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Find the energy eigenvalues and eigenfunctions for a particle in a box, with  coordinates of the boundary given to be \(x=L\) and \(x=2L\). \[ V(x) =  \begin{cases} 0 , & L \le x \le 2L \\ \infty, & x < L \text{ or } x > 2L .\end{cases}\]

ANSWER: \[ E_n= \frac{\hbar^2k_n^2}{2m}, \quad u_n(x) = \sqrt{\tfrac{2}{L}} \sin k_n(x-L)\] where \( k_n = \frac{n\pi\hbar }{L}\)