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

Hypervirial Theorems for Variational Wave Functions

Saul T. Epstein, Joseph O. Hirschfelder

Published Aug 15, 1961• 111 citations
Publisher

Abstract

It is shown that a sufficient condition for an optimal energy variational wave function ${\ensuremath{\psi}}_{0}$ to satisfy the hypervirial relation $({\ensuremath{\psi}}_{0}, [H, W]{\ensuremath{\psi}}_{0})=0$ is for the trial function $\ensuremath{\psi}$ to admit variations of the form $\frac{\ensuremath{\partial}\ensuremath{\psi}}{\ensuremath{\partial}a}=(\frac{i}{\ensuremath{\hbar}})W\ensuremath{\psi}$. Here $H$ is the Hamiltonian, $W$ is a Hermitian operator, and $a$ is a variational parameter. Explicit forms of such trial functions are exhibited for several $W'\mathrm{s}$. The case in which $W$ generates a point transformation of the coordinates is discussed in detail. Conditions are given for the existence of simultaneous hypervirial theorems.

DOI
10.1103/physrev.123.1495

Related papers

Applications of quantum mechanical hypervirial theorems

John C. Nash, Nash, John C.

Two applications of the quantum mechanical hypervirial theorem are discussed. This states that for any system having eigenstates A and B of Hamiltonian H, an operator W has the matrix element relation (A,(HW andminus; WH) B) = (EA andminus; EB) (A, W B) providing W is such that H is Hermitean between A and W B. The first application uses the operator W = eitx to derive a further relationship between certain matrix elements. For one dimensional systems in which the potential V(x) involves only positive powers of x, this relationship becomes a differential equation in f(t) = (A, eitx B) This differential equation has been solved analytically for the harmonic oscillator and in series for the quartic oscillator which were taken as example systems. The second application is the off-diagonal hypervirial method, which seeks to require a wave function A to satisfy a number of hypervirial relations. In particular the wave function A is forced to obey as many conditions as it has variable parameters. To accomplish this a computational algorithm was developed which allowed one of these parameters to be non-linear. The results for oscillator test systems are not encouraging. The systems tested were the quartic and sixth power oscillators with a trial basis of harmonic oscillator eigenfunctions. In general the results were not as good as those obtainable by the variational method.

•Jan 1, 1972•0 citations

Variational Approach to the Many-Body Problem

R. U. Ayres

Certain problems connected with the choice of trial variational functions for the quantum many-body problem are discussed. It is convenient to make use of the reduced-density-matrix formulation developed by L\"owdin and Mayer, since a system whose Hamiltonian contains only 1- and 2-body interactions is characterized by the two simplest reduced density matrices ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. It is known that symmetry, Hermiticity, and normalizability conditions are not sufficient to ensure a physically realizable choice of the functions ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. However an additional nontrivial restriction is imposed by the fact that such functions must be possible variational extrema. The implications of this condition are investigated in some detail and it turns out that all the nonphysical choices of ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$ which have been exhibited by various authors are eliminated. The condition is essentially nonquantum inasmuch as an analogous argument should be possible for a classical system. Moreover one of the consequences of this condition is that the system (classical or quantum) may exhibit macroscopic behavior, i.e., order-disorder transitions. The analysis leads to apparently reasonable choices of trial forms for ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$, although it is still not known whether any further restriction must be imposed to complete the sufficiency argument.

Physical Review•Sep 15, 1958•39 citations

Tensor Virial Theorems for Variational Wave Functions

Dave Pandres

It is shown that the tensor virial theorem and certain generalizations of it are satisfied by optimum energy variational wave functions in which the variational parameters are elements of a square matrix that scales the column matrix of particle coordinates.

Physical Review•Jul 15, 1963•14 citations

Hamiltonian formulation of quantum mechanics with semiclassical implications. II. Variational treatment

Kenneth G. Kay

It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.

