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

Australian Journal of Physics

Barrier Theory Based on Half-barrier Penetration

John Heading

Published Apr 1, 1980• 1 citation
Publisher

Abstract

Special methods are investigated for. the calculation of reflection and transmission coefficients for a real symmetric potential barrier. These methods, involving half-barrier penetration only, are both of physical and mathematical interest, and they also afford a considerable saving in time when numerical procedures are necessary in the case of analytically nonsoluble models.

DOI
10.1071/ph800307a

Related papers

The Penetration of a Potential Barrier by Electrons

Carl Eckart

A potential barrier of the kind studied by Fowler and others may be represented by the analytic function $V$ (Eq. (1)). The Schr\"odinger equation associated to this potential is soluble in terms of hypergeometric functions, and the coefficient of reflection for electrons approaching the barrier with energy $W$ is calculable (Eq. (15)). The approximate formula, $1\ensuremath{-}\ensuremath{\rho}=\mathrm{exp}{\ensuremath{-}\ensuremath{\int}\frac{4\ensuremath{\pi}}{h}{(2m(V\ensuremath{-}W))}^{\frac{1}{2}}\mathrm{dx}}$ is shown to agree very well with the exact formula when the width of the barrier is great compared to the de Broglie wave-length of the incident electron, and $W<{V}_{max}$.

Physical Review•Jun 1, 1930•2,463 citations

Analytical considerations of barrier transmission

John Heading

Abstract The success of approximate methods in the calculation of transmission coefficients of waves penetrating potential barriers rests, not so much on any satisfactory approximation theory, but on a comparison with a particular barrier, namely the parabolic barrier. The paucity of exact analytical solutions of barrier problems has prevented an explicit detailed analysis of the errors involved, but two recently devised barriers with analytical solutions are investigated here, in order to compare the exact values of the transmission coefficients (both as to their mathematical structure and numerically) with their approximate forms. An estimate is also made of the effect of neglecting poles existing in the complex propagation plane.

Mathematical Proceedings of the Cambridge Philosophical Society•Mar 1, 1972•5 citations

On Quantum-Mechanical Reflection Coefficients and their Numerical Determination

Vladimir Rojansky, W. Wetzel

A method, based on the use of Milne's function $w$, for the numerical determination of quantum-mechanical reflection coefficients in one-dimensional problems to any preassigned degree of accuracy is outlined. The case in which a "barrier" is both preceded and followed by a field-free space is considered in detail, sections 2-4, and a numerical example is worked out, section 5. The procedure in the case in which a "barrier" is preceded by a field-free space and is followed by a potential for which Schroedinger's equation can be solved analytically is outlined, sections 6-7, and the special case in which a "barrier" is followed by a uniform field is considered in some detail, section 8. The case in which a "barrier" is both preceded and followed by potentials for which Schroedinger's equation can be solved analytically is mentioned, section 8. Formulas are given for the evaluation in terms of $\ensuremath{\psi}$ and its first derivative, of the respective densities of the dextral and of the sinistral current flowing past any point $x$ in regions where the total energy is greater than the potential energy, section 6. A procedure is given for finding solutions representing unidirectional beams at infinity, section 6.

Physical Review•Dec 1, 1931•3 citations

A numerical study of quantum barrier penetration in one dimension

Charles W. Leming, Andrew Van Smith

Elementary numerical methods are used to evaluate solutions to Schrödinger’s equation for a variety of one-dimensional potential barriers and to determine the transmission coefficients of free particles. These techniques are used to study the oscillatory variations in transmission coefficients for both continuous and discontinuous potentials. The amplitude of the variations is found to be strongly increased by discontinuities in the potential function but small variations exist even for a smooth, continuous potential function. The techniques described are suitable for intermediate-level undergraduate physics students.

American Journal of Physics•May 1, 1991•11 citations

Quantum-Mechanical Tunneling

J. H. Fermor

An elementary method is given for the exact determination of the quantum mechanical transmission and reflection coefficients for a rectangular one-dimensional potential barrier. The method also finds application in approximate solutions for multistep barriers.

American Journal of Physics•Dec 1, 1966•39 citations

Tunnelling through asymmetric parabolic potential barriers

Zafar Ahmed

An exact transmission coefficient for the asymmetric parabolic barrier potential, i.e. , where and , is rederived.

Journal of Physics A Mathematical and General•Jul 21, 1997•5 citations
Special methods are investigated for. the calculation of reflection and transmission coefficients for a real symmetric potential barrier. These methods, involving half-barrier penetration only, are both of physical and mathematical interest, and they also afford a considerable saving in time when numerical procedures are necessary in the case of analytically nonsoluble models.
A potential barrier of the kind studied by Fowler and others may be represented by the analytic function VVV (Eq. (1)). The Schr\"odinger equation associated to this potential is soluble in terms of hypergeometric functions, and the coefficient of reflection for electrons approaching the barrier with energy WWW is calculable (Eq. (15)). The approximate formula, $1\ensuremath{-}\ensuremath{\rho}=\mathrm{exp}{\ensuremath{-}\ensuremath{\int}\frac{4\ensuremath{\pi}}{h}{(2m(V\ensuremath{-}W))}^{\frac{1}{2}}\mathrm{dx}}$ is shown to agree very well with the exact formula when the width of the barrier is great compared to the de Broglie wave-length of the incident electron, and W<VmaxW<{V}_{max}W<Vmax​.
Abstract The success of approximate methods in the calculation of transmission coefficients of waves penetrating potential barriers rests, not so much on any satisfactory approximation theory, but on a comparison with a particular barrier, namely the parabolic barrier. The paucity of exact analytical solutions of barrier problems has prevented an explicit detailed analysis of the errors involved, but two recently devised barriers with analytical solutions are investigated here, in order to compare the exact values of the transmission coefficients (both as to their mathematical structure and numerically) with their approximate forms. An estimate is also made of the effect of neglecting poles existing in the complex propagation plane.
A method, based on the use of Milne's function www, for the numerical determination of quantum-mechanical reflection coefficients in one-dimensional problems to any preassigned degree of accuracy is outlined. The case in which a "barrier" is both preceded and followed by a field-free space is considered in detail, sections 2-4, and a numerical example is worked out, section 5. The procedure in the case in which a "barrier" is preceded by a field-free space and is followed by a potential for which Schroedinger's equation can be solved analytically is outlined, sections 6-7, and the special case in which a "barrier" is followed by a uniform field is considered in some detail, section 8. The case in which a "barrier" is both preceded and followed by potentials for which Schroedinger's equation can be solved analytically is mentioned, section 8. Formulas are given for the evaluation in terms of $\ensuremath{\psi}$ and its first derivative, of the respective densities of the dextral and of the sinistral current flowing past any point xxx in regions where the total energy is greater than the potential energy, section 6. A procedure is given for finding solutions representing unidirectional beams at infinity, section 6.
Elementary numerical methods are used to evaluate solutions to Schrödinger’s equation for a variety of one-dimensional potential barriers and to determine the transmission coefficients of free particles. These techniques are used to study the oscillatory variations in transmission coefficients for both continuous and discontinuous potentials. The amplitude of the variations is found to be strongly increased by discontinuities in the potential function but small variations exist even for a smooth, continuous potential function. The techniques described are suitable for intermediate-level undergraduate physics students.