Ehrenfest theorem

The Ehrenfest theorem, named after Austrian theoretical physicist Paul Ehrenfest, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force $F=-V'(x)$ on a massive particle moving in a scalar potential $V(x)$ ,

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

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

${\frac {d}{dt}}\langle A\rangle ={\frac {1}{i\hbar }}\langle [A,H]\rangle +\left\langle {\frac {\partial A}{\partial t}}\right\rangle ~,$

where $A$ is some quantum mechanical operator and $⟨ A ⟩$ is its expectation value.

It is most apparent in the Heisenberg picture of quantum mechanics, where it amounts to just the expectation value of the Heisenberg equation of motion. It provides mathematical support to the correspondence principle.

The reason is that Ehrenfest's theorem is closely related to Liouville's theorem of Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator. Dirac's rule of thumb suggests that statements in quantum mechanics which contain a commutator correspond to statements in classical mechanics where the commutator is supplanted by a Poisson bracket multiplied by $iħ$ . This makes the operator expectation values obey corresponding classical equations of motion, provided the Hamiltonian is at most quadratic in the coordinates and momenta. Otherwise, the evolution equations still may hold approximately, provided fluctuations are small.

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