We derive the virial theorem for Hamiltonians whose domain of definition is not invariant under spatial dilatations. This generalization is important for problems where the support space is noncompact.
Given an eigenstate of a time-independent Hamiltonian, an equation is derived which generates the corresponding energy by means of a pointwise application of only the kinetic operator. Satisfaction of the equation at all points in configuration space, for arbitrary values of a scale factor, constitutes a necessary eigenstate condition. Explicit knowledge of the potential operator, other than knowledge of its homogeneity, is not required. In addition, integration of the pointwise equation yields a generalization of the familiar virial theorem.
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.
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.