Simple derivation of the quantum mechanical virial theorem

H. A. Gersch

Herbert Leinfelder

Domagoj Kuić

L.D. Faddeev, L.A. Khalfin, I.V. Komarov

The validity of the virial theorem was proved by Born, Heisenberg and Jordan1 on the ground of matrix calculation and after that by Finkelstein2 on the ground of the Schro¨dinger equation. Both proofs essentially reproduced ideas of the classical approach. In the present note, a new and purely quantum mechanical derivation of this relation is given from the variational principle. The derivation is very simple and allows one to get some interesting consequences.

Y. İpekoğlu, S. Turgut

Abstract A simple proof of the quantum virial theorem that can be used in undergraduate courses is given. The proof proceeds by first showing that the energy eigenvalues of a Hamiltonian remain invariant under a scale transformation. Then invoking the Hellmann–Feynman theorem produces the final statement of the virial theorem.

Aritra Ghosh, Malay Bandyopadhyay