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Journal of Mathematical Physics

The semiclassical expansion of the anharmonic-oscillator propagator

Maurice M. Mizrahi

Published May 1, 1979• 29 citations
Publisher

Abstract

This paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum-mechanical propagator corresponding to the quartic anharmonic-oscillator potential, V=mω2q2/2 +λq4/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt-Morette, which does not entail the usual time-slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small-disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tends to its harmonic counterpart when λ→0.

DOI
10.1063/1.524157

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The general method of path integrals in quantum mechanics introduced by R.P.Feynman is applied in this paper to discuss the amplitude of transition probability in quantum mechanics, that is, the subject of the propagator.Mauriel M.Mizrahi's semiclassical (WKB) approximation method has been used in studying the general expansion in detail for the propagator corresponding to an arbitrary Hamiltonian operator H. ?? where S_c is the action functional evaluated at the classical path (q_c, p_c) , ρ is a constant, ?? Maurice M.Mizrahi's solution of the K_(WKB) expansion has been generalized, and another linear independent particular solution is obtained.This is the solution when 0~2 = det M equals zero.The K_(WKB) can be applied to all systems corresponding to an arbitrary Hamiltonian operator.

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Reports on Progress in Physics•Jan 1, 1972•1,424 citations
This paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum-mechanical propagator corresponding to the quartic anharmonic-oscillator potential, V=mω2q2/2 +λq4/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt-Morette, which does not entail the usual time-slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small-disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tends to its harmonic counterpart when λ→0.
The general method of path integrals in quantum mechanics introduced by R.P.Feynman is applied in this paper to discuss the amplitude of transition probability in quantum mechanics, that is, the subject of the propagator.Mauriel M.Mizrahi's semiclassical (WKB) approximation method has been used in studying the general expansion in detail for the propagator corresponding to an arbitrary Hamiltonian operator H. ?? where S_c is the action functional evaluated at the classical path (q_c, p_c) , ρ is a constant, ?? Maurice M.Mizrahi's solution of the K_(WKB) expansion has been generalized, and another linear independent particular solution is obtained.This is the solution when 0~2 = det M equals zero.The K_(WKB) can be applied to all systems corresponding to an arbitrary Hamiltonian operator.
In quantum mechanics, the propagator is basically the most important; It contains all physical properties, Feynman's path integration is widely accepted to be the formal method for finding the propagator. However, the method is limited to only some type of problems. One challenging and interesting problem which has not been able to be solved since the time of Feynman is potential with quadratic and cubic terms. The problem is analysed in detail by using Baker - Hausdorff lemma in order to find the possibility of the best approximation; It is found that the best practical one is the first-order semiclassical approximation by using condition the cubic potential term has value very smaller than the quadratic potential term. The approximated propagator has been obtained for quadratic and cubic potential terms when it is analogous to the propagator with quadratic potential term, we have increased prefactor and exponential terms due to the influence of cubic potential term which expresses the anharmonic part of the propagator. The approximated propagator is reduced to the well-known propagator for the simple harmonic oscillator when the anharmomc part is zero.
The evaluation of the average value of the position coordinate, 〈x〉, of a particle moving in a harmonic oscillator potential (V(x)=kx2/2) with a small anharmonic piece (V(x)=−λkx3) is a standard calculation in classical Newtonian mechanics and statistical mechanics where the problem has relevance to thermal expansion. In each case, the calculation is most easily done using a perturbative expansion. In this note, we perform the same computation of 〈x〉 in quantum mechanics using time-independent perturbation theory and the ladder operator formalism to show how similar results are obtained. We also indicate how a semiclassical calculation using a classical probability distribution can also be used to obtain the same result.
The time-dependent WKB approximation for a coherent state is expanded to third order around a guiding real trajectory, allowing for the novel treatment of nonlinearity in its semiclassical dynamics, which is generally present in all physical systems far from the classical regime. The result is a closed-form solution consisting of a linear combination of Airy functions and their derivatives multiplied by an exponential. The expression's ability to capture nonlinearity is demonstrated by examining the autocorrelation of initial coherent states in anharmonic systems with few to many contributing periodic orbits. Its accuracy is compared to the quadratic expansion and found to be superior in regimes of ℏ where the curvature begins to be significant, as expected. Moreover, the expression is shown to be a real-trajectory uniformization over two coalescing saddle points that are emblematic of significant curvature. This extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time and establishes their regime of validity.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.