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Reviews in Mathematical Physics

On Time Development of a Quasi-Quantum Particle in Quartic Potential (x2-a2)2/2g

Shigeki Matsutani

Published Nov 1, 1997• 4 citations

Abstract

In this article, I have precisely considered the time development of a quantum particle (of excited states) in the quartic potential (x2-a2)2/2g by means of the semiclassical path integral method. Using the elliptic functions, I have evaluated the tunneling phenomena and the quasi-quantum fluctuation around the quasi-classical paths. I found that the quasi-quantum fluctuation is expressed by the Lamé equation and was exactly solved. Then I have shown that the obtained kernel function is in agreement with exact solutions of the linear potential and the quadratic potential under certain limits as no time-development kernel function of the quartic potential has ever been found which contains the exact solution of the linear and the quadratic potential. It is natural because the classical motion in the quartic potential becomes those of the linear and the quadratic potential under the limits. Thus the obtained time-development kernel function also consists of the energy representation of the Green function of the quartic potential in the semiclassical path integral method given by Carlitz and Nicole (Ann. Phys.164 (1985) 411), which agrees with that of the WKB method in the operator formalism.

DOI: 10.1142/s0129055x97000336

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In this article, I have precisely considered the time development of a quantum particle (of excited states) in the quartic potential (x2-a2)2/2g by means of the semiclassical path integral method. Using the elliptic functions, I have evaluated the tunneling phenomena and the quasi-quantum fluctuation around the quasi-classical paths. I found that the quasi-quantum fluctuation is expressed by the Lamé equation and was exactly solved. Then I have shown that the obtained kernel function is in agreement with exact solutions of the linear potential and the quadratic potential under certain limits as no time-development kernel function of the quartic potential has ever been found which contains the exact solution of the linear and the quadratic potential. It is natural because the classical motion in the quartic potential becomes those of the linear and the quadratic potential under the limits. Thus the obtained time-development kernel function also consists of the energy representation of the Green function of the quartic potential in the semiclassical path integral method given by Carlitz and Nicole (Ann. Phys.164 (1985) 411), which agrees with that of the WKB method in the operator formalism.

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 quantum tunneling rate through an inverted parabola potential in an oscillating external field is studied. The tunneling rate is calculated within quasi-classical accuracy as a function of the frequency of the field. In this calculation, the complex-time path integral method, which has been previously applied to the case of tunneling through static potentials, is used after being appropriately extended for application to the case of tunneling through explicitly time-dependent potentials. Since the Feynman kernel for the case considered in the present paper can be exactly obtained, we are able to calculate the tunneling rate over a considerably wider range of frequencies of the oscillating field than in the case that this quantity is calculated through a perturbative treatment of the external field, as done in the previous studies. The present analysis reveals that the perturbative analysis for the present problem is insufficient even in the region where it has been considered to be appropriate. From the numerical calculation of the quasi-classical tunneling rate, we find that, as the frequency of the external field increases, all of the tunneling rates are initially enhanced by a significant amount, but beyond a certain value, all tunneling rates become rapidly decaying functions of this frequency. This fact may imply that the consideration beyond quasi-classical accuracy is needed in the high frequency regime. Tunneling processes that do not conserve energy are also considered. They cannot be neglected in this time-dependent problem, and we find that tunneling rates become large when tunneling particles acquire energy during tunneling.

The complex-time formalism is developed in the framework of the path-integral formalism, to be used for analysis of the quantum tunneling phenomena. We show that subleading complex-time saddle-points do not account for the right WKB result. Instead, we develop a reduction formula, which enables us to construct Green functions from simple components of the potential, for which saddle-point method is applicable. This method leads us to the valid WKB result, which incorporates imaginary-time instantons and bounces, as well as the real-time boundary conditions.

We investigate the quasi-exact solutions of the Schr?dinger wave equation for two-dimensional non-hermitian complex Hamiltonian systems within the frame work of an extended complex phase space characterized by x = x1 + ip3, y = x2 + ip4, px = p1 + ix3, py = p2 + ix4. Explicit expressions of the energy eigenvalues and the eigenfunctions for ground and first excited states for a complex quartic potential are obtained. Eigenvalue spectra of some variants of the complex quartic potential, including PT-symmetric one, are also worked out.

In order to study the validity of the functional integral formula of the Green's function, the large time limit of the time evolution kernel exp (-iHt) of any quantum system is discussed. With the Feynman prescription, giving an infinitesimally small negative imaginary part to quadratic term of the potential in the Hamiltonian, the correct formula for this limit is obtained. It differs from the naive formula and depends on the detailed form of the potential. The usual formula holds for the perturbative expansion in terms of the potential, the proof of which is given by the path integral method. The new expression for the ground state wave function is also obtained in the course of the proof.

It is shown that, in general, full agreement between the semiclassical approximation of the path integral and the WKB method of ordinary quantum mechanics is achieved only by the use of complex-time classical solutions. This technique is applied to compute the shift in energy levels due to tunneling and to calculate the lifetimes of virtual states. It is emphasized that the exponential time decay of the kernel is only approximately valid for finite time and that for $T\ensuremath{\rightarrow}\ensuremath{\infty}$ a power-law behavior applies. The Gaussian approximation of the kernel corroborates these statements.