Wavefunctions corresponding to states of different energy of a particle in a box are mutually “orthogonal” in the sense that, if the two wavefunctions are multiplied together and then integrated over the length of the box, the outcome is zero.

(a) Confirm that the wavefunctions for n = 1 and n = 2 are orthogonal. 

(b) Demonstrate, without doing a calculation, that all wavefunctions with even n are orthogonal to all wavefunctions with odd n.