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## Homework Statement

Part (a): Find wavefunction and energy levels.

Part (b): Find a possible wavefunction. Is this wavefunction unique?

Part (c): What is the probability of finding it in the ground state?

Part (d): What's the probability of finding it in the second excited state?

## Homework Equations

## The Attempt at a Solution

__Part (a)__I have found the wavefunction and energy levels:

[tex]\phi = \sqrt{\frac{2}{a}}\sqrt{\frac{2}{a}}sin\left(\frac{n_x \pi x}{a}\right)sin\left(\frac{n_y \pi y}{a}\right)[/tex]

[tex]E = \frac{\hbar^2}{8ma^2}(n_x^2 + n_y^2)[/tex]

If a = b, there is degeneracy in the first excited state: |n

_{x},n

_{y}> |1,0> and |0,1> give the same energy.

__Part (b)__[tex]E = E_0 + \frac{(E_1-E_0)}{4} = \frac{3}{4}E_0 + \frac{1}{4}E_1[/tex]

I'm guessing the wavefunction will be a linear combination of ground and first excited state:

[tex] |\psi \rangle = \frac{3}{4}|\psi_0\rangle + \frac{1}{4}|\psi_1\rangle[/tex]

[tex] |\psi \rangle = \sqrt{\frac{2}{a}}\sqrt{\frac{2}{b}}\left[\frac{3}{4} sin(\frac{\pi x}{a})sin(\frac{\pi y}{a}) +\frac{1}{4}sin(\frac{2\pi x}{a})sin(\frac{2\pi y}{a})\right][/tex]

But the thing is, this wavefunction isn't normalized, as ##(\frac{1}{4})^2 + (\frac{3}{4})^2 = \frac{10}{16}##.