[UPSC CSE 2007] Schrodinger’s Wave equation and applications (Q209, 20M)

Determine the discrete energy levels and the corresponding eigenfunctions for a particle in an infinitely deep potential well inside a cube of dimension L, i.e., assume \begin{gathered} V(x,y,z) = 0 \quad \text{for } 0 < x < L;\\ 0 < y < L;\\ 0 < z < L = \infty \quad \text{elsewhere} \end{gathered}