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# Ehrenfest's Theorem

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Consider a 1-D free particle, describable as a wave packet at initial time t0.

a) Show, applying Ehrenfest's theorem, that <X> is a linear function of time and <P> is a constant.

b) Write the equations of motion for the mean values <X^2> and <XP + PX>. Integrate these equations.

c) Show that, with a suitable choice of the time origin, the rms deviation <(X-<X>)^2> is given by...

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#### Solution Preview

a) Ehrenfest's Theorem:

d<x>/dt = <p>/m (1)

d<p>/dt = -<grad V> (2)

d<p>/dt = 0, because grad V is zero for a free particle. Therefore p is constant. From (1) it then follows that <x> is a linear function of t.

b) For a general time independent operator A:

d<A>/dt = i/h-bar <[H,A]>

In our case H = p^2/2m:

d<A>/dt = i/(2 m h-bar) <[p^2,A]> (3)

By using that [x,p] = i h-bar, you find after some ...

#### Solution Summary

Expressions for d<x>/dt, d<x^2>/dt, d<x p+p x>/dt are derived using Ehrenfest's theorem.

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