Product eigenstate eigenvalues

[quasar_4]

Homework Statement

What are the eigenvalues of the set of operators (L1^2, L1z, L2^2, L2z) corresponding to the product eigenstate $$\left\langle$$m1 l1 | m2 l2 $$\right\rangle$$?

PS: If you have Liboff's quantum book, this is problem #9.30.

Homework Equations

We've also been learning about the Clepsch Gorden coefficients, so they might play a role. I can't figure out enough of the tex on this website to type them (It doesn't seem to like my LaTeX commands, even though I'm usually pretty well-versed on it!).

We also have that | l1l2m1m2> = |l1m1> |l2m2>.

The Attempt at a Solution

I'm very, very confused about these eigenvalues. I am not even sure what the eigenvalue equation looks like. I thought that since | l1l2m1m2> = |l1m1> |l2m2>, maybe the eigenvalues are just the same as | l1l2m1m2> eigenvalues, so that's my current best guess, but I'm not sure. I am confused as to how or if this is related to somehow finding Clebsch-Gordon coefficients. Basically, anything you can tell me will help. In fact, I'd love help just interpeting the problem (I'm not really sure what it is I'm looking for to find these eigenvalues). Any hints? Anyone?

 

[Jhero]

Thank you for your question. It seems like you are on the right track in terms of thinking about the eigenvalues of the given set of operators. The eigenvalue equation for an operator is given by:

A|ψ⟩ = λ|ψ⟩

Where A is the operator, |ψ⟩ is the eigenstate, and λ is the corresponding eigenvalue. In this case, we have four operators: L1^2, L1z, L2^2, and L2z. These operators correspond to the angular momentum of two particles, with L1^2 and L2^2 representing the total angular momentum squared, and L1z and L2z representing the z-components of the angular momentum.

Since we are given the product eigenstate |m1l1⟩|m2l2⟩, we can use the properties of angular momentum to determine the possible values of the eigenvalues. As you mentioned, the Clebsch-Gordan coefficients will play a role in this, as they relate the eigenstates of the individual particles to the overall eigenstate.

To find the eigenvalues, we can use the fact that:

L1^2 |m1l1⟩ = l1(l1+1)|m1l1⟩
L1z |m1l1⟩ = m1|m1l1⟩
L2^2 |m2l2⟩ = l2(l2+1)|m2l2⟩
L2z |m2l2⟩ = m2|m2l2⟩

From this, we can see that the eigenvalues for L1^2 and L2^2 will be l1(l1+1) and l2(l2+1) respectively. For L1z and L2z, the eigenvalues will be m1 and m2 respectively. So the eigenvalues for the given set of operators will be (l1(l1+1), m1, l2(l2+1), m2).

I hope this helps clarify the problem for you. If you have any further questions, please don't hesitate to ask. Good luck with your studies!