A particle of mass \(m\) is trapped inside a three-dimensional box with sides of length \(L\). The potential energy of the particle is zero for \(0

(a) Find the corresponding energy eigenvalue \(E_{1}\).

(b) Find all other energy eigenfunctions \(\psi_{n}\) and corresponding energy eigenvalues \(E_{n}\) in terms of the "quantum number" \(n\).

(c) Compare these allowed energy values with those predicted by the "old quantum theory", showing that they agree in the limit of large \(n\).