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Physical Review

Virial Theorem and its Generalizations in Scattering Theory

Peter D. Robinson, Joseph O. Hirschfelder

Published Feb 1, 1963• 33 citations
Publisher

Abstract

Heretofore, the hypervirial theorems recently introduced by Hirschfelder have only been applied to bound-state systems. In this paper it is shown that, with certain modifications, these theorems can also be applied to free systems, and, in particular, to scattering problems. The new form of the general hypervirial theorem is derived, and the theory is illustrated with the problem of a particle scattered by a central field. The ordinary virial theorem is deduced, together with other results of physical interest. Both classical and quantum-mechanical formalisms are considered, and in some cases the semiclassical approximation links corresponding results.

DOI
10.1103/physrev.129.1391

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Heretofore, the hypervirial theorems recently introduced by Hirschfelder have only been applied to bound-state systems. In this paper it is shown that, with certain modifications, these theorems can also be applied to free systems, and, in particular, to scattering problems. The new form of the general hypervirial theorem is derived, and the theory is illustrated with the problem of a particle scattered by a central field. The ordinary virial theorem is deduced, together with other results of physical interest. Both classical and quantum-mechanical formalisms are considered, and in some cases the semiclassical approximation links corresponding results.
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.
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.
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.
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.