Solving 2D Oscillator: Hamiltonian Op & Eigenvalue Analysis

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    2d Oscillator
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I would be eternally grateful if I could get some help. The question is: Write down the Haniltonian operator for the 2D harmonic oscillator with the potential V(x,y)=1/2(x^2+y^2).

By using teh separation of variables and theform of the eigenvalue for the 1D harmoinc oscillator, find the energy eigenvalues for the 2D oscillator.
 
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show your attempt to solution!

What is your ansatz for eigenfunction with separation of variables?
 

Thread 'Number of quantum accessible states of a particle given T, N?'

It's given a gas of particles all identical which has T fixed and spin S. Let's ##g(\epsilon)## the density of orbital states and ##g(\epsilon) = g_0## for ##\forall \epsilon \in [\epsilon_0, \epsilon_1]##, zero otherwise. How to compute the number of accessible quantum states of one particle? This is my attempt, and I suspect that is not good. Let S=0 and then bosons in a system. Simply, if we have the density of orbitals we have to integrate ##g(\epsilon)## and we have...
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