Virial Theorem

$2⟨T⟩=⟨\overrightarrow{r}\cdot \overrightarrow{\mathrm{\nabla }}\hat{V}⟩$.

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

$H|{\phi }_e⟩=E|{\phi }_E⟩$. $\frac{d}{dt}⟨A⟩=\frac{d}{dt}⟨{\phi }_E|A|{\phi }_E⟩=0=\frac{-i}{\hslash }⟨[A,H]⟩$

$[A,H]=[\overrightarrow{r}\cdot \overrightarrow{p},H]=[\overrightarrow{r}\cdot \overrightarrow{p},\frac{{\hat{\overrightarrow{p}}}^2}{2m}+\hat{V}(x,y,z)]=[x{p}_x+y{p}_y+z{p}_z,\frac{p_x^2+p_y^2+p_z^2}{2m}+\hat{V}(x,y,z)]=2i\hslash T-i\hslash \overrightarrow{r}\cdot \overrightarrow{\mathrm{\nabla }}\hat{V}$.