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arXiv:hep-ph/9503225

An Anti-WKB Approximation

John B. Bronzan

Published Mar 3, 1995• 0 citations
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Abstract

In the WKB approximation the $\nabla^2S$ term in Schrodinger's equation is subordinate to the |\nabla S|^2 term. Here we study an anti-WKB approximation in which the $\nabla^2 S$ term dominates (after a guess for S is supplied). Our approximation produces only the nodeless ground state wavefunction, but it can be used in potential problems where the potential is not symmetric, and in problems where there are many degrees of freedom. As a test, we apply the method to potential problems, including the hydrogen and helium atoms and to $ϕ^4$ field theory.

DOI
10.48550/arxiv.hep-ph/9503225

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In the WKB approximation the ∇2S\nabla^2S∇2S term in Schrodinger's equation is subordinate to the |\nabla S|^2 term. Here we study an anti-WKB approximation in which the ∇2S\nabla^2 S∇2S term dominates (after a guess for S is supplied). Our approximation produces only the nodeless ground state wavefunction, but it can be used in potential problems where the potential is not symmetric, and in problems where there are many degrees of freedom. As a test, we apply the method to potential problems, including the hydrogen and helium atoms and to ϕ4ϕ^4ϕ4 field theory.
The relative convergences of the WKB and SWKB approximations are examined by calculating the eigenenergies for four potentials by one- and two-term WKB and SWKB approximations. Exact eigenenergies for these four potentials are also calculated by a numerical integration of the Schrodinger equation. Varied results are found for the four potentials. It is found that in general the effect of the second term in the WKB and SWKB approximations depends on the potential, the parameters involved and the quantum number of the state. No simple generalizations are possible.
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.
The WKB wavefunctions for one-dimensional nonrelativistic problems are derived by a simple transformation method. By applying the Liouville substitution to Schrödinger's equation and then neglecting certain “small” terms in the transformed equation, one arrives at the WKB solutions. The terms neglected in obtaining the WKB solutions are explicit. No asymptotic expansions are used in the derivation. Use of a known criterion from the theory of differential equations gives assurance that the WKB solutions are asymptotic to exact solutions of Schrödinger's equation in certain cases.
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.
In this chapter we study another very important and powerful approximation method, the so-called eikonal approximation or Wentzel-Kramer-Brillouin (WKB) method, which is used in studying quantum mechanical systems that are subjected to complicated potentials. This method can be used to find approximate solutions to differential equations where the highest derivative is multiplied by a small parameter. The WKB method gives approximate solution to the Schrödinger equation irrespective of its complicated potential and is mostly used to study one-dimensional systems and also the radial equations for higher-dimensional systems with rotational symmetry.