My name is Alexander FufaeV and here I write about:

What is the Ehrenfest Theorem?

In general, the Ehrenfest theorem states:

$$ \begin{align} \frac{\text{d}}{\text{d}t} \langle\hat{A}\rangle ~=~ \frac{i}{\hbar} \langle [\hat{H}, \hat{A}] \rangle ~+~ \langle \frac{\partial \hat{A}}{\partial t} \rangle \end{align} $$

Statement of the Ehrenfest theorem

The Ehrenfest Theorem states how classical mechanics is related to quantum mechanics, namely, that under certain conditions, classical equations apply to the expectation values of quantum mechanical quantities.

Here $ \hat{A} $ is any quantum mechanical operator and $ [\hat{H}, \hat{A}] $ is the commutator of the energy operator and the operator $ \hat{A} $. $ \langle\hat{A}\rangle $ is the mean value of the operator.

This quantum mechanical equation is analogous to the total time derivative of a classical function $ f(q,p,t) $ (with $q,p,t$ being generalized coordinates in the Hamiltonian formalism):

$$ \begin{align} \frac{\text{d}}{\text{d}t} f(q,p,t) ~=~ - \{H,f\} ~+~ \frac{\partial f}{\partial t} \end{align} $$

Here, $ \{H,f\} $ is the Poisson bracket of the Hamilton function $ H $ and the function $ f $.