Linear Combos of Y; eigenfunction of L_x

[lleee]

Homework Statement

I have to find the linear combinations of Y_10, Y_11, and Y_1-1 that are eigenfunctions of L_y. There are three such combinations...

Homework Equations

The Attempt at a Solution

Starting from (L_y)(psi_y)=(alpha)(psi_y),
Using the relationshiP: psi_y= aY_11 + bY_10 + cY_1-1
And: L_y=(i/2)(L_(-) - L_(+))

I solved it to the point where I got to:
(alpha)(psi_y) = (alpha)aY_11 + (alpha)bY_10 + (alpha)cY_1-1 = (i/sqrt(2))(bY_1-1 - bY_11 + (a-c)Y_10)

What I have to solve is this system of equations for when alpha=0, 1, -1:
(ib/sqrt(2)) = (alpha)c
(-ib/sqrt(2) = (alpha)a
i(a-c)/sqrt(2) = (alpha)b

So for instance, when alpha=-1, the answers I know are: a=1/2, c=-1/2, b=-i/sqrt(2).

Which gives one of the three linear combinations: psi_y=(1/2)Y_11 - (i/sqrt(2))Y_10 - (1/2)Y_1-1.



However, I have no idea how to solve that system of equations, though I really feel like I should :( So I guess it really boils down to me not knowing the math... help..?

 

[jdstokes]

Do you know how to find eigenvalues and eigenvectors? You have three simultaneous relations between a,b and c. Write down the whole set as matrix multiplying the column vector (a,b,c)^t. Then use the fact that a matrix equation of the form $A \mathbf{a} = \mathbf{0}$ has nontrivial solutions iff det A = 0. The solution of the charcteristic polynomial gives you the possible values of alpha. Substituting those back into your matrix equation will give you the appropriate values of a,b and c.

 

[lleee]

Can you lead me to a website that explains how to solve it that way? I think I've heard of that method but never done it myself...

 

[lleee]

Actually, unless I misunderstood you, I guess I already know the possible values of alpha, 0, 1, and -1. I'm just having a brain freeze and cannot figure out how to solve this set of equations for values of a, b, and c after substituting in each value of alpha...

 

[jdstokes]

You need to prove that alpha = -1,0,1. You do this by finding the eigenvalues of that matrix.

Do you know how to write the system of equations

(ib/sqrt(2)) = (alpha)c
(-ib/sqrt(2) = (alpha)a
i(a-c)/sqrt(2) = (alpha)b

as a matrix?

 

[lleee]

No :(