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

Quantum Gaussian wells and barriers

Francisco Marcelo Fernández

Published Jun 17, 2011• 30 citations
Publisher

Abstract

We apply the variational method to the ground-state and first-excited-state energies of a Gaussian well and show that the expansion of the approximate energy when the potential strength approaches zero does not agree with the exact result. We argue that a recent semiclassical approach to tunneling through a Gaussian barrier does not agree with an accurate calculation of the transmission probability. We point out the advantages of converting the Schrödinger equation into a dimensionless eigenvalue equation.

DOI
10.1119/1.3574505

Related papers

Gaussian effective potential: Computation of eigenvalues

Arjendu K. Pattanayak, William C. Schieve

The Gaussian effective potential (GEP) is an approximation to quantum mechanics that semiquantitatively estimates quantum effects such as zero-point fluctuations and tunneling. We show how to use the GEP to compute semiclassical eigenvalues. It is well known that the GEP provides very accurate variational approximations to the ground-state energy of simple quantum systems; our results show that it also provides accurate excited-state energies. We also show how this method is related to the first-order Bargmann-Voros coherent-state representation of quantum mechanics, and that our results improve upon those derived from the latter method. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Jul 1, 1996•2 citations

Exact solution to the Schrödinger’s equation with pseudo-Gaussian potential

Felix Iacob, Marina Lute

We consider the radial Schrödinger equation with the pseudo-Gaussian potential. By making an ansatz to the solution of the eigenvalue equation for the associate Hamiltonian, we arrive at the general exact eigenfunction. The values of energy levels for the bound states are calculated along with their corresponding normalized wave-functions. The case of positive energy levels, known as meta-stable states, is also discussed and the magnitude of transmission coefficient through the potential barrier is evaluated.

arXiv:1411.5162•Journal of Mathematical Physics•Dec 1, 2015•15 citations

Variational approach to the stationary states of the double-oscillator potential

Behzad Mirhashem, Richard A. Ferrell

A variational approach to the calculation of the two lowest eigenvalues of Schrödinger’s equation for the double-oscillator potential is presented. A family of two-parameter double-Gaussian trial functions provides an alternative to the exact series expansions for the energy eigenfunctions and yields energy estimates of high accuracy.

American Journal of Physics•May 1, 1991•4 citations

Approximations for the Lowest Energy Eigenstates in a Double Square Well Potential

Don Nelson Page

Highly accurate closed-form approximations are given for the ground state and first excited state wavefunctions and energies for a nonrelativistic particle in a one-dimensional double square well potential with a square barrier in between (so that the potential is a sequence of five constant pieces that alternate in value from being above and below the ground state and first excited state energies), under the assumption that the barrier is sufficiently wide that the tunneling across it is very small.

arXiv:1711.05745•Nov 15, 2017•0 citations

Tunneling of Bound Systems at Finite Energies: Complex Paths Through Potential Barriers

Gian Franco Bonini, Andrew G. Cohen, Claudio Rebbi B +1 more

We adapt the semiclassical technique, as used in the context of instanton transitions in quantum field theory, to the description of tunneling transmissions at finite energies through potential barriers by complex quantum mechanical systems. Even for systems initially in their ground state, not generally describable in semiclassical terms, the transmission probability has a semiclassical (exponential) form. The calculation of the tunneling exponent uses analytic continuation of degrees of freedom into a complex phase space as well as analytic continuation of the classical equations of motion into the complex time plane. We test this semiclassical technique by comparing its results with those of a computational investigation of the full quantum mechanical system, finding excellent agreement.

arXiv:quant-ph/9901062•Jan 21, 1999•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
We apply the variational method to the ground-state and first-excited-state energies of a Gaussian well and show that the expansion of the approximate energy when the potential strength approaches zero does not agree with the exact result. We argue that a recent semiclassical approach to tunneling through a Gaussian barrier does not agree with an accurate calculation of the transmission probability. We point out the advantages of converting the Schrödinger equation into a dimensionless eigenvalue equation.
The Gaussian effective potential (GEP) is an approximation to quantum mechanics that semiquantitatively estimates quantum effects such as zero-point fluctuations and tunneling. We show how to use the GEP to compute semiclassical eigenvalues. It is well known that the GEP provides very accurate variational approximations to the ground-state energy of simple quantum systems; our results show that it also provides accurate excited-state energies. We also show how this method is related to the first-order Bargmann-Voros coherent-state representation of quantum mechanics, and that our results improve upon those derived from the latter method. \textcopyright{} 1996 The American Physical Society.
We consider the radial Schrödinger equation with the pseudo-Gaussian potential. By making an ansatz to the solution of the eigenvalue equation for the associate Hamiltonian, we arrive at the general exact eigenfunction. The values of energy levels for the bound states are calculated along with their corresponding normalized wave-functions. The case of positive energy levels, known as meta-stable states, is also discussed and the magnitude of transmission coefficient through the potential barrier is evaluated.
Highly accurate closed-form approximations are given for the ground state and first excited state wavefunctions and energies for a nonrelativistic particle in a one-dimensional double square well potential with a square barrier in between (so that the potential is a sequence of five constant pieces that alternate in value from being above and below the ground state and first excited state energies), under the assumption that the barrier is sufficiently wide that the tunneling across it is very small.
We adapt the semiclassical technique, as used in the context of instanton transitions in quantum field theory, to the description of tunneling transmissions at finite energies through potential barriers by complex quantum mechanical systems. Even for systems initially in their ground state, not generally describable in semiclassical terms, the transmission probability has a semiclassical (exponential) form. The calculation of the tunneling exponent uses analytic continuation of degrees of freedom into a complex phase space as well as analytic continuation of the classical equations of motion into the complex time plane. We test this semiclassical technique by comparing its results with those of a computational investigation of the full quantum mechanical system, finding excellent agreement.
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.