# Derivation of quantum virial theorem

The quantum virial thoerem is derived by arguing that the left-hand-side of the following expression is zero for stationary/bound states: $$\frac{d}{dt}\langle{\bf{r} \cdot \bf{p}}\rangle = \bigg\langle\frac{\bf{p}^2}{m}\bigg\rangle - \langle \bf{r} \cdot \nabla{V} \rangle$$ In the classical case this follows from a consideration of the time average of this expression, which tends to $$0$$. Why is the same true, intuitively, in the quantum case?

• WP. Hint: go to the Heisenberg picture. – Cosmas Zachos Sep 29 '19 at 23:58

For any operator $$O$$, the expectation value in a stationary (eigen-) state $$|\psi(t) \rangle$$ is constant, $$\langle O \rangle = \langle \psi(t) | O | \psi(t) \rangle = \langle \psi(0) | e^{i E t} O e^{-i E t} | \psi(0) \rangle = e^{(i E - i E)t} \langle \psi(0) | O | \psi(0) \rangle = \langle \psi(0) | O | \psi(0) \rangle$$ where $$E$$ is the energy of that state. Since this expectation value doesn't depend on time, the time derivative is identically 0.