Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Ehrenfest's Theorem

A fundamental theorem in statistical mechanics stating that the time evolution of the mean values of operators in a Hamiltonian system is governed by the Hamiltonian itself.
📜

The statement of the theorem

Let H\mathcal{H} be the Hamiltonian operator governing the system's time evolution, and let A^\hat{A} be an observable operator in the Hilbert space H\mathcal{H}. The time evolution of the expectation value ⟨A^⟩\langle \hat{A} \rangle is governed by the equation:\n\nddt⟨A^⟩=1iℏ⟨[A^,H]⟩+∂⟨A^⟩∂t\frac{d}{dt} \langle \hat{A} \rangle = \frac{1}{i\hbar} \langle [\hat{A}, \mathcal{H}] \rangle + \frac{\partial \langle \hat{A} \rangle}{\partial t}\n\nFor an operator A^\hat{A} that depends on position x^\hat{x} and momentum p^\hat{p}, the commutator [A^,H][\hat{A}, \mathcal{H}] can be expanded using the canonical commutation relations [x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar. This leads to the generalized form:\n\nddt⟨A^⟩=1ℏ⟨{A^,H}P⟩+∂⟨A^⟩∂t\frac{d}{dt} \langle \hat{A} \rangle = \frac{1}{\hbar} \langle \{ \hat{A}, \mathcal{H} \}_{P} \rangle + \frac{\partial \langle \hat{A} \rangle}{\partial t}\n\nwhere 1ℏ⟨{A^,H}P⟩\frac{1}{\hbar} \langle \{ \hat{A}, \mathcal{H} \}_{P} \rangle represents the expectation value of the Poisson bracket structure, linking the quantum expectation value to the classical dynamics derived from the Hamiltonian H\mathcal{H}.
Source: Wikipedia