Ehrenfest theorem ↩

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created: 2021-12-22 19:09:57
modified: 2022-01-10 04:13:04

Suppose is a solution to the Schrödinger equation for a potential and initial condition .
Then the expected position and momentum in the state satisfy:

Here we assumed the Hamiltonian operator to be the standard:

Proof

These are easy to obtain using the quantum mechanical time derivative.

Position

Since there is no explicit time dependence:

And the commutator is:

Where we used the commutator relation of and . Then

QED

Momentum

We use a very similar procedure here as well.

The commutator is a little trickier though:

This has to be observed on a wave function: