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Physical Review

A WKB-Type Approximation to the Schrödinger Equation

Stanley C. Miller, Robert H. Good

Published Jul 1, 1953• 495 citations
Publisher

Abstract

A method of approximating solutions of the one-dimensional Schr\"odinger equation is presented in this paper. The method closely resembles the usual WKB approximation. Whereas in the ordinary WKB method the exponential function is used as the basis of the approximation, in this paper the solutions of an arbitrary Schr\"odinger equation are used. The general advantage is that by proper choice of the arbitrary equation an improved approximation can be obtained. The method is illustrated by treating the potential well and potential barrier problems when there are two turning points. The approximations to the wave functions are continuous even across the turning points. The barrier transmission problem is treated uniformly for energies above and below the peak of the barrier.

DOI
10.1103/physrev.91.174

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A method of approximating solutions of the one-dimensional Schr"odinger equation is presented in this paper. The method closely resembles the usual WKB approximation. Whereas in the ordinary WKB method the exponential function is used as the basis of the approximation, in this paper the solutions of an arbitrary Schr"odinger equation are used. The general advantage is that by proper choice of the arbitrary equation an improved approximation can be obtained. The method is illustrated by treating the potential well and potential barrier problems when there are two turning points. The approximations to the wave functions are continuous even across the turning points. The barrier transmission problem is treated uniformly for energies above and below the peak of the barrier.
Solution of the Schrödinger's equation in the zero order WKB approximation is analyzed. We observe and investigate several remarkable features of the WKB0_00​ method. Solution in the whole region is built with the help of simple connection formulas we derive from basic requirements of continuity and finiteness for the wave function in quantum mechanics. We show that, for conservative quantum systems, not only total energy, but also momentum is the constant of motion. We derive the quantization conditions for two and more turning point problems. Exact energy eigenvalues for solvable and some ``insoluble'' potentials are obtained. The eigenfunctions have the form of a standing wave, Ancos⁡(knx+δn)A_n\cos(k_nx+δ_n)An​cos(kn​x+δn​), and are the asymptote of the exact solution.
The Schrödinger equation is a fundamental equation in physics for describing quantum mechanical behavior. It is a partial differential equation that shows how a wave function of a physical system evolves over time. In addition, it is associated with the kinetic energy and the potential energy, both of which contribute towards the total energy. This paper explains mathematical aspect of the Schrödinger equation. The concept of the WKB approximation has also been reviewed, which is an important method in finding the approximate solutions for the wave function. In addition, the paper also shows how to derive the transmission and reflection probabilities in relation to a one-dimensional quantum problem, called a “tunneling problem”, which occurs when a classical particle passes through a region where total energy is less than its potential energy. The WKB approximation can be used as a basis for formally writing down the exact solutions. Furthermore, it gives a high accuracy for the propagation of waves with high frequency or short wavelength.
A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.
The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The WKB method is due to Gregor Wentzel, Hendrik Anthony Kramers and Leon Nicolas Brillouin. The method is applicable for slowly varying potentials, that is, the potentials that change not more than a few wavelengths. The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The Schrödinger equation is solved to find the penetration of a square potential barrier. The WKB method is applied to obtain a transmission coefficient for penetration through a barrier.