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## Homework Statement

The unperturbed Hamiltonian H

_{0}of two independent one-dimensional operators is

[tex]H_0=a^{\dagger}a+2b^{\dagger}b[/tex]

where a and b are operators such that [tex][a,a^{\dagger}]=1=[b,b^{\dagger}][/tex]

Find the degeneracies of the eigenvalues of H

_{0}with energies E

_{0}= 0, 1, 2, 3, 4.

## The Attempt at a Solution

As I understand it, the eigenvalues of H

_{0}ARE the energies E

_{0}= 0, 1, 2, 3, 4. So that we have the equation H

_{0}|n>=E

_{0}|n>. But I'm not sure how to evaluate H

_{0}|n> as all we know about the operators a and b is the commutation relations they satisfy.