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:1101.3209 · European Journal of Physics

Wronskian method for bound states

Francisco Marcelo Fernández

Published Mar 23, 2011• 17 citations
arXivHTMLPDF

Abstract

We propose a simple and straightforward method based on Wronskians for the calculation of bound-state energies and wavefunctions of one-dimensional quantum-mechanical problems. We explicitly discuss the asymptotic behaviour of the wavefunction and show that the allowed energies make the divergent part vanish. As illustrative examples we consider an exactly solvable model, the Gaussian potential well, and a two-well potential proposed earlier for the interpretation of the infrared spectrum of ammonia.

DOI
10.1088/0143-0807/32/3/008

Related papers

Wronskian method for bound states

Francisco Marcelo Fernández

We propose a simple and straightforward method based on Wronskians for the calculation of bound-state energies and wavefunctions of one-dimensional quantum-mechanical problems. We explicitly discuss the asymptotic behaviour of the wavefunction and show that the allowed energies make the divergent part vanish. As illustrative examples we consider an exactly solvable model,the Gaussian potential well, and a two-well potential proposed earlier for the interpretation of the infrared spectrum of ammonia.

•Jan 1, 2011•20 citations

Quantum scattering by Wronskians

Francisco Marcelo Fernández

We show that Wronskians between properly chosen linearly independent solutions of the Schrödinger equation greatly facilitate the study of quantum scattering in one dimension. They enable one to obtain the necessary relationships between the coefficients that determine the asymptotic behavior of the wavefunction. As illustrative examples we calculate the transmission probability for the penetration of a Gaussian barrier and the scattering resonances for a Gaussian well.

arXiv:1101.0957•Jan 5, 2011•3 citations

Wronskian method for one-dimensional quantum scattering

Francisco Marcelo Fernández

The use of Wronskians with properly chosen linearly independent solutions of the Schrödinger equation greatly facilitates the calculation of the transmission probability for scattering in one dimension and also provides a reliable test for the accuracy of the calculation. We apply the approach to a Gaussian barrier and a Gaussian well, and test its accuracy for an exactly solvable model.

American Journal of Physics•Jul 26, 2011•15 citations

Quantum mechanics without potential function

A. D. Alhaidari, M. E. H. Ismail

In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states, and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and three-parameter systems.

arXiv:1408.4003•Journal of Mathematical Physics•Jul 1, 2015•41 citations

Possible scenarios transgressing the nondegeneracy theorem

Luiz G. M. Ramos, Antonio S. de Castro

In contrast to the nondegeneracy theorem, we present various scenarios in one-dimensional quantum mechanics that demonstrate how the Wronskian of two bound-state eigenfunctions with the same energy eigenvalue can be zero without implying that the eigenfunctions are linearly dependent. It is shown that the nondegeneracy theorem fails only when the potential makes different bound-state solutions corresponding to the same energy vanish at the singular point or region of singularity.

Revista Brasileira de Ensino de Física•Jan 1, 2023•1 citation

A novel application of a Fourier integral representation of bound states in quantum mechanics

Guillermo Palma, Ulrich Raff

A method is developed to find the bound state eigenvalues and eigenfunctions of one-dimensional and rotational symmetric three-dimensional quantum mechanical problems. The method is based on the fact that eigenfunctions of bound states are square integrable. We use this property and a judicious ansatz inspired on its asymptotic behavior to obtain a differential equation that can be solved straightforwardly using a Fourier transform. The main advantage of the method is that it avoids the traditional and tedious convergence analysis using a series representation. The method leads to an integral representation of the wave function and provides insight into concepts such as energy quantization, spectrum degeneracy, and bound states.

American Journal of Physics•Jan 24, 2011•12 citations
We propose a simple and straightforward method based on Wronskians for the calculation of bound-state energies and wavefunctions of one-dimensional quantum-mechanical problems. We explicitly discuss the asymptotic behaviour of the wavefunction and show that the allowed energies make the divergent part vanish. As illustrative examples we consider an exactly solvable model, the Gaussian potential well, and a two-well potential proposed earlier for the interpretation of the infrared spectrum of ammonia.
We propose a simple and straightforward method based on Wronskians for the calculation of bound-state energies and wavefunctions of one-dimensional quantum-mechanical problems. We explicitly discuss the asymptotic behaviour of the wavefunction and show that the allowed energies make the divergent part vanish. As illustrative examples we consider an exactly solvable model,the Gaussian potential well, and a two-well potential proposed earlier for the interpretation of the infrared spectrum of ammonia.
In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states, and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and three-parameter systems.
A method is developed to find the bound state eigenvalues and eigenfunctions of one-dimensional and rotational symmetric three-dimensional quantum mechanical problems. The method is based on the fact that eigenfunctions of bound states are square integrable. We use this property and a judicious ansatz inspired on its asymptotic behavior to obtain a differential equation that can be solved straightforwardly using a Fourier transform. The main advantage of the method is that it avoids the traditional and tedious convergence analysis using a series representation. The method leads to an integral representation of the wave function and provides insight into concepts such as energy quantization, spectrum degeneracy, and bound states.