Word Count : 350

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We need to determine the first-order perturbation correction to the energy eigenvalues for the two-fold degenerate first excited state of a two-dimensional infinite potential well. The unperturbed Hamiltonian is: \[ H_0 = -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \] with the potential: \[ V(x, y) = \begin{cases} 0, & 0 \leq x \leq L, \quad 0 \leq y \leq L \\ \infty, & \text{otherwise} \end{cases} \] The wavefunctions for the unperturbed system are given by: \[ \psi_{n_x, n_y} (x,y) = \frac{2}{L} \sin\left(\frac{n_x \pi x}{L}\right) \sin\left(\frac{n_y _______ _________ ______ _______ _________ ______ ____.
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