Find the wavefunction for a 1 dimensional wave packet

[maverick280857]

Hi,

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

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

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

I started by finding $A$:

$$\int_{-\infty}^{\infty}\frac{|\Phi(p)|^2}{2\pi\hbar}dp = 1$$

This gives $A = \sqrt{\pi d}$.

Now,

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

which gives

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

(There may be an algebraic error here..)

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

 

[maverick280857]

Anyone?

 

[maverick280857]

Solved

Got it.

$$\Phi(p,t) = \Phi(p)e^{-\frac{p^{2}t}{2m\hbar}}$$

and

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