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

Revista Mexicana de Física E

Impenetrable barriers in quantum mechanics

Salvatore De Vincenzo

Published Jan 6, 2008• 2 citations
OpenAlex

Abstract

"We derive the expression V (x) u(x) = c δ(x − a) + v(x) u(x) (where V (x) is the potential, u(x) the wave function, c a constant and v(x) a finite potential function for x ≤ a), which is present in the one-dimensional Schr ̈ dinger equation on the whole real line when we have o an impenetrable barrier at x ≥ a, that is, an infinite step potential there. By studying the solution of this equation, we identify, connect and discuss three different Hamiltonian operators that describe the barrier. We extend these results by constructing an infinite square-well potential from two impenetrable barriers."

Related papers

Quantum Standing Waves and Tunneling Through a Finite Range Potential

Haiduke Sarafian

We consider a time independent one dimensional finite range and repulsive constant potential barrier between two impenetrable walls. For a nonrelativistic massive particle projected towards the potential with energies less than the barrier and irrespective of the spatial positioning of the potential allowing for quantum tunneling, analytically we solve the corresponding Schrodinger equation. For a set of suitable parameters utilizing Mathematica we display the wave functions along with their associated probabilities for the entire region. We investigate the sensitivity of the probability distributions as a function of the potential range and display a gallery of our analysis. We extend our analysis for bound state particles confined within constant attractive potentials.

Journal of Modern Physics•Jan 1, 2011•2 citations

Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect

Gilbert A. Ibitola, Olanrewaju Ajanaku

In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.

World Journal of Applied Physics•Dec 22, 2016•0 citations

One-Dimensional Potential Steps, Barriers, and Wells

Ana Marı́a Cetto, Luis de la Peña

Chapter 5 familiarizes the student with some of the most basic features of quantum systems, by applying the Schrödinger equation to simple, one-dimensional problems. It shows that it is possible to obtain qualitatively correct conclusions about the behavior of quantum particles in the presence of more or less arbitrary potentials by replacing them with piecewise-constant potentials, which greatly simplifies the mathematics. Replacing the real potential with a simpler one that retains its basic characteristics has the great advantage of allowing us to study the essence of the physical situation, as illustrated by the examples of the one-dimensional potential step, the square well, the square potential barrier and the symmetric double well. These examples are used to discuss specific features such as quantum tunneling, and applications such as masers and molecular clocks.

Cambridge University Press eBooks•Dec 18, 2025•0 citations

The electron

Laszlo Solymar, D. Walsh, R. R. A. Syms

Abstract Discusses with some rigour the properties of electrons, based on the Schrodinger equation. Introduces the concepts of wave function, quantum-mechanical operators, and wave packets. Examples cover the electron meeting an infinitely long potential barrier and the passage of electrons through a finite barrier (which leads to the phenomenon of tunnelling).The electron in a potential well is also discussed, solving the problem both for a finite and for an infinite well, and finding the permissible energy levels. The chapter is concluded with the philosophical implications that arise from the quantum-mechanical approach. Two limericks relevant to the subject are quoted.

•Sep 27, 2018•0 citations

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

Hwasung Lee, Y J Lee

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.

arXiv:quant-ph/0611197•Journal of Physics A Mathematical and Theoretical•Mar 14, 2007•3 citations

Analysis of the Solution of Schrdinger Equation for Free Particle

Luo Guan

Based on the theory of Schrdinger equation for the free particle,the solution of that equation was discussed in detail. It is found that the equation only exist an insignificant solution. And the mediocre solution can be also obtained from the one-dimensional infinitely deep potential well in a potential well width tending to infinite and from one-dimensional square potential barrier in the barrier height tending to zero. Although the solution is insignificant,but it describes some ideal that is consistent with that of classical mechanics. It explains the connection between the classical mechanics and quantum mechanics. The mediocre solutions show that 1) For infinite free particles,it is followed by Newton' s first motion law,2) For infinite free particle,it does not exist matter wave characteristics. Only if there is some potential( i. e. interactions),the Wave-particle duality be embodied.

Chongqing Shifan Daxue xuebao. Ziran kexue ban•Jan 1, 2014•0 citations
"We derive the expression V (x) u(x) = c δ(x − a) + v(x) u(x) (where V (x) is the potential, u(x) the wave function, c a constant and v(x) a finite potential function for x ≤ a), which is present in the one-dimensional Schr ̈ dinger equation on the whole real line when we have o an impenetrable barrier at x ≥ a, that is, an infinite step potential there. By studying the solution of this equation, we identify, connect and discuss three different Hamiltonian operators that describe the barrier. We extend these results by constructing an infinite square-well potential from two impenetrable barriers."
We consider a time independent one dimensional finite range and repulsive constant potential barrier between two impenetrable walls. For a nonrelativistic massive particle projected towards the potential with energies less than the barrier and irrespective of the spatial positioning of the potential allowing for quantum tunneling, analytically we solve the corresponding Schrodinger equation. For a set of suitable parameters utilizing Mathematica we display the wave functions along with their associated probabilities for the entire region. We investigate the sensitivity of the probability distributions as a function of the potential range and display a gallery of our analysis. We extend our analysis for bound state particles confined within constant attractive potentials.
In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.
Chapter 5 familiarizes the student with some of the most basic features of quantum systems, by applying the Schrödinger equation to simple, one-dimensional problems. It shows that it is possible to obtain qualitatively correct conclusions about the behavior of quantum particles in the presence of more or less arbitrary potentials by replacing them with piecewise-constant potentials, which greatly simplifies the mathematics. Replacing the real potential with a simpler one that retains its basic characteristics has the great advantage of allowing us to study the essence of the physical situation, as illustrated by the examples of the one-dimensional potential step, the square well, the square potential barrier and the symmetric double well. These examples are used to discuss specific features such as quantum tunneling, and applications such as masers and molecular clocks.
Abstract Discusses with some rigour the properties of electrons, based on the Schrodinger equation. Introduces the concepts of wave function, quantum-mechanical operators, and wave packets. Examples cover the electron meeting an infinitely long potential barrier and the passage of electrons through a finite barrier (which leads to the phenomenon of tunnelling).The electron in a potential well is also discussed, solving the problem both for a finite and for an infinite well, and finding the permissible energy levels. The chapter is concluded with the philosophical implications that arise from the quantum-mechanical approach. Two limericks relevant to the subject are quoted.
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.
Based on the theory of Schrdinger equation for the free particle,the solution of that equation was discussed in detail. It is found that the equation only exist an insignificant solution. And the mediocre solution can be also obtained from the one-dimensional infinitely deep potential well in a potential well width tending to infinite and from one-dimensional square potential barrier in the barrier height tending to zero. Although the solution is insignificant,but it describes some ideal that is consistent with that of classical mechanics. It explains the connection between the classical mechanics and quantum mechanics. The mediocre solutions show that 1) For infinite free particles,it is followed by Newton' s first motion law,2) For infinite free particle,it does not exist matter wave characteristics. Only if there is some potential( i. e. interactions),the Wave-particle duality be embodied.