Ehrenfest Theorem - Derivation

Derivation

Suppose some system is presently in a quantum state Φ. If we want to know the instantaneous time derivative of the expectation value of A, that is, by definition

where we are integrating over all space. If we apply the Schrödinger equation, we find that

and

Note, because the Hamiltonian is Hermitian. Placing this into the above equation we have

Often (but not always) the operator A is time independent, so that its derivative is zero and we can ignore the last term.