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In Ehrenfest theorem we know that $$m\frac{d\left< x\right>}{dt}=\left< p\right>+m\left<\frac{\partial x}{\partial t}\right>.$$ So how can I exactly calculate a specific $\left<\frac{\partial x}{\partial t}\right>$ in coordinate representation?

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The "partial time derivative symbol" $\frac{\partial }{\partial t}$ denotes the explicit time derivative. The fundamental phase space variables (such as, e.g. $\hat{x}$) don't depend explicitly on time, neither in the Heisenberg picture nor in Schrödinger picture.

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