Tuesday, 6 August 2013

Ehrenfest's Theorem

From Ladyman & Ross 2007 "Every Thing Must Go" (p. 95: review by Cian Dorr here):
Similarly, the equation (known as Ehrenfest’s theorem), $gradV(\langle r \rangle) = m(d^2\langle r \rangle /dt^2)$, where V is the potential and r is the position operator, exhibits continuity between classical and quantum mechanics.
This formula is not quite correct. The expectation value brackets $\langle . \rangle$ are misplaced.

There are many ways of putting it, but in this context, Ehrenfest's Theorem states:
$\frac{d}{dt} \langle p \rangle = - \langle \nabla V \rangle$.
This is very similar though to the usual law from classical mechanics, for the rate of change of momentum of a particle in a field $V$:
$\frac{dp}{dt} = - \nabla V$.

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