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

Physical Review E

Virial theorem for an inhomogeneous medium, boundary conditions for the wave functions, and stress tensor in quantum statistics

Viktor Borisovich Bobrov, S. A. Trigger, G. J. F. van Heijst, Pieter P. J. M. Schram

Published Jul 19, 2010• 16 citations
Publisher

Abstract

On the basis of the stationary Schrödinger equation, the virial theorem in an inhomogeneous external field for the canonical ensemble is proved. It is shown that the difference in the form of virial theorem is conditioned by the value of the wave-function derivative on the surface of the volume, surrounding the system under consideration. The stress tensor in such a system is determined by the average values of the wave-function space derivatives.

DOI
10.1103/physreve.82.010102

Related papers

Virial theorem in quantum statistical mechanics for a charged particle in an electromagnetic field

Donald H. Kobe

The virial theorem is derived in quantum statistical mechanics for a time-dependent Hamiltonian that describes a charged particle in a time-dependent electromagnetic field. The time derivative of the expectation value of the Clausius operator satisfies an equation which is the same form as the classical equation for the Clausius function (r⋅mv). By taking a time average, the virial theorem of quantum mechanics for time-dependent states is obtained. By taking a statistical average and a time average, the virial theorem of quantum statistical mechanics is obtained.

American Journal of Physics•Feb 1, 1995•3 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

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

On the statistical thermodynamics of quantum gases

Viktor Borisovich Bobrov

On the basis of the grand canonical Gibbs ensemble, using the virial theorem and the equations of motion for Green’s functions, it was shown that the thermodynamic functions of a one-component quantum gas are uniquely determined by the two-particle Green’s function outside the framework of perturbation theory. A thermodynamic self-consistency condition is formulated, which makes it possible to select approximate expressions for the two-particle Green’s function.

Low Temperature Physics•Jan 1, 2019•0 citations

Application of Modified Hypervirial and Ehrenfest Theorems and Several Their Consequences

Anzor Khelashvili, Teimuraz Nadareishvili

Abstract It is well-known that if the physical system is located in the restricted area, additional “surface terms” emerge in the traditional form of hypervirial and/or Erenfest theorems. In the current literature mainly one-dimensional Schrodinger equation was considered in this respect. Our observation consists in that this situation emerges automatically in spherical coordinates, as well as one of the coordinates, namely radial distance, is restricted by a half-line. In particular, these considerations are clearly manifested, when one consider spherically symmetric potentials and operators, depended only on distance. Evidently, these additional terms give rise owing the boundary conditions for wave functions and the behavior of the considered operators at the origin of coordinates. We have analyzed the role of these additional terms for various model- potentials in the Schrodinger equation. We consider regular, as well as soft-singular potentials and show that the inclusion of these terms is very important for obtaining correct physical results. Some complicated integrals for hypergeometric functions are also derived. The modified virial relations, derived below, is converted into the usual relations, when the additional terms are absent and when present, they give reasonable corrections in correct direction. This fact also provides its legitimacy.

arXiv:1912.04842•Physics of Particles and Nuclei•Jan 1, 2021•2 citations

Some Quantum-Mechanical Relations in case of singular operators and potentials

Anzor Khelashvili, Teimuraz Nadareishvili

Elaboration of some fundamental relations in three dimensional quantum mechanics is considered in connection to taking into account the restriction on domains in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. In particular, we derive the relation between the average value of operator time derivative and the time derivative of mean value of this operator, which usually is considered as the same by definition. The deviation from known result is deduced manifested by extra term, which depends on boundary behavior mentioned above. The general form for this extra term participates to the hypervirial like theorems. As a particular case, the virial theorem for a Coulomb potential is considered and correction to the Kramers sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced also and its consistency with real physical picture is demonstrated.

•Jun 2, 2018•0 citations
On the basis of the stationary Schrödinger equation, the virial theorem in an inhomogeneous external field for the canonical ensemble is proved. It is shown that the difference in the form of virial theorem is conditioned by the value of the wave-function derivative on the surface of the volume, surrounding the system under consideration. The stress tensor in such a system is determined by the average values of the wave-function space derivatives.
The virial theorem is derived in quantum statistical mechanics for a time-dependent Hamiltonian that describes a charged particle in a time-dependent electromagnetic field. The time derivative of the expectation value of the Clausius operator satisfies an equation which is the same form as the classical equation for the Clausius function (r⋅mv). By taking a time average, the virial theorem of quantum mechanics for time-dependent states is obtained. By taking a statistical average and a time average, the virial theorem of quantum statistical mechanics is obtained.
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.
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.
Abstract It is well-known that if the physical system is located in the restricted area, additional “surface terms” emerge in the traditional form of hypervirial and/or Erenfest theorems. In the current literature mainly one-dimensional Schrodinger equation was considered in this respect. Our observation consists in that this situation emerges automatically in spherical coordinates, as well as one of the coordinates, namely radial distance, is restricted by a half-line. In particular, these considerations are clearly manifested, when one consider spherically symmetric potentials and operators, depended only on distance. Evidently, these additional terms give rise owing the boundary conditions for wave functions and the behavior of the considered operators at the origin of coordinates. We have analyzed the role of these additional terms for various model- potentials in the Schrodinger equation. We consider regular, as well as soft-singular potentials and show that the inclusion of these terms is very important for obtaining correct physical results. Some complicated integrals for hypergeometric functions are also derived. The modified virial relations, derived below, is converted into the usual relations, when the additional terms are absent and when present, they give reasonable corrections in correct direction. This fact also provides its legitimacy.
Elaboration of some fundamental relations in three dimensional quantum mechanics is considered in connection to taking into account the restriction on domains in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. In particular, we derive the relation between the average value of operator time derivative and the time derivative of mean value of this operator, which usually is considered as the same by definition. The deviation from known result is deduced manifested by extra term, which depends on boundary behavior mentioned above. The general form for this extra term participates to the hypervirial like theorems. As a particular case, the virial theorem for a Coulomb potential is considered and correction to the Kramers sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced also and its consistency with real physical picture is demonstrated.