Physical Review A•Aug 1, 1992•24 citations

Hypervirial and Ehrenfest Theorems in Spherical Coordinates: Systematic Approach

Anzor Khelashvili, Teimuraz Nadareishvili

Abstract Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas 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. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case, the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover, the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.

arXiv:1806.01268•Physics of Particles and Nuclei•Jan 1, 2020•3 citations

A variational calculation exhibiting both minima and maxima

Mingli Zhou, Kenneth R. Brownstein

A simple example of a variational calculation for the quantum mechanical energies of a system is presented. The trial function employed has only one (nonlinear) variational parameter. We point out a rather unexpected result for the energy extrema associated with this parameter.

European Journal of Physics•Nov 1, 1999•0 citations
It is shown that a sufficient condition for an optimal energy variational wave function \ensuremathψ0{\ensuremath{\psi}}_{0}\ensuremathψ0​ to satisfy the hypervirial relation (\ensuremathψ0,[H,W]\ensuremathψ0)=0({\ensuremath{\psi}}_{0}, [H, W]{\ensuremath{\psi}}_{0})=0(\ensuremathψ0​,[H,W]\ensuremathψ0​)=0 is for the trial function \ensuremathψ\ensuremath{\psi}\ensuremathψ to admit variations of the form \ensuremath∂\ensuremathψ\ensuremath∂a=(i\ensuremathℏ)W\ensuremathψ\frac{\ensuremath{\partial}\ensuremath{\psi}}{\ensuremath{\partial}a}=(\frac{i}{\ensuremath{\hbar}})W\ensuremath{\psi}\ensuremath∂a\ensuremath∂\ensuremathψ​=(\ensuremathℏi​)W\ensuremathψ. Here HHH is the Hamiltonian, WWW is a Hermitian operator, and aaa is a variational parameter. Explicit forms of such trial functions are exhibited for several W′sW'\mathrm{s}W′s. The case in which WWW generates a point transformation of the coordinates is discussed in detail. Conditions are given for the existence of simultaneous hypervirial theorems.
Two applications of the quantum mechanical hypervirial theorem are discussed. This states that for any system having eigenstates A and B of Hamiltonian H, an operator W has the matrix element relation (A,(HW andminus; WH) B) = (EA andminus; EB) (A, W B) providing W is such that H is Hermitean between A and W B. The first application uses the operator W = eitx to derive a further relationship between certain matrix elements. For one dimensional systems in which the potential V(x) involves only positive powers of x, this relationship becomes a differential equation in f(t) = (A, eitx B) This differential equation has been solved analytically for the harmonic oscillator and in series for the quartic oscillator which were taken as example systems. The second application is the off-diagonal hypervirial method, which seeks to require a wave function A to satisfy a number of hypervirial relations. In particular the wave function A is forced to obey as many conditions as it has variable parameters. To accomplish this a computational algorithm was developed which allowed one of these parameters to be non-linear. The results for oscillator test systems are not encouraging. The systems tested were the quartic and sixth power oscillators with a trial basis of harmonic oscillator eigenfunctions. In general the results were not as good as those obtainable by the variational method.
Certain problems connected with the choice of trial variational functions for the quantum many-body problem are discussed. It is convenient to make use of the reduced-density-matrix formulation developed by L\"owdin and Mayer, since a system whose Hamiltonian contains only 1- and 2-body interactions is characterized by the two simplest reduced density matrices ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. It is known that symmetry, Hermiticity, and normalizability conditions are not sufficient to ensure a physically realizable choice of the functions ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. However an additional nontrivial restriction is imposed by the fact that such functions must be possible variational extrema. The implications of this condition are investigated in some detail and it turns out that all the nonphysical choices of ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$ which have been exhibited by various authors are eliminated. The condition is essentially nonquantum inasmuch as an analogous argument should be possible for a classical system. Moreover one of the consequences of this condition is that the system (classical or quantum) may exhibit macroscopic behavior, i.e., order-disorder transitions. The analysis leads to apparently reasonable choices of trial forms for ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$, although it is still not known whether any further restriction must be imposed to complete the sufficiency argument.
It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.
Abstract Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas 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. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case, the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover, the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.