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Find the wavefunction for a 1 dimensional wave packet

  • #1
1,789
4
Hi,

I'm teaching myself quantum mechanics (so this isn't homework). I came across the following question:

[tex]\Phi(p) = A\Theta\left[\frac{\hbar}{d}-|p-p_{0}|\right][/tex]

I have to find the constant of normalization, [itex]\psi(x)[/itex], and the coordinate space wave function [itex]\psi(x,t)[/itex] in the limit [itex]\frac{h/d}{p_{0}} << 1[/itex].

I started by finding [itex]A[/itex]:

[tex]\int_{-\infty}^{\infty}\frac{|\Phi(p)|^2}{2\pi\hbar}dp = 1[/tex]

This gives [itex]A = \sqrt{\pi d}[/itex].

Now,

[tex]\psi(x) = \frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}dp'\Phi(p')e^{-ip'x/\hbar}[/tex]

which gives

[tex]\psi(x) = \sqrt{\frac{d}{\pi}}e^{ip_{0}x/\hbar}\frac{\sin(x/d)}{x}[/tex]

(There may be an algebraic error here..)

My problem is: how do I find [itex]\psi(x,t)[/itex]? I am not sure how to proceed here.
 

Answers and Replies

  • #2
1,789
4
Anyone?
 
  • #3
1,789
4
Solved

Got it.

[tex]\Phi(p,t) = \Phi(p)e^{-\frac{p^{2}t}{2m\hbar}}[/tex]

and

[tex]\psi(x,t) = \int \Phi(p,t)e^{-ipx/\hbar}dp[/tex]
 

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