Question on spin 1/2 states in arbitrary direction

[Shivy G]

Homework Statement

The problem is given above. I am struggling on the first two parts, where I am tasked with finding gamma and beta.

Homework Equations

For spin one half states in arbitrary directions, I know that psi = a*|n;+> + b*|n;-> .

|n;+> = cos(theta/2)|+> +sin(theta/2)*e^i*phi|->

|n;-> = -sin(theta/2)e^-i*phi|+> + cos(theta/2)|->

The Attempt at a Solution

I have attached my attempt at a solution. I am not sure what I am doing wrong at this point. Any help for the problem would be greatly appreciated :)

 

[blue_leaf77]

There are multiple mistakes in your calculation. I would suggest that you start over again from the first step by first normalizing the given $|\psi\rangle$.

 

[Shivy G]

I ended up getting the solution. I was making a mistake by splitting it up into |n;+> and |n;->. Instead, I factored out a phase and then normalized the whole thing. The answer came fairly quickly from there. Thanks.