Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

Comment on “Exactly solvable models to illustrate supersymmetry and test approximation methods in quantum mechanics,” Am. J. Phys. 79, 755–761 (2011)

Donald W. L. Sprung, Akbar Safari, Nicolas Sator

Published Jul 18, 2012• 4 citations
Publisher

Abstract

In the paper under discussion, it was observed that the accuracy of WKB approximate binding energies for the exponential potential deteriorates as the energy increases, an apparent violation of the correspondence principle. We show that a careful application of the WKB approximation as it applies to the centrifugal barrier potential leads to improved approximate binding energies of the exponential potential. Our analysis supports the Friedrich and Trost “reflection phase” version of the WKB quantization condition.

DOI
10.1119/1.3699066

Related papers

Some Recent Developments in WKB Approximation

Akbar Safari

WKB theory provides a plausible link between classical mechanics and quantum mechanics in its semi-classical limit. Connecting the WKB wave function across the turning points in a quantum well, leads to the WKB quantization condition. In this work, I focus on some improvements and recent developments related to the WKB quantization condition. First I discuss how the combination of super-symmetric quantum mechanics and WKB, gives the SWKB quantization condition which is exact for a large class of potentials called shape invariant potentials. Next I turn to the fact that there is always a probability of refection when the potential is not constant and the phase of the wave function should account for this refection. WKB theory ignores refection except at turning points. I explain the work of Friedrich and Trost who showed that by including the correct "refection phase" at a turning point, the WKB quantization condition can be made to give exact bound state energies. Next I discuss the work of Cao and collaborators which takes refection of the wave function into account everywhere. We show that Cao's method provides a way to compute the F-T refection phase. Finally I discuss a paper of Fabre and Guery-Odelin who used the exponential potential to study the accuracy of WKB. In their results the accuracy deteriorates as the energy increases, which is inconsistent with Bohr's correspondence principle. Using the Friedrich and Trost method, we resolved this problem.

•Sep 18, 2013•3 citations

A WKB-Type Approximation to the Schrödinger Equation

Stanley C. Miller, Robert H. Good

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.

Physical Review•Jul 1, 1953•495 citations

Accuracy of the WKB Approximation: The Case of General Quartic Potential

Marko Vranivcar, Marko Robnik

We analyse the accuracy of the approximate WKB quantization for the case of general one-dimensional quartic potential. In particular, we are interested in the validity of semiclassically predicted energy eigenvalues when approaching the limit E→∞, and in the accuracy of low lying energy levels below the potential barrier in the case of generally asymmetric double-well quartic potential. In the latter case, using the standard WKB quantization an unnatural localization of eigenstates due to the negligence of tunneling is implied and thus the validity of semiclassics is uncertain. In all computations the higher order corrections to the leading semiclassical approximation are included using the complex contour integration technique. We show that these corrections can improve accuracy of semiclassical approximation greatly by many orders of magnitude.

arXiv:nlin/0003051•Progress of Theoretical Physics Supplement•Jan 1, 2000•8 citations

High Order WKB Prediction of the Energy Splitting in the Symmetric Double Well Potential

Marko Robnik, Luca Salasnich, Marko Vrani\v car

The accuracy of the WKB approximation when predicting the energy splitting of bound states in a double well potential is the main subject of this paper. The splitting of almost degenerate energy levels below the top of the barrier results from the tunneling and is thus supposed to be exponentially small. By using the standard WKB quantization we deduce an analytical formula for the energy splitting, which is the usual Landau formula with additional quantum corrections. We also examine the accuracy of our and Landau formula numerically for the case of the symmetric double well quartic potential.

arXiv:nlin/0003050•Progress of Theoretical Physics Supplement•Jan 1, 2000•1 citation

Zeroth WKB Approximation in Quantum Mechanics

М. Н. Сергеенко

Solution of the Schrödinger's equation in the zero order WKB approximation is analyzed. We observe and investigate several remarkable features of the WKB$_0$ 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, $A_n\cos(k_nx+δ_n)$, and are the asymptote of the exact solution.

arXiv:quant-ph/0206179•Jun 25, 2002•3 citations

Bohr–Sommerfeld, WKB, and modified semiclassical quantization rules

D. J. W. Geldart, David Kiang

The WKB approximation for energy eigenvalues can be advantageously introduced to students when the Bohr–Sommerfeld quantization rule is discussed. A simple heuristic derivation of these quantization rules is given. Exceptional cases occur when conditions necessary for the validity of semiclassical approximations are violated, but appropriate modifications can still provide useful semiclassical approximations even in such cases. Various situations are illustrated by examples.

American Journal of Physics•Feb 1, 1986•11 citations
In the paper under discussion, it was observed that the accuracy of WKB approximate binding energies for the exponential potential deteriorates as the energy increases, an apparent violation of the correspondence principle. We show that a careful application of the WKB approximation as it applies to the centrifugal barrier potential leads to improved approximate binding energies of the exponential potential. Our analysis supports the Friedrich and Trost “reflection phase” version of the WKB quantization condition.
WKB theory provides a plausible link between classical mechanics and quantum mechanics in its semi-classical limit. Connecting the WKB wave function across the turning points in a quantum well, leads to the WKB quantization condition. In this work, I focus on some improvements and recent developments related to the WKB quantization condition. First I discuss how the combination of super-symmetric quantum mechanics and WKB, gives the SWKB quantization condition which is exact for a large class of potentials called shape invariant potentials. Next I turn to the fact that there is always a probability of refection when the potential is not constant and the phase of the wave function should account for this refection. WKB theory ignores refection except at turning points. I explain the work of Friedrich and Trost who showed that by including the correct "refection phase" at a turning point, the WKB quantization condition can be made to give exact bound state energies. Next I discuss the work of Cao and collaborators which takes refection of the wave function into account everywhere. We show that Cao's method provides a way to compute the F-T refection phase. Finally I discuss a paper of Fabre and Guery-Odelin who used the exponential potential to study the accuracy of WKB. In their results the accuracy deteriorates as the energy increases, which is inconsistent with Bohr's correspondence principle. Using the Friedrich and Trost method, we resolved this problem.
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.
We analyse the accuracy of the approximate WKB quantization for the case of general one-dimensional quartic potential. In particular, we are interested in the validity of semiclassically predicted energy eigenvalues when approaching the limit E→∞, and in the accuracy of low lying energy levels below the potential barrier in the case of generally asymmetric double-well quartic potential. In the latter case, using the standard WKB quantization an unnatural localization of eigenstates due to the negligence of tunneling is implied and thus the validity of semiclassics is uncertain. In all computations the higher order corrections to the leading semiclassical approximation are included using the complex contour integration technique. We show that these corrections can improve accuracy of semiclassical approximation greatly by many orders of magnitude.
The accuracy of the WKB approximation when predicting the energy splitting of bound states in a double well potential is the main subject of this paper. The splitting of almost degenerate energy levels below the top of the barrier results from the tunneling and is thus supposed to be exponentially small. By using the standard WKB quantization we deduce an analytical formula for the energy splitting, which is the usual Landau formula with additional quantum corrections. We also examine the accuracy of our and Landau formula numerically for the case of the symmetric double well quartic potential.
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.