The spatial part of the ground state energy eigenfunction for a one$-$dimensional simple harmonic oscillator is

${}_0(x)=(\frac{m}{})\frac{1}{4}e-\frac{m}{2h}x2.$

Consider a quantum particle of mass $m$ that moves in two dimensions under the influence of a a spring force directed towards the origin, e$.$g$.$ a simple harmonic oscillator potential $($SHO$2).$

$($a$)$ What is the Hamiltonian in this case? What is the time$-$independent Schr$$dinger equation?

$($b$)$ Write down an expression for the spatial part of the ground$-$state energy eigenfunction, ${}_{00}(x,y).$ Show that it is correctly normalise.

$($c$)$ List the energies and degeneracies of the six lowest$-$lying energy levels.