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Paul Ehrenfest connected quantum expectation values with classical-looking equations of motion. For suitable quantum states evolving under the Hamiltonian $H=p^2/(2m)+V(x)$, the Ehrenfest theorem gives $d⟨x⟩/dt=⟨p⟩/m$ and $d⟨p⟩/dt=-⟨V^{\prime }(x)⟩$. Here $V$ is potential energy. The mean force need not equal the force evaluated at the mean position. If $V$ is a quadratic function, it does; for more general potential energy functions, a sufficiently narrow wave packet can justify an approximation. The theorem therefore states precisely which comparison between quantum mechanics and classical mechanics is exact and which step needs another assumption.^[ref]

A physicist associated with the Ehrenfest theorem and, with Tatyana Afanasyeva-Ehrenfest, the Ehrenfest model.