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

European Journal of Physics

A linear variational exercise with a simple non-orthogonal basis for the particle-in-the-box problem

Vı́ctor Luaña, Alberto Otero‐de‐la‐Roza, M. A. Blanco, J M Recio

Published Nov 26, 2009• 4 citations
Publisher

Abstract

The particle-in-the-box, with or without an additional potential, is proposed as an excellent laboratory to teach and explore the details of the linear variational method using a non-orthogonal basis. The x n ( a − x ) n and x n ( a /2 − x )( a − x ) n polynomials are shown to form a complete basis for the even and odd states, respectively, of the particle confined to the x ∊ [0, a ] interval. A short and simple Octave code is presented as the natural extension to the hand calculations when the basis set grows in size.

DOI
10.1088/0143-0807/31/1/010

Related papers

Wide Effectiveness of a Sine Basis for Quantum-Mechanical Problems in d Dimensions

Richard L. Hall, Alexandra Lemus Rodríguez

It is shown that the spanning set for L 2 ( [ 0,1 ] ) provided by the eigenfunctions { 2 sin ( n π x ) } n = 1 ∞ of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to [ a , b ] , where a and b are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in R d turns out to be appropriate also for problems that are softly confined by U -shaped potentials, including those with strong singularities at r = 0 . Specific examples are discussed in detail, along with some bound N -boson systems.

arXiv:1212.4516•Advances in Mathematical Physics•Jan 1, 2013•3 citations

Exploring the Rayleigh-Ritz Variational Principle

Emili Besalú, Josep Martí

The particle-in-a-box (PIB) system is used as a simple example in order to illustrate the basis of the quantum variational principle. After the general exposition of the variational principle demonstration, a quadratic polynomial trial function is applied to the PIB hamiltonian and the different steps of the demonstration are followed and reproduced for this particular case.

Journal of Chemical Education•Jan 1, 1998•9 citations

Application of the variational method to the particle-in-the box problem

David D. Keeports

It is the purpose of this paper to consider a more general application of the variational method to the particle-in-the-box problem with polynomial trial functions.

Journal of Chemical Education•Apr 1, 1989•6 citations

Wide Effectiveness of a Sine Basis for Quantum-Mechanical Problems in Dimensions

Richard L. Hall, Alexandra Lemus Rodríguez

It is shown that the spanning set for provided by the eigenfunctions of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to , where and are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in turns out to be appropriate also for problems that are softly confined by U -shaped potentials, including those with strong singularities at . Specific examples are discussed in detail, along with some bound -boson systems.

•Jan 1, 2013•0 citations

On the use of direct variational methods to study confined quantum systems

J. L. Marin, Salvador A. Cruz

The direct variational method is used to study two simple confined systems, namely, the hydrogen atom and the harmonic oscillator within impenetrable spherical boxes. The trial wave functions have been assumed as the product of the ‘‘free’’ solutions of the corresponding Schrödinger equation and a simple function that satisfies the respective boundary conditions. The energy levels obtained in this way are extremely close to the exact ones, thus proving the utility of the proposed method. This might constitute a complementary exercise for undergraduate students since usual quantum mechanics textbooks do not explore the use of the variational method to study confined systems.

American Journal of Physics•Oct 1, 1991•64 citations

Quantum mechanics with orthogonal polynomials

Abdulaziz D Alhaidari

We present a formulation of quantum mechanics based on orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis in configuration space where the expansion coefficients are orthogonal polynomials in the energy. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable non-relativistic quantum mechanical problems becomes larger than in the conventional formulation (see, for example, Table 1 in the text).

arXiv:1709.07652•Sep 22, 2017•5 citations
The particle-in-the-box, with or without an additional potential, is proposed as an excellent laboratory to teach and explore the details of the linear variational method using a non-orthogonal basis. The x n ( a − x ) n and x n ( a /2 − x )( a − x ) n polynomials are shown to form a complete basis for the even and odd states, respectively, of the particle confined to the x ∊ [0, a ] interval. A short and simple Octave code is presented as the natural extension to the hand calculations when the basis set grows in size.
It is shown that the spanning set for L 2 ( [ 0,1 ] ) provided by the eigenfunctions { 2 sin ( n π x ) } n = 1 ∞ of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to [ a , b ] , where a and b are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in R d turns out to be appropriate also for problems that are softly confined by U -shaped potentials, including those with strong singularities at r = 0 . Specific examples are discussed in detail, along with some bound N -boson systems.
It is shown that the spanning set for provided by the eigenfunctions of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to , where and are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in turns out to be appropriate also for problems that are softly confined by U -shaped potentials, including those with strong singularities at . Specific examples are discussed in detail, along with some bound -boson systems.
The direct variational method is used to study two simple confined systems, namely, the hydrogen atom and the harmonic oscillator within impenetrable spherical boxes. The trial wave functions have been assumed as the product of the ‘‘free’’ solutions of the corresponding Schrödinger equation and a simple function that satisfies the respective boundary conditions. The energy levels obtained in this way are extremely close to the exact ones, thus proving the utility of the proposed method. This might constitute a complementary exercise for undergraduate students since usual quantum mechanics textbooks do not explore the use of the variational method to study confined systems.
We present a formulation of quantum mechanics based on orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis in configuration space where the expansion coefficients are orthogonal polynomials in the energy. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable non-relativistic quantum mechanical problems becomes larger than in the conventional formulation (see, for example, Table 1 in the text).