To find the energy eigenvalues in the 3D Hilbert space

[Double_Helix]

A fictitious system having three degenerate angular momentum states with $\ell=1$ is described by the Hamiltonian $$\hat H=\alpha (\hat L^2_++\hat L^2_-)$$ where $\alpha$ is some positive constant. How to find the energy eigenvalues of $\hat H$?

 

[Orodruin]

How would you usually find the eigenvalues of an operator?

Also, the end tag is [/tex], not [\tex]. Or you can just use double hashes (##) for both tags instead.

 

[Double_Helix]

1. How would you usually find the eigenvalues of an operator?

   I tried to write the matrix form of the operator.

 

[Orodruin]

1. 
   I tried to write the matrix form of the operator.

And that looked like? What did you do after that?