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.

Ehrenfest, P. (1927). Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Zeitschrift für Physik. 45 (7-8): 455-457.

relates the time derivative of the expectation values of the position and momentum operators

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 ) {\displaystyle F=-V'(x)} on a massive particle moving in a scalar potential V ( x ) {\displaystyle V(x)} , 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 where A is some quantum mechanical operator and ⟨A⟩ is its expectation value.