The Virial Theorem

Consider

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} &\frac{d}{dt}<xp>\\ &=\frac{1}{i\hbar}<[xp,H]> \\ &=\frac{ 2<p^{2}> }{2m}+\frac{1}{i\hbar}<xpV-xVp>\\ &=\frac{2<p^{2}>}{2m}+ \frac{1}{i\hbar}\int_{-\infty}^{\infty}dx[\psi^{*}x\frac{\hbar}{i}\frac{\partial}{\partial x} (V\psi)-xV\frac{\hbar}{i}\frac{\partial}{\partial x}\psi] \\ &= \frac{2<p^{2}>}{2m}+<x\frac{\partial V}{\partial x}> \end{align} }

Taking time average at both sides, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{<xp>_{t=T}-<xp>_{t=0}}{T}=2\frac{1}{T}\int_{0}^{T}dt<\frac{p^{2}}{2m}>-\frac{1}{T} \int_{0}^{T}dt <x\frac{\partial V}{\partial x}> }

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T \rightarrow \infty , LHS \rightarrow 0 } .

For stationary state, the expectation values are constant in time, so we arrive $2<KE>=<x{\frac {\partial V}{\partial x}}>$, which is known as the Virial Theorem.

In 3D, it is modified to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2<KE>=<\mathbf{r} \cdot \nabla V> = -<\mathbf{r} \cdot \mathbf{F} > } .