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Applications of quantum mechanical hypervirial theorems

John C. Nash, Nash, John C.

Published Jan 1, 1972• 0 citations

Abstract

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.

DOI: 10.5287/ora-pv6ea0g69

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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.

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.

Abstract We discuss the application of operator methods to one-dimensional systems other than the harmonic oscillator. In Section 3.1 the method of ‘oneway factorization’ is used to obtain generalized quantum harmonic potentials. In Section 3.2 factorization and shift operators for an arbitrary one-dimensional potential are considered, and a classification is presented of the problems that can be treated by this method [1]. The use of the hypervirial theorem in the calculation of expectation values for one-dimensional systems is discussed in Section 3.3. In Sections 3.4–3.7 four examples are used to illustrate the applications of the preceding material. In these examples we consider in some detail how shift operators for energy can be derived and we also use these shift operators to calculate wavefunctions. Supersymmetric quantum mechanics and shape invariance are discussed in Sections 3.8 and 3.9.

Abstract Quantum mechanical problems capable of exact solution are traditionally solved in a few instances only (such as the harmonic oscillator and angular momentum) by operator methods, but mainly by means of Schrodinger's wave mechanics. The present volume shows that a large range of one- and three-dimensional problems, including certain relativistic ones, are solvable by algebraic, representation-independent methods using commutation relations, shift operators, the virial, hypervirial, and Hellman-Feyman theorems. Applications of these operator methods to the calculation of eigenvalues, matrix elements, and wavefunctions are discussed in detail. This book is an introduction to the use of operator methods in quantum mechanics and also a reference work with numerous problmes solved. It is suitable for use by students of intermediate quantum mechanics and also more advanced postgraduate students who wish to study the algebraic method of solving quantum mechanical problems.

Sometimes it is useful and necessary to consider quantum-mechanical systems enclosed in "boxes" of different shapes with impenetrable or semipermeable walls, in contrast to the usual requirement that wave functions vanish at infinity. This kind of model has been widely applied to interpret, analyze, and discuss several physical phenomena. In this communication we describe and study the hypervirial relationships for finite systems in order to generalize some previous works related to the virial theorem. We apply our formal results to various simple models and discuss their potential applications to solid-state theory.

Abstract In both classical and quantum physics the harmonic oscillator is one of the most important physical systems [1–4]. Furthermore, because of the simplicity and symmetry of the Hamiltonian operator, the quantum mechanical oscillator is particularly suited to analysis by abstract operator methods. For this reason the study of the one dimensional oscillator is a useful introduction to the application of operator methods in quantum mechanics. In Sections 2.1–2.4 we use these methods to determine energy eigenvalues and eigenkets, and matrix elements; the analysis is extended to isotropic multidimensional oscillators in Section 2.10. Quasiclassical states of the oscillator are discussed in Section 2.5. In Section 2.6 shift operators are used to calculate wave functions. In Sections 2.7–2.11 we discuss some applications of the theory of the oscillator.