Quantum mechanical virial theorem in systems with translational and rotational symmetry

TL;DR

This paper derives a generalized quantum virial theorem for systems with translational and rotational symmetry, expressed through commutators involving the dilation generator and the Hamiltonian, applicable to many quantum systems.

Contribution

It introduces a new formulation of the virial theorem in quantum mechanics using commutators, applicable to both nonrelativistic and relativistic systems with symmetry.

Findings

Matrix elements of [G, H] vanish under symmetry conditions

Eigenvectors form an orthonormal basis in the symmetric subspace

The theorem applies to a wide class of quantum N-particle systems

Abstract

Generalized virial theorem for quantum mechanical nonrelativistic and relativistic systems with translational and rotational symmetry is derived in the form of the commutator between the generator of dilations G and the Hamiltonian H. If the conditions of translational and rotational symmetry together with the additional conditions of the theorem are satisfied, the matrix elements of the commutator [G, H] are equal to zero on the subspace of the Hilbert space. Normalized simultaneous eigenvectors of the particular set of commuting operators which contains H, J^{2}, J_{z} and additional operators form an orthonormal basis in this subspace. It is expected that the theorem is relevant for a large number of quantum mechanical N-particle systems with translational and rotational symmetry.