The WKB-method for a potential minimum

Jaroslav Zamastil

In this paper we present a method for obtaining the WKB approximation for non-separable multidimensional potentials. At its leading order one has to solve the classical Hamilton-Jacobi equation with zero energy for which a very efficient method is proposed. The essence of this method lies in the recognition that in the vicinity to the potential minimum the solution of Schrödinger equation has to approach that of coupled harmonic oscillators. Quantum corrections to the semiclassical result are then obtained very easily and to an arbitrarily high order. The method is applied to the calculation of the tunneling splitting and tunneling lifetime. We show that classical turning points are part of the dynamical problem; they could not be determined simply from looking at the form of the potential, but are obtained from the solution of the pertinent equations of motion.

K. T. Hecht

Michael Martin Nieto, L. M. Simmons, Vincent P. Gutschick

The minimum-uncertainty coherent states empirically provide an approximation to the motion of a classical particle. This can be understood conceptually. One can obtain a relationship between the WKB approximation and the definition of the minimum-uncertainty coherent states. This definition is in terms of the "natural" quantum operators, which connect only adjacent energy eigenstates. The classical form of these operators is also related to the WKB approximation. In the Appendix we comment on the origin of "exact" WKB results.

Brian C. Hall

X. W. Wang, D. L. Lin, F. T. Hioe

Mean switching time of a two-mode laser is calculated in two different approaches: the first-passage-time treatment by the generalized Kramers method and the lifetime approach by the two-dimensional WKB method. In both cases, the most probable path is generally curved. Numerical study in the WKB calculation reveals that the general curved most probable path can be replaced with high accuracy by the path specified by the minimum potential for many cases of practical interest, that is, when the mode-coupling constant \ensuremath{\xi} is within the range 1<\ensuremath{\xi}<5. Thus, we are able to carry out calculations of the mean switching time that are accurate (within the framework of the WKB approximation) throughout the entire range, above threshold, of the pump parameter a. Compared with the analytical formula which is valid only for large a, our result yields perfect agreement in the large-a limit, but shows that the previously used formula consistently overestimates the mean switching time when a is not large where the experiments are performed.

Pedro Pereyra Padilla