Quantum energy of a particle in a 2 dimensional space

[Apashanka]

Homework Statement

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Homework Equations

Doing this problem like e.g setting the determinant of potential matrix and the ω2*kinetic matrix equal to 0 ,det(V-ω2T)=0,I got the frequency of the normal modes of vibration to be 2ω0 and ω0 where ω0 is the natural frequency,
But sir how to treat this problem quantum mechanically?
The term z is a typographical error..[/B]

The Attempt at a Solution

 

[DrClaude]

I imagine that you know the solution to the quantum harmonic oscillator. In that case, you should try to map the problem to that of uncoupled harmonic oscillators. If you notice the symmetry of the problem, that gives you a hint that you can achieve this by performing a certain rotation.