Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / Section B / 9 / b / Solution

For a nonzero integer mode, normalize the wavefunction:Thus $|A|=\sqrt{2/L}$. Choosing an arbitrary overall phase makes $A=\sqrt{2/L}$ real and positive. The physical modes are conventionally indexed $n=1,2,\dots$; $n=0$ would be the zero function and cannot be normalized.