If a quantum mechanical particle has definite energy \(E\) we can write its wave function in the form \(\Psi(\mathbf{r}, t)=\psi(\mathbf{r}) e^{-i E t / \hbar}\).

(a) Substitute this into the full Schrödinger equation to show that the time independent Schrödinger equation for \(\psi(\mathbf{r})\) may be written

\[abla^{2} \psi+\frac{2 m}{\hbar^{2}}(E-U) \psi=0\]

where both \(\psi\) and \(U\) are functions of position.

(b) A particle of mass \(m\) is trapped inside a one-dimensional box of width \(L\). The potential energy of the particle is zero for \(0

(c) Find all other energy eigenfunctions \(\psi_{n}\) of the particle, and the corresponding energy eigenvalues \(E_{n}\).