Homework Help: Gaussian Wave Packet

Hi,

I've got the right answer in this problem, but I'm not sure if I've got the correct reasoning

1. The problem statement, all variables and given/known data

A Gaussian wave packet travels through free space. Which of the following statements about the packet are correct for all such wave packets?

I. The average momentum of the wave packet is zero.
II. The width of the wave packet increases with time, as t [itex] \rightarrow \infty [/itex].
III. The amplitude of the wave packet remains constant with time.
IV. The narrower the wave packet is in momentum space, the wider it is in coordinate space.
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3. The attempt at a solution

Here was my reasoning:

I. (INCORRECT) [itex] \mathbf{p} = \hbar\mathbf{k} [/itex], and the Gaussian need not be peaked at [itex] \mathbf{k} = 0 [/itex].

II. (CORRECT). [itex] \mathbf{v}_{\textrm{phase}} = \frac{\hbar\mathbf{k}}{m}[/itex], a function of k, therefore dispersion occurs.

III. (INCORRECT) contradicts II.

IV. (CORRECT) This is a basic result of Fourier theory.
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Ans: II and IV only are correct.

 

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