# The Virial Theorem

### From FSUPhysicsWiki

We will now derive the quantum mechanical virial theorem. For a Hamiltonian of the form,

this theorem gives the expectation value of the kinetic energy in a stationary state in terms of the potential energy. To derive this relation, we consider the expectation value of The time derivative of this expectation value is

For a stationary state, the expectation value of is constant in time. This gives us the relation,

This is the virial theorem.

As an example of its application, let us consider the isotropic three-dimensional harmonic oscillator,

The right-hand side of the virial theorem is given by

Therefore,