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