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

arXiv:quant-ph/0611197 · Journal of Physics A Mathematical and Theoretical

Determinable solutions for one-dimensional quantum potentials: scattering, quasi-bound and bound-state problems

Hwasung Lee, Y J Lee

Published Mar 14, 2007• 3 citations
arXivHTMLPDF

Abstract

We derive analytic expressions of the recursive solutions to the Schrödinger's equation by means of a cutoff potential technique for one-dimensional piecewise constant potentials. These solutions provide a method for accurately determining the transmission probabilities as well as the wave function in both classically accessible region and inaccessible region for any barrier potentials. It is also shown that the energy eigenvalues and the wave functions of bound states can be obtained for potential-well structures by exploiting this method. Calculational results of illustrative examples are shown in order to verify this method for treating barrier and potential-well problems.

DOI
10.1088/1751-8113/40/13/017

Related papers

General recursive solution for one dimensional quantum potentials

Sérgio Luiz Morelhão, André Perrotta

A revision of the recursive method proposed by S.A. Shakir [Am. J.Phys. \textbf{52}, 845 (1984)] to solve bound eigenvalues of the Schrodinger equation is presented. Equations are further simplified and generalized for computing wave functions of any given one-dimensional potential, providing accurate solutions not only for bound states but also for scattering and resonant states, as demonstrated here for a few examples.

arXiv:quant-ph/0605115•May 15, 2006•0 citations

General recursive solution for one-dimensional quantum potentials: a simple tool for applied physics

Sérgio Luiz Morelhão, André Perrotta

A revision of the recursive method proposed by S.A. Shakir [Am. J. Phys. 52, 845 (1984)] to solve bound eigenvalues of the Schrödinger equation is presented. Equations are further simplified and generalized for computing wave functions of any given one-dimensional potential, providing accurate solutions not only for bound states but also for scattering and resonant states, as demonstrated here for a few examples.

Revista Brasileira de Ensino de Física•Jan 1, 2007•0 citations

General recursive solution for one-dimensional quantum potentials: a simple tool for applied physics

Sérgio Luiz Morelhão, André Perrotta

A revision of the recursive method proposed by S.A. Shakir [Am. J. Phys. 52, 845 (1984)] to solve bound eigenvalues of the Schrödinger equation is presented. Equations are further simplified and generalized for computing wave functions of any given one-dimensional potential, providing accurate solutions not only for bound states but also for scattering and resonant states, as demonstrated here for a few examples.

Revista de ensino de física•Jan 1, 2007•0 citations

Exact solution of the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential barrier

Wayne W. Lui, Masao Fukuma

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.

Journal of Applied Physics•Sep 1, 1986•208 citations

Solvable models of an open well and a bottomless barrier: one-dimensional exponential potentials

Zafar Ahmed, D. C. Ghosh, Sachin Kumar +1 more

Abstract We present one-dimensional potentials V ( x ) = V 0 ( e 2 ∣ x ∣ / a − 1 ) as solvable models of a well ( V 0 > 0 ) and a barrier ( V 0 < 0 ). Apart from being a new addition to solvable models, these models are instructive for finding bound and scattering states from the analytic solutions of the Schrödinger equation. The exact analytic (semi-classical and quantal) forms for bound states of the well and reflection/transmission ( R / T ) coefficients for the barrier have been derived. Interestingly, the crossover energy E c where R ( E c ) = 1 / 2 = T ( E c ) may occur above, below or at the barrier-top. A connection between the poles of these coefficients and the bound-state eigenvalues of the well is also demonstrated.

arXiv:1706.05275•European Journal of Physics•Sep 13, 2017•9 citations

Quantum wave impedance calculation for an arbitrary piesewise constant potential

Orest Hryhorchak

A method for determining the quantum wave impedance function for a quantum-mechanical system with an arbitrary piecewise constant potential has been developed.Using this method, both the well-known iterative formula [A.N. Khondker, M. R. Khan, A. F. M. Anwar, J. Appl.Phys.63, 5191 (1988)] and an alternative approach for calculating quantum wave impedance were derived.This approach was applied to both scattering and bound state cases.The general form of the wave functions for bound states, as well as the transmission and reection coecients in the scattering case, were obtained for systems with a potential consisting of an arbitrary number of rectangular barriers and wells.The application of the developed method was demonstrated by describing symmetric rectangular double-well and double-barrier quantum-mechanical systems.

Journal of Physical Studies•Jan 1, 2024•1 citation
We derive analytic expressions of the recursive solutions to the Schrödinger's equation by means of a cutoff potential technique for one-dimensional piecewise constant potentials. These solutions provide a method for accurately determining the transmission probabilities as well as the wave function in both classically accessible region and inaccessible region for any barrier potentials. It is also shown that the energy eigenvalues and the wave functions of bound states can be obtained for potential-well structures by exploiting this method. Calculational results of illustrative examples are shown in order to verify this method for treating barrier and potential-well problems.
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.
Abstract We present one-dimensional potentials V ( x ) = V 0 ( e 2 ∣ x ∣ / a − 1 ) as solvable models of a well ( V 0 > 0 ) and a barrier ( V 0 < 0 ). Apart from being a new addition to solvable models, these models are instructive for finding bound and scattering states from the analytic solutions of the Schrödinger equation. The exact analytic (semi-classical and quantal) forms for bound states of the well and reflection/transmission ( R / T ) coefficients for the barrier have been derived. Interestingly, the crossover energy E c where R ( E c ) = 1 / 2 = T ( E c ) may occur above, below or at the barrier-top. A connection between the poles of these coefficients and the bound-state eigenvalues of the well is also demonstrated.
A method for determining the quantum wave impedance function for a quantum-mechanical system with an arbitrary piecewise constant potential has been developed.Using this method, both the well-known iterative formula [A.N. Khondker, M. R. Khan, A. F. M. Anwar, J. Appl.Phys.63, 5191 (1988)] and an alternative approach for calculating quantum wave impedance were derived.This approach was applied to both scattering and bound state cases.The general form of the wave functions for bound states, as well as the transmission and reection coecients in the scattering case, were obtained for systems with a potential consisting of an arbitrary number of rectangular barriers and wells.The application of the developed method was demonstrated by describing symmetric rectangular double-well and double-barrier quantum-mechanical systems.