Virial Theorem

\(2\langle T\rangle = \langle \vec{r}\cdot \vec{\nabla} \hat{V}\rangle\).

Proof. \(\hat{H}=\frac{\hat{\vec{p}}^2}{2m}+\hat{V}(\vec{r})\), \(\hat{A}=\vec{r}\cdot\vec{p}\).

\(H|\varphi_e\rangle = E|\varphi_E\rangle\). \(\frac{d}{dt}\langle A\rangle = \frac{d}{dt}\langle \varphi_E|A|\varphi_E\rangle = 0 = \frac{-i}{\hbar}\langle [A,H]\rangle\)

\([A,H] = [\vec{r}\cdot\vec{p},H] = [\vec{r}\cdot\vec{p},\frac{\hat{\vec{p}}^2}{2m}+\hat{V}(x,y,z)] = [xp_x+yp_y+zp_z, \frac{p_x^2+p_y^2+p_z^2}{2m}+\hat{V}(x,y,z)]=2i\hbar T - i\hbar \vec{r}\cdot\vec{\nabla}\hat{V}\).