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.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.

Physical review. EJul 19, 20185 citations

THE WKB APPROXIMATION AND ELECTRON TUNNELING

Albert Thomas Fromhold

Elsevier eBooksJan 1, 19811 citation

WKB Approximation in Quantum Mechanics

Boris Mikhailovich Karnakov, В. П. Крайнов

Springer eBooksAug 28, 20121 citation

Multidimensional WKB approximation for particle tunneling

Jaroslav Zamastil

A method for obtaining the WKB wave function describing the particle tunneling outside of a two-dimensional potential well is suggested. The Cartesian coordinates $(x,y)$ are chosen in such a way that the $x$ axis has the direction of the probability flux at large distances from the well. The WKB wave function is then obtained by simultaneous expansion of the wave function in the coordinate $y$ and the parameter determining the curvature of the escape path. It is argued, both physically and mathematically, that these two expansions are mutually consistent. It is shown that the method provides systematic approximation to the outgoing probability flux. Both the technical and conceptual advantages of this approach in comparison with the usual approach based on the solution of classical equations of motion are pointed out. The method is applied to the problem of the coupled anharmonic oscillators and verified through the dispersion relations.A method for obtaining the WKB wave function describing the particle tunneling outside of a two-dimensional potential well is suggested. The Cartesian coordinates $(x,y)$ are chosen in such a way that the $x$ axis has the direction of the probability flux at large distances from the well. The WKB wave function is then obtained by simultaneous expansion of the wave function in the coordinate $y$ and the parameter determining the curvature of the escape path. It is argued, both physically and mathematically, that these two expansions are mutually consistent. It is shown that the method provides systematic approximation to the outgoing probability flux. Both the technical and conceptual advantages of this approach in comparison with the usual approach based on the solution of classical equations of motion are pointed out. The method is applied to the problem of the coupled anharmonic oscillators and verified through the dispersion relations.

Physical Review AAug 10, 20058 citations

The probability of potential barrier penetration in the generalized wkB approximation

Н. И. Жирнов

Russian Physics JournalJan 1, 19693 citations

A method for obtaining the WKB wave function describing the particle tunneling outside of a two-dimensional potential well is suggested. The Cartesian coordinates

(x,y)

are chosen in such a way that the

x

axis has the direction of the probability flux at large distances from the well. The WKB wave function is then obtained by simultaneous expansion of the wave function in the coordinate

y

and the parameter determining the curvature of the escape path. It is argued, both physically and mathematically, that these two expansions are mutually consistent. It is shown that the method provides systematic approximation to the outgoing probability flux. Both the technical and conceptual advantages of this approach in comparison with the usual approach based on the solution of classical equations of motion are pointed out. The method is applied to the problem of the coupled anharmonic oscillators and verified through the dispersion relations.