 Solution

Solving $-i\hslash u^{\prime }(\varphi )=\lambda u(\varphi )$ gives an exponential. The periodic boundary condition quantizