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Physical Review

Extension of the WKB Equation

Charles E. Hecht, Joseph E. Mayer

Published Jun 15, 1957• 44 citations
Publisher

Abstract

The WKB form suitable for the classical region of potential energy less than the total energy, $\ensuremath{\psi}(x)={(\frac{\mathrm{dz}}{\mathrm{dx}})}^{\ensuremath{-}\frac{1}{2}}sinz$, is used to obtain a function $z(x)$ that reproduces the time-independent Schr\"odinger equation solutions $\ensuremath{\psi}(x)$, to any desired degree of accuracy through the turning point and into the nonclassical region.

DOI
10.1103/physrev.106.1156

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The WKB form suitable for the classical region of potential energy less than the total energy, \ensuremathψ(x)=(dzdx)\ensuremath−12sinz\ensuremath{\psi}(x)={(\frac{\mathrm{dz}}{\mathrm{dx}})}^{\ensuremath{-}\frac{1}{2}}sinz\ensuremathψ(x)=(dxdz​)\ensuremath−21​sinz, is used to obtain a function z(x)z(x)z(x) that reproduces the time-independent Schr"odinger equation solutions \ensuremathψ(x)\ensuremath{\psi}(x)\ensuremathψ(x), to any desired degree of accuracy through the turning point and into the nonclassical region.
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.
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.
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.