riccatiequation, we find the sequence of WKB equations(∇S0 )2 − p2∇2 S0 + 2∇S0 · ∇S1∇2 S1 + (∇S1 )2 + 2∇S0 · ∇S2∇2 S2 + 2∇S1 · ∇S2 + 2∇S0 · ∇S3...∇ 2 Sn +n+1Xm=0====0,0,0,0,∇Sm · ∇Sn+1−m = 0,...(4.15).Note that these equations involve only the vectorsqn = ∇Sn(4.16)and allow for a successive determination of S0 , S1 , S2 , . . . , giving higher and highercorrections to the eikonal S(x).In one dimension we recognize in (4.5) the√ Riccati differential equation (2.531)fulfilled by ∂τ S(x), if we identify x with τ 2M, and v(τ ) with E − V (τ ), where(4.5) reads−ih̄∂τ [∂τ S(τ )] + [∂τ S(τ )]2 = p2 (τ ) = v(τ ).(4.17)If we re-express the expansion terms (2.535) in terms of w(τ ) =replace w(τ ), w 0(τ ), . .
. by p( x), p0 (x), . . . , and find directly1 p0 (x),q0 = ±p(x), q1 = −2 p(x)"#1 p00 (x) 3 p02 (x)q2 = ±−≡ ∓p(x)(x),4 p2 (x) 8 p3 (x)qv(τ ), we may1q3 = 0 (x), . . . .2(4.18)The equation for q2 (x) defines also the quantity g(x) used in subsequent equations.The eikonal has the expansionS(x) = ±Zdx p(x)[1 + h̄2 g(x)]i+h̄ [log p(x) + h̄2 g(x)] ± . . . .2(4.19)H. Kleinert, PATH INTEGRALS4.1 Wentzel-Kramers-Brillouin (WKB) Approximation371Keeping only terms up to the order h̄, which is possible if h̄2 |(x)| 1, we find the(as yet unnormalized) WKB wave functionRx 0 01ψWKB (x) = qe±(i/h̄) dx p(x ) .p(x)(4.20)In the classically accessible regime V (x) ≤ E, this is an oscillating wave function;in the inaccessible regime V (x) ≥ E, it decreases or increases exponentially. Thetransition from one to the other is nontrivial since for V (x) ≈ E, the WKB approximation breaks down.
After some analytic work1 , however, it is possible to derivesimple connection rules for the linearly independent solutions. Let k(x) ≡ p(x)/h̄in the oscillating regime and κ(x) ≡ |p(x)|/h̄ in the exponential regime. Supposethat there is a crossover at x = a connecting an inaccessible regime on the left ofx = a with an accessible one on the right. Then the connection rules areV (x) > E1 − R a dx0 κ√ e xκ1 R a dx0 κ√ exκV (x) < Ex2√ cos←−−−−−→−→dx0 k −akZ x1dx0 k −←−←−−−−→ − √ sinakZπ,4π.4(4.21)(4.22)In the opposite situation at the point x = b, they turn intoV (x) < Eb2√ cosdx0 k −xkZ b1dx0 k −− √ sinxkZ!π4!π4←−←−−−−→←−−−−−→−→V (x) > E1 − R x dx0 κ√ e b,κ2 Rx 0√ e b dx κ .κ(4.23)(4.24)The connection rules can be used safely only along the direction of the double arrows.For their derivation one solves the Schrödinger equation exactly in the neighborhood of the turning points where the potential rises or falls approximately linearly.These solutions are connected with adjacent WKB wave functions.
The connectionformulas can also be found directly by a formal trick: When approaching the dangerous turning points, one escapes into the complex x-plane and passes around thesingularities at a finite distance. This has to be sufficiently large to preserve theWKB condition (4.7), but small enough to allow for the linear approximation of thepotential near the turning point. Take for example the rule (4.23). When approaching the turning√ point at x = b from the right, the function κ(x) is approximately proportional to x − b. Going around this zero in the upper complex half-plane takes1R.E.
Langer, Phys. Rev. 51 , 669 (1937). See also W.H. Furry, Phys. Rev. 71 , 360 (1947),and the textbooks S. Flügge, Practical Quantum Mechanics, Springer, Berlin, 1974; L.I. Schiff,Quantum Mechanics, McGraw-Hill, New York, 1955; N. Fröman and P.O. Fröman, JWKBApproximation, North-Holland, Amsterdam, 1965.3724 Semiclassical Time Evolution Amplitude√ −1 −i R x dx0 kκ(x) into ik(x) and the wave function κ ebecomes ek e Rb.√−1 i x dx0 kiπ/4Going around the turning point in the lower half-plane produces ek e b.√ −1Rb0The sum of the two terms is 2 k cos( x dx k − π/4). The argument does notshow why one should use the sum rather than the average.
This becomes clear onlyafter a more detailed discussion found in quantum-mechanical textbooks2 . The simplest derivation of the connection formulas is based on the large-distance behaviors(2.601) and (2.606) of wave the function to the right and left of a linearly risingpotential and applying this to the linearly rising section of the general potentialnear the turning point.In a simple potential well, the function p(x) has two zeros, say one at x = a andone at x = b.
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