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?