For a particle in a two-dimensional box, the total energy eigenfunctions are $\psi_{n_x,n_y}(x,y) = N \sin \frac{n_x \pi x}{a} \sin \frac{n_y \pi y}{b}$.
Obtain an expression for $E_{n_x,n_y}$ in terms of $n_x$, $n_y$, $a$, and $b$ by substituting this wave function into the following equation:
$-\frac{\hbar^2}{2m} (\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) \psi(x,y) = E \psi(x,y)$.
Match the items in the left column to the appropriate blanks in the sentences and the equations on the right. Make certain each sentence and equation is complete before submitting your answer.
