Probability Density for Interferometer

[Einstein2nd]

Hey guys. All of the info for the problem is in a picture (sorry for not using the template).

I've tried working on this for ours and I still can't seem to get the trig identities right :(

http://img208.imageshack.us/img208/1770/assignmentquestion2.jpg

NOTE THAT THERE SHOULD BE ANOTHER BRACKET ON THE VERY END OF THE EQUATION FOR THE PROBABILITY DENSITY. IT SHOULD HAVE sin(delta)), NOT sin(delta) AS IT CURRENTLY HAS.

from that final step, I've done many things by both hand and scientific notebook and I just can't seem to get things to simplify down properly. There is no way I could possibly post all of the different things I've tried but don't worry, I'm not simply looking for a copy-paste answer into homework. I want to be able to understand the working.

Please clarify my initial working and steer me in the correct direction. I'm pretty sure that I understand the physics, it's just the maths...

 

[Einstein2nd]

Any chance of some help?

I was told to use Euler's formula (which I have tried in the past) and then separate the real and imaginary parts and then add the squares. I'm not getting the answer correct thoguh by doing this :(

 

[nlightner]

Hello,

Thank you for reaching out for assistance with your problem. It looks like you have been working hard on this and I can understand how frustrating it can be when the math isn't working out.

Firstly, I would recommend double checking your trig identities and making sure you are using the correct ones for the problem. It may also be helpful to write out each step of your working to identify where the mistake may be occurring.

Additionally, a helpful tip for simplifying trigonometric expressions is to use the identities for sine and cosine squared, as well as the double angle formulas. This can often make the expressions easier to work with.

Another approach you could take is to use a graphing calculator or software to plot the probability density function and see if it matches with your expected result. This can help you identify any errors in your working.

In terms of understanding the physics behind the problem, it may be helpful to review the principles of interferometry and how it relates to the probability density function. This could provide some insight into how the math is connected to the physics.

I hope this helps guide you in the right direction. Keep persevering and I'm sure you will be able to solve the problem. Best of luck!