Consider the 2-dimensional harmonic oscillator with Hamiltonian 1 P + 2m + = mw (x...

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Consider the 2-dimensional harmonic oscillator with Hamiltonian 1 P² +² 2m ²+ = mw ³² (x² + y²³) Hº i) Find the energy eigenstates and eigenvalues of this Hamiltonian. Verify that the ground state is unique, while at the first excited level there are two degenerate states (Hint: you do not need to do any long computation here, you just need to use the results for the 1-dimensional harmonic oscillator in the correct way.) ii) Consider the perturbation H = H + \H' H' = ₁² + ₂xy + c3y² and C₁, C2, C3 are constants. Find the change of the energy of the ground state to first order in perturbation theory in A. where Find the change of the energy of the first excited level (there are two degenerate states at this level) to first order in perturbation theory in A. Consider the 2-dimensional harmonic oscillator with Hamiltonian 1 P² +² 2m ²+ = mw ³² (x² + y²³) Hº i) Find the energy eigenstates and eigenvalues of this Hamiltonian. Verify that the ground state is unique, while at the first excited level there are two degenerate states (Hint: you do not need to do any long computation here, you just need to use the results for the 1-dimensional harmonic oscillator in the correct way.) ii) Consider the perturbation H = H + \H' H' = ₁² + ₂xy + c3y² and C₁, C2, C3 are constants. Find the change of the energy of the ground state to first order in perturbation theory in A. where Find the change of the energy of the first excited level (there are two degenerate states at this level) to first order in perturbation theory in A.

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