My name is Alexander FufaeV and here I write about:

# What is the Ehrenfest Theorem?

In general, the Ehrenfest theorem states:

0

Formula: Ehrenfest theorem

\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

0

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):

0

Total time derivative of a classical function

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

0

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