Particle in a box: possible momentum and probability

[C. Darwin]

Homework Statement

$$\Psi(x,0) = \frac{1}{\sqrt{L}}, ~~~~~~ \left|x\right| < L/2$$

At the same instant, the momentum of the particle is measured, what are the possible values, and with what probability?

Homework Equations

The Attempt at a Solution

Well, I know that $$\Delta{}x = L$$ so can I then say that since $$\Delta{}p \geq \frac{\hbar}{2L}$$ p must be greater than the same amount?

As far as finding the probability goes, I think I need to do the Fourier transform $$a(k) = \int_{-L/2}^{L/2} \frac{1}{\sqrt{L}} e^{-ikx} dx = \frac{2}{k\sqrt{L}}sin(\frac{L}{2}k)$$

Now if I take the square of a(k), how do I normalize it? What are the limits of the integral? If I normalize Psi(x) before I do the Fourier transform, will it be normalized after?

 

[lanedance]

been a while since I've done these, but if psi is normalised, & you perform your Fourier transform with the correct constants, the momentum expression will also be normalised. (could always check on an easy function to integrate)

the momentum probability integral will have limits from -infinity to infinity.

The position integration is in essence the same, however you know psi is zero outside the box