Hello everyone! For my quantum mechanics class I have to study the problem of two quantum oscillator coupled to each other and in particular to find the eigenstates and eigenergies for a subspace of the Fock space. I know that, in general, to solve this kind of problem I have to diagonalize the hamiltonian of the system that in this case is the following one: $$ H=\hbar\omega_0 (a^+a+b^+b)+\hbar J(a^+b+b^+a) $$ with a and b bosonic creation and annhilation operator for the two harmonic oscillator. What I do not understand is how to write the matrix in a given subspace. For example in the case of one quanta of energy present in the oscillators my two eigenstate would be: |00> a superposition of |01> and |10> (one quanta of energy in the first oscillator and 0 in the second one and viceversa). Is that correct? I don't know how to set the problem, thank to everyone for the help.