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

American Journal of Physics

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

Donald H. Kobe

Published Feb 1, 1995• 3 citations
Publisher

Abstract

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.

DOI
10.1119/1.17978

Related papers

A geometric approach to a generalized virial theorem

José F. Cariñena, Fernando Falceto, Manuel F. Rañada

The virial theorem, introduced by Clausius in the field of statistical mechanics and later applied in both classical mechanics and quantum mechanics, is studied by making use of symplectic formalism as an approach, in the case of both the Hamiltonian and Lagrangian systems. The possibility of establishing virial-like theorems from one-parameter groups of non-strictly canonical transformations is analysed; the case of systems with a position-dependent mass is also discussed. Using the modern symplectic approach to quantum mechanics, we arrive at the quantum virial theorem in full analogy with the classical case.

arXiv:1209.4584•Journal of Physics A Mathematical and Theoretical•Sep 13, 2012•17 citations

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 +1 more

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.

Physical Review E•Jul 19, 2010•16 citations

Time-dependent quantum statistical inequalities

Sidney Golden

The following is established. Any system having a temporal evolution which is determined by its Hamiltonian and which has an initial statistical operator that is a completely monotonic function of a given nonnegative and time-independent observable will evolve in time so that the expectation value of that observable is never less than the initial value. In a suitable time-averaged sense, for systems having time-independent Hamiltonians, the velocity and acceleration of the expectation value are shown to be nonnegative. The results obtained depend in no way on the system being macroscopic. A theorem dealing with the energy of a system and another dealing with the virial of a system are derived to illustrate the use of the time-dependent inequalities.

International Journal of Quantum Chemistry•Jun 18, 2009•1 citation

Quantum Statistical Mechanics in Classical Phase Space. Expressions for the Multi-Particle Density, the Average Energy, and the Virial Pressure

Phil Attard

Quantum statistical mechanics is formulated as an integral over classical phase space. Some details of the commutation function for averages are discussed, as is the factorization of the symmetrization function used for the grand potential and for the multi-particle density. Three binary choices (eight routes) for the average energy are shown to be mutually consistent. An expression for the phase space function that gives the average virial pressure is derived.

arXiv:1811.00730•Nov 2, 2018•0 citations

Gauge-invariant non-relativistic limit of an electron in a time-dependent electromagnetic field

Donald H. Kobe, Kuo‐Ho Yang

The Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.

Journal of Physics A Mathematical and General•Oct 1, 1980•34 citations

Quantum power in de Broglie–Bohm theory

Donald H. Kobe

In the de Broglie–Bohm approach to quantum mechanics for a charged particle in a time-dependent electromagnetic field the time derivative of the energy is equal to the classical power plus a quantum power. We show that the average of the quantum power is zero. The de Broglie–Bohm energy is obtained from the quantum mechanical energy operator, which is the Hamiltonian with the gauge-dependent scalar potential subtracted. The time derivative of the average de Broglie–Bohm energy is shown to be equal to Ehrenfest's theorem for the quantum energy operator.

Journal of Physics A Mathematical and Theoretical•Apr 24, 2007•48 citations
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.
The virial theorem, introduced by Clausius in the field of statistical mechanics and later applied in both classical mechanics and quantum mechanics, is studied by making use of symplectic formalism as an approach, in the case of both the Hamiltonian and Lagrangian systems. The possibility of establishing virial-like theorems from one-parameter groups of non-strictly canonical transformations is analysed; the case of systems with a position-dependent mass is also discussed. Using the modern symplectic approach to quantum mechanics, we arrive at the quantum virial theorem in full analogy with the classical case.
The following is established. Any system having a temporal evolution which is determined by its Hamiltonian and which has an initial statistical operator that is a completely monotonic function of a given nonnegative and time-independent observable will evolve in time so that the expectation value of that observable is never less than the initial value. In a suitable time-averaged sense, for systems having time-independent Hamiltonians, the velocity and acceleration of the expectation value are shown to be nonnegative. The results obtained depend in no way on the system being macroscopic. A theorem dealing with the energy of a system and another dealing with the virial of a system are derived to illustrate the use of the time-dependent inequalities.
The Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.
In the de Broglie–Bohm approach to quantum mechanics for a charged particle in a time-dependent electromagnetic field the time derivative of the energy is equal to the classical power plus a quantum power. We show that the average of the quantum power is zero. The de Broglie–Bohm energy is obtained from the quantum mechanical energy operator, which is the Hamiltonian with the gauge-dependent scalar potential subtracted. The time derivative of the average de Broglie–Bohm energy is shown to be equal to Ehrenfest's theorem for the quantum energy operator.