What Are the Allowed Values of m_n in Angular Momentum?

[fizixgal]

Need help with angular momentum!

Hi - I have been working on this homework problem for 3 days now and I'm so stumped! Any help would be much appreciated!

A particles with total angular momentum J = 1 is in an eigenstate |j,m> of the operators J^2, J_z. This problem concerns the probability of finding the particle in an eigenstate |j,m_n> of the operator K(hat) = n (dot) J which is the projection of ang. mom. along n.

n = sin(theta)cos(phi) e_x + sin(theta)sin(phi) e_y + cos(theta) e_z ---> spherical polar angles

What are allowed values of m_n?

How would I go about finding the probability of finding the particle in eigenstate |j,m_n> for any value of m_n when the particle is prepared in the state |j,m=1> where I think j is always 1 for this problem.

Thanx!

 

[StatMechGuy]

What do you think the allowed values of $$m_n$$ are? Is there anything particularly sacred about your choice of the z direction?

Also, there are two good ways to look at this problem. One is to compute said rotated operator eigenstates, the other is to rotate the z-direction eigenstate problem into the n-direction. Either way works.