Consider
d d t < x p > = 1 i ℏ < [ x p , H ] > = 2 < p 2 > 2 m + 1 i ℏ < x p V − x V p > = 2 < p 2 > 2 m + 1 i ℏ ∫ − ∞ ∞ d x [ ψ ∗ x ℏ i ∂ ∂ x ( V ψ ) − x V ℏ i ∂ ∂ x ψ ] = 2 < p 2 > 2 m + < x ∂ V ∂ x > {\displaystyle {\begin{aligned}&{\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{aligned}}}
Taking time average at both sides, we have
< x p > t = T − < x p > t = 0 T = 2 1 T ∫ 0 T d t < p 2 2 m > − 1 T ∫ 0 T d t < x ∂ V ∂ x > {\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 T → ∞ , L H S → 0 {\displaystyle T\rightarrow \infty ,LHS\rightarrow 0} .
For stationary state, the expectation values are constant in time, so we arrive 2 < K E >=< x ∂ V ∂ x > {\displaystyle 2<KE>=<x{\frac {\partial V}{\partial x}}>} , which is known as the Virial Theorem.
In 3D, it is modified to
2 < K E >=< r ⋅ ∇ V >= − < r ⋅ F > {\displaystyle 2<KE>=<\mathbf {r} \cdot \nabla V>=-<\mathbf {r} \cdot \mathbf {F} >} .
![{\displaystyle {\begin{aligned}&{\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1995920f0293ffd897ccabe258218ace9c7b57ef)
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, which is known as the Virial Theorem.
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