1. The problem statement, all variables and given/known data A two-dimensional isotropic harmonic oscillator of mass μ has an energy of 2hω. It experiments a perturbation V = xy. What are its energies and eigenkets to first order? 2. Relevant equations The energy operator / Hamiltonian: H = -h²/2μ(Px² + Py²) + μω(x² + y²) 3. The attempt at a solution The only eigenstates with such an energy are |1 0> and |0 1>, so now I have to find an operator that a) conmutes with H and V and b) has non-degenerate eigenvalues whose eigenkets are linear combination of these two states. I've been trying some operators, but none seems to fulfill all the conditions. Which one would you use? Thank you for your time.