Ehrenfest theorem

The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force F = −dV/dx on a massive particle moving in a scalar potential,

${\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ,\;\;{\frac {d}{dt}}\langle p\rangle =-\left\langle {\frac {\partial V(x)}{\partial x}}\right\rangle ~.}$

Loosely speaking, one can thus say that "quantum mechanical expectation values obey Newton’s classical equations of motion". (This loose statement needs some caveats, see.)

The Ehrenfest theorem is a special case of a more general relation between the expectation of any quantum mechanical operator and the expectation of the commutator of that operator with the Hamiltonian of the system

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