Ehrenfest's Theorem

Ehrenfest's theorem expresses the time derivative of the expectation value of a quantum observable in terms of its commutator with the Hamiltonian (Ehrenfest 1927). Let be a normalized vector satisfying , where is a self-adjoint operator and is the reduced Planck constant. For an observable , write . Under differentiability and domain hypotheses that justify the operations, the theorem gives

(1)

For unbounded operators, the state must lie in common domains on which the products and derivatives used in this identity are defined.

For a particle on a line with and , the position and momentum expectation values satisfy

			(2)
			(3)

These identities resemble the classical equations of motion, but need not equal . Consequently, the expectation value of position does not in general follow an exact classical trajectory. Equality holds when the force is an affine function of position, as in a potential of degree at most two.