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

CRC Press eBooks

1 A Comment on the Virial Relation

L.D. Faddeev, L.A. Khalfin, I.V. Komarov

Published May 21, 2004• 0 citations
Publisher

Abstract

The validity of the virial theorem was proved by Born, Heisenberg and Jordan1 on the ground of matrix calculation and after that by Finkelstein2 on the ground of the Schro¨dinger equation. Both proofs essentially reproduced ideas of the classical approach. In the present note, a new and purely quantum mechanical derivation of this relation is given from the variational principle. The derivation is very simple and allows one to get some interesting consequences.

DOI
10.1201/9780203643204-17

Related papers

Proof of the quantum virial theorem

Zhu Dongpei

Based on the Feynman–Hellmann theorem, a new proof of the quantum virial theorem is presented. The same approach is also used to derive the virial formulation in relativistic quantum mechanics and for the case where the vector potential exists.

American Journal of Physics•Mar 1, 1986•5 citations

On the virial theorem for a particle in a box: Accounting for Cauchy's boundary condition

R. Cabrera‐Trujillo, Oriol Vendrell

Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this work, we address this issue by solving the particle in a finite box and show that the virial theorem is fulfilled when the appropriate Cauchy boundary conditions are taken into account. We also illustrate how, in the limit of the infinite potential box, the virial theorem holds as well. As a consequence, it is possible to determine the averaged force exerted by the walls on the particle. Finally, a discussion of these results in the classical limit is provided.

American Journal of Physics•Nov 19, 2020•4 citations

Generalised Virial theorems in Classical and Quantum Physics

C V Sukumar

Generalisations of the virial theorm in Classical Mechanics and Quantum Mechanics are examined. It is shown that the generalised virial theorem in Quantum Mechanics leads to certain relations between matrix elements. The differences between the generalisations in Classical and Quantum Mechanics are identified. Some results arising from the radial Schrödinger equation in Quantum Mechanics are discussed. It is also shown that the generalisations of the virial theorem may be extended to arbitrary number of dimensions.

arXiv:1410.5592•Oct 21, 2014•0 citations

A generalized quantum virial equation

Jan‐Åke Schweitz

A method, similar to the one previously used by the author in his derivation of a generalized classical virial theorem (see ibid., vol.10, no.4, p.507 (1977)), is used to derive a generalized quantum virial equation. This equation is applicable to any part of a larger system of particles. Furthermore, by the introduction of a flux density operator, it is possible to express the quantum surface flux virial in two alternative forms, as easily interpreted as the classical form. As an intermediate result, an equation of continuity for a general one-particle observable is obtained. In an appendix, an equation of motion of the reduced density matrix of the first order is derived.

Journal of Physics A Mathematical and General•Apr 1, 1977•8 citations

An elementary derivation of the quantum virial theorem from Hellmann–Feynman theorem

Y. İpekoğlu, S. Turgut

Abstract A simple proof of the quantum virial theorem that can be used in undergraduate courses is given. The proof proceeds by first showing that the energy eigenvalues of a Hamiltonian remain invariant under a scale transformation. Then invoking the Hellmann–Feynman theorem produces the final statement of the virial theorem.

European Journal of Physics•Jun 1, 2016•6 citations

The Wavefunctions and Energy Eigenvalues of the Schrodinger Equation for Different Potentials Due to the Virial Theorem

Hasan Arslan, N. Hulaguhanoglu

The derivation of the virial theorem is presented both in classical and quantum mechanical approach.The kinetic energy and potential energy of the mechanical energy is converted to each other due to the virial theorem.Some of the dierent potentials are considered.For some of these potentials, the wavefunctions and energy eigenvalues of the Schrödinger Equation are derived.

Acta Physica Polonica A•Apr 1, 2015•0 citations
The validity of the virial theorem was proved by Born, Heisenberg and Jordan1 on the ground of matrix calculation and after that by Finkelstein2 on the ground of the Schro¨dinger equation. Both proofs essentially reproduced ideas of the classical approach. In the present note, a new and purely quantum mechanical derivation of this relation is given from the variational principle. The derivation is very simple and allows one to get some interesting consequences.
Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this work, we address this issue by solving the particle in a finite box and show that the virial theorem is fulfilled when the appropriate Cauchy boundary conditions are taken into account. We also illustrate how, in the limit of the infinite potential box, the virial theorem holds as well. As a consequence, it is possible to determine the averaged force exerted by the walls on the particle. Finally, a discussion of these results in the classical limit is provided.
Generalisations of the virial theorm in Classical Mechanics and Quantum Mechanics are examined. It is shown that the generalised virial theorem in Quantum Mechanics leads to certain relations between matrix elements. The differences between the generalisations in Classical and Quantum Mechanics are identified. Some results arising from the radial Schrödinger equation in Quantum Mechanics are discussed. It is also shown that the generalisations of the virial theorem may be extended to arbitrary number of dimensions.
A method, similar to the one previously used by the author in his derivation of a generalized classical virial theorem (see ibid., vol.10, no.4, p.507 (1977)), is used to derive a generalized quantum virial equation. This equation is applicable to any part of a larger system of particles. Furthermore, by the introduction of a flux density operator, it is possible to express the quantum surface flux virial in two alternative forms, as easily interpreted as the classical form. As an intermediate result, an equation of continuity for a general one-particle observable is obtained. In an appendix, an equation of motion of the reduced density matrix of the first order is derived.