Understanding Gaussian Wave Packets: Momentum, Width, and Amplitude Explained

[cepheid]

Hi,

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

Homework Statement

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 $\rightarrow \infty$.
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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The Attempt at a Solution

Here was my reasoning:

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

II. (CORRECT). $\mathbf{v}_{\textrm{phase}} = \frac{\hbar\mathbf{k}}{m}$, 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.

 

[cepheid]

bump for this thread

 

[TheDestroyer]

I know this is a late answer, but I just saw this topic and wanted to answer.

I. is correct, because the wave packet isn't centered at k=0, check Schwabl Quantum Mechanics book page 16. And also it doesn't make sense at all to be around zero, because that will cause the wave packet to include negative wave vectors.

III. With wave packets, you should use Group velocity not phase velocity, which ensures it being constant as you have said (Through the derivative).

Thanks for reading