Multidimensional WKB approximation for particle tunneling.

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.

Quantum effects and quantum chaos in multidimensional tunneling.

The ground-state energy splitting due to tunneling in two-dimensional double wells of the form V(x,y)=(x^{2}-R^{2})^{2}/8R^{2}+x^{2}-R^{2}/R^{2}γy+ω^{2}/2y^{2} is calculated. Several results are reported. First, we give a systematic WKB expansion of the splitting in series in powers of R^{-2}, each term of the series being a finite polynomial in γ^{2}. We find an ascending sequence of the values of the parameter γ characterizing the curvature of the classical path, for which the successive corrections to the leading order vanish. This effect arises because curvature of the path and quantum nature of motion cancel each other; it does not appear for one-dimensional double wells. Second, we find that for large curvatures, such as for those describing the proton transfer in a malonaldehyde and hydroxalate anion, this expansion is of no practical use. Thus, the WKB expansion is reordered to a strong coupling form, each term of the series in powers of R^{-2} being an infinite series in powers of γ[over ¯]^{2}, γ[over ¯]=γ/R. Third, we find that the radius of convergence of the series is determined by the singularity at γ[over ¯]_{s}=ω/2. At the singularity the system changes its character from being a double well to become a single well. Close to this singularity the classical action and its first quantum correction are found to be nonanalytic functions of γ[over ¯], most likely of the form [1-(γ[over ¯]/γ[over ¯]_{s})^{2}]^{α}, where α=1/2 and α=-1/2 for the classical action and its first quantum correction, respectively. Since in the semiclassical regime of large R the splitting is exponentially dependent on the value of the classical action and its first quantum correction, close to the singularity we establish strong sensitivity of the splitting on slight variations of the parameter γ[over ¯] entering the Hamiltonian linearly.

WKB approach to calculating the lifetime of quasistationary states.

A novel WKB approach to calculating the lifetime of quasistationary states in the potential wells of the form V(x)=P(x)-muQ(x), where P(x) is the radial part of the potential for the spherically symmetric harmonic oscillator or the hydrogen atom and Q(x) is a polynomial, is suggested. In this approach, the usual explicit procedure of the asymptotic matching of the perturbative and WKB wave functions is avoided and a simple formula for the imaginary part of the energy is found. The leading and the first correction terms for the imaginary part of the energy and the related lifetime are analytically calculated.