Finding the probability density of a recombined beam.

[Whistlekins]

Homework Statement

So a neutron beam is split into two components, one by reflection, the other by transmission. The phase shift undergone by the reflected beam is $\pi$ radians, and the phase shift of the transmitted beam is $\Delta$.

What is the equation of the probability density of the recombined wave?

Homework Equations

Wave equation psi = A*e^(i(kx - wt))

The Attempt at a Solution

I'm assuming that the expressions for the reflected wave is psi_r = psi * e^i*pi and the transmitted wave is psi_t = psi * e^i*del

Then it would just be a simple case of adding them, and squaring the absolute of the sum to find the probability density? Which seems to be a simple algebra problem. But I can't for the life of me seem to arrive at the correct expression, given to be:

rho = 2|psi|^2 * sin(del/2)^2, where psi is the equation of the original beam.

A point in the right direction would be great, even if it's just affirming that my expressions for the reflected and transmitted waves are correct.

Thanks!

 

[BruceW]

I'm assuming that the expressions for the reflected wave is psi_r = psi * e^i*pi and the transmitted wave is psi_t = psi * e^i*del

I think you've got it almost right. But does it make sense that the 'magnitudes' of the transmitted and reflected waves are each as great as the incoming wave?