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

Canadian Journal of Physics

WKB quantization rules for three-dimensional confinement

Anjana Sinha, Rajkumar Roychoudhury, Y. P. Varshni

Published Jan 1, 2001• 4 citations
Publisher

Abstract

Confined quantum systems have been studied by various authors over the past decades, by using various mathematical techniques. In this work, we derive the WKB quantization rules for quantum systems confined in an impenetrable spherical box of radius r0. We apply the proposed method to two systems explicitly, viz., the confined harmonic oscillator and the confined hydrogen atom. The results are found to be in reasonable agreement with those obtained by other methods. PACS No.: 03.65

DOI
10.1139/cjp-79-6-939

Related papers

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

SHIFTED 1/N EXPANSION AND ENCLOSED QUANTUM SYSTEMS

Anjana Sinha, Rajkumar Roychoudhury

A modified version of the shifted 1/N expansion method is formulated for constrained quantum mechanical system. This method is applied to boxed-in hydrogen atom and spherical harmonic oscillator. Results from the shifted 1/N expansion method are compared with the exact numerical results in case of hydrogen atom, and with approximate analytical results in case of the spherical harmonic oscillator. Agreement between the results is found to be good in both the cases.

Fizika B•Jan 1, 1994•1 citation

Notes on Application of WKB Method

Gua Xiao-Yan, Jianqiang Sun

The notes here presented are of the modifications introduced in the application of WKB method. The problems of two- and three-dimensional harmonic oscillator potential are revisited by WKB and the new formulation of quantization rule respectively. It is found that the energy spectrum of the radial harmonic oscillator, which is reproduced exactly by the standard WKB method with the hanger modification, is also reproduced exactly without the hanger modification via the new quantization rule approach. An alternative way to obtain the non-integral Maslov index for three-dimensional harmonic oscillator is proposed.

Communications in Theoretical Physics•Oct 1, 2008•2 citations

Exact solutions for the D-dimensional spherical isotropic confined harmonic oscillator

Henry E Montgomery Jr, G. Campoy, N. Aquino

We study the size effect on the energy levels of the D-dimensional isotropic harmonic oscillator confined within a box of radius $r_c$ with impenetrable walls. Two different approaches are used to obtain the energy eigenvalues and eigenfunctions for D=1,2,...,5. In the first we solve the Schroedinger equation exactly. In the second we use a series expansion of the wave function. The numerical results obtained are extremely accurate; these values are reported with 50 decimal places.

arXiv:0803.4029•Mar 28, 2008•1 citation

Quantum mechanical virial-like theorem for confined quantum systems

Neetik Mukherjee, Amlan K. Roy

Confinement of atoms inside impenetrable (hard) and penetrable (soft) cavities has been studied for nearly eight decades. However, a unified virial theorem for such systems has not yet been found. Here we provide a general virial-like equation in terms of mean square and expectation values of potential and kinetic energy operators. It appears to be applicable in both free and confined situations. Apart from that, we have derived an equation using the time-independent Schr\"odinger equation, which can be treated as a sufficient condition for a given stationary quantum state. A change of boundary condition does not affect these virial equations. In the hard confining condition, the perturbing (confining potential) does not affect the expression; it merely shifts the boundary from infinity to a finite region. In the soft case, on the contrary, the final expression includes contributions from the perturbing term. These are demonstrated numerically for several representative enclosed systems like harmonic oscillators (one-dimensional and three-dimensional) and hydrogen atoms. Its applicability in various other confinements (including angular) has been discussed. In essence, a virial equation has been proposed for free and confined quantum systems, from simple arguments.

arXiv:1904.02386•Physical Review A•Feb 22, 2019•16 citations

Shannon and Fisher Entropies for the 2P and 3D States of the Hydrogen Atom Confined in a Penetrable Spherical Box

Andrei Solorzano Perez, José Manuel Nieto-Jalil

A straightforward way to study the electronic structure and certain physical properties of atoms and molecules under high pressures is through the confinement (caging) model, in which the atom is placed at the center of a spherical box with impenetrable walls. The hydrogen atom confined inside penetrable and impenetrable spherical cavities is perhaps the most studied compressed system, and exact solutions exist for both types of confinement. The physical properties of the hydrogen atom confined within impenetrable-wall cavities (AHCI) have been widely analyzed; however, the same system embedded in penetrable spherical cavities (AHCP) has received less attention. A more physical model suggests using a spherical box with penetrable walls. Therefore, in this work, we compute via the variational method, the Shannon and Fisher information entropy in position space,$S_{r}$and$I_{r}$, for the$2 \mathbf{p}$and 3d states of a hydrogen atom subjected to spatial confinement by penetrable and impenetrable spherical cavities. Additionally, we calculate the tunneling probability for the penetrable confinement and seek to relate it to the overlaps of the radial probability densities as a function of the cavity radius and the height of the confining barrier. To model the penetrable confinement, we employ a logistic potential which, unlike earlier reports that used a discontinuous step potential, is a continuous function. The wavefunction is expanded in terms of Slater functions that satisfy appropriate boundary conditions; in some cases, we find notable differences with respect to the impenetrable potential.

Mexican International Conference on Computer Science•Nov 10, 2025•0 citations
Confined quantum systems have been studied by various authors over the past decades, by using various mathematical techniques. In this work, we derive the WKB quantization rules for quantum systems confined in an impenetrable spherical box of radius r0. We apply the proposed method to two systems explicitly, viz., the confined harmonic oscillator and the confined hydrogen atom. The results are found to be in reasonable agreement with those obtained by other methods. PACS No.: 03.65
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.
The notes here presented are of the modifications introduced in the application of WKB method. The problems of two- and three-dimensional harmonic oscillator potential are revisited by WKB and the new formulation of quantization rule respectively. It is found that the energy spectrum of the radial harmonic oscillator, which is reproduced exactly by the standard WKB method with the hanger modification, is also reproduced exactly without the hanger modification via the new quantization rule approach. An alternative way to obtain the non-integral Maslov index for three-dimensional harmonic oscillator is proposed.
We study the size effect on the energy levels of the D-dimensional isotropic harmonic oscillator confined within a box of radius rcr_crc​ with impenetrable walls. Two different approaches are used to obtain the energy eigenvalues and eigenfunctions for D=1,2,...,5. In the first we solve the Schroedinger equation exactly. In the second we use a series expansion of the wave function. The numerical results obtained are extremely accurate; these values are reported with 50 decimal places.
Confinement of atoms inside impenetrable (hard) and penetrable (soft) cavities has been studied for nearly eight decades. However, a unified virial theorem for such systems has not yet been found. Here we provide a general virial-like equation in terms of mean square and expectation values of potential and kinetic energy operators. It appears to be applicable in both free and confined situations. Apart from that, we have derived an equation using the time-independent Schr\"odinger equation, which can be treated as a sufficient condition for a given stationary quantum state. A change of boundary condition does not affect these virial equations. In the hard confining condition, the perturbing (confining potential) does not affect the expression; it merely shifts the boundary from infinity to a finite region. In the soft case, on the contrary, the final expression includes contributions from the perturbing term. These are demonstrated numerically for several representative enclosed systems like harmonic oscillators (one-dimensional and three-dimensional) and hydrogen atoms. Its applicability in various other confinements (including angular) has been discussed. In essence, a virial equation has been proposed for free and confined quantum systems, from simple arguments.
A straightforward way to study the electronic structure and certain physical properties of atoms and molecules under high pressures is through the confinement (caging) model, in which the atom is placed at the center of a spherical box with impenetrable walls. The hydrogen atom confined inside penetrable and impenetrable spherical cavities is perhaps the most studied compressed system, and exact solutions exist for both types of confinement. The physical properties of the hydrogen atom confined within impenetrable-wall cavities (AHCI) have been widely analyzed; however, the same system embedded in penetrable spherical cavities (AHCP) has received less attention. A more physical model suggests using a spherical box with penetrable walls. Therefore, in this work, we compute via the variational method, the Shannon and Fisher information entropy in position space,SrS_{r}Sr​andIrI_{r}Ir​, for the2p2 \mathbf{p}2pand 3d states of a hydrogen atom subjected to spatial confinement by penetrable and impenetrable spherical cavities. Additionally, we calculate the tunneling probability for the penetrable confinement and seek to relate it to the overlaps of the radial probability densities as a function of the cavity radius and the height of the confining barrier. To model the penetrable confinement, we employ a logistic potential which, unlike earlier reports that used a discontinuous step potential, is a continuous function. The wavefunction is expanded in terms of Slater functions that satisfy appropriate boundary conditions; in some cases, we find notable differences with respect to the impenetrable potential.