Help Finding eigenvalues of angular momentum operators

[thebigstar25]

urgent help!.. Finding eigenvalues of angular momentum operators

the question is asking to find the eigenvalues of:

3/5 Lx - 4/5 Ly ...


I feel that i have to connect it with the L^2 and Lz operators but i just have no idea how to start .. any suggestions will be greatly appreciated ..

 

[Mindscrape]

Why is there a ..., is there more to the problem?

If you know the matrix forms of the operators (are you in 2D, 3D, what?), then all you have to do is subtract and diagonalize.

 

[thebigstar25]

... R just dots , and I am supposed to do it without considering the matrix form :(

 

[Mindscrape]

Ah, okay, take a general ket and work with it then.

You should know L^2|l,m> and Lz|l,m> and that L^2=Lx^2+Ly^2+Lz^2

 

[thebigstar25]

that what I was confused about .. how would I apply L^2=Lx^2+Ly^2+Lz^2 in order to find 3/5 Lx - 4/5 Ly , since the expression I have without "square" .. I am not sure how to connect them together..

 

[vela]

Once you know the answer, it seems like it should have been obvious. One suggestion is to think about rotations.

A different approach would be to write L_x and L_y in terms of the ladder operators L₊ and L_-, but I'm not sure how this way works out.

 

[thebigstar25]

i tried ur second suggestion but things got messy there .. How would i make use of the symmetry along z axis..

 

[nickjer]

Does it only ask for eigenvalues, or eigenvectors as well? Because eigenvalues are very simple to get using rotations as vela noted.

 

[thebigstar25]

it is just the eigenvalues .. My problem is that I am not sure how to apply that to my question .. Anyway, this question from an old homework and i just wanted to know how to solve it ..

 

[nickjer]

First, do you know what the rotation operator for angular momentum is? Since that will help you solve this problem.

 

[thebigstar25]

no i don't know, or maybe i do but i don't know that it is called rotation operator?.. I am not sure :/