Based on the Feynman–Hellmann theorem, a new proof of the quantum virial theorem is presented. The same approach is also used to derive the virial formulation in relativistic quantum mechanics and for the case where the vector potential exists.
Abstract The Hellmann–Feynman, virial and comparison theorems are three fundamental theorems of quantum mechanics. For the first two, counterparts exist for classical mechanics with relativistic or nonrelativistic kinetic energy. It is shown here that these three theorems are valid for classical mechanics with a nonstandard kinetic energy. This brings some information about the connections between the quantum and classical worlds. Constraints about the functional form of the kinetic energy are also discussed.