Ehrenfest's Theorem

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"When we speak of regaining classical mechanics, we refer to the numerical aspects. Qualitatively we know that the deterministic world of classical mechanics does not exist. Once we have bitten the quantum apple, our loss of innocence is permanent." --R. Shankar, Principles of Quantum Mechanics

[edit] Theorem

If $\hat{\Omega}$ is an operator with no explicit time dependence, then

$\frac{d}{dt} \langle \hat{\Omega} \rangle = \left({\frac{-i}{\hbar}}\right) \langle \phi | [\hat{\Omega}, \hat{H}] | \phi \rangle + \langle \phi | \hat{\dot{\Omega}} | \phi \rangle$

[edit] Discussion

Ehrenfest's Theorem describes the time evolution of the expected value of dynamic variables. The relations that derive from this system share a striking similarity to the relations of classical physics. For the majority of systems, as the size of the system increases the relative fluctuations decrease, and so the systems begin to resemble, more and more, their classical counterparts.

[edit] Derivation

By the product rule

$\frac{d}{dt} \langle \hat{\Omega} \rangle = \langle \dot{\phi} | \hat{\Omega} | \phi \rangle + \langle \phi | \hat{\dot{\Omega}} | \phi \rangle + \langle \phi | \hat{\Omega} | \dot{\phi} \rangle$

By the Schrödinger Equation

$\frac{d}{dt} \langle \hat{\Omega} \rangle = i\hbar \langle \phi | \hat{H}\hat{\Omega} | \phi \rangle + \langle \phi | \hat{\dot{\Omega}} | \phi \rangle - i \hbar \langle \phi | \hat{\Omega}\hat{H} | \phi \rangle$

Finally

$\frac{d}{dt} \langle \hat{\Omega} \rangle = \left({\frac{-i}{\hbar}}\right) \langle \phi | [\hat{\Omega}, \hat{H}] | \phi \rangle \langle \phi | \hat{\dot{\Omega}} | \phi \rangle$

Note the striking similarity to the classical equation, written in terms of Poisson Brackets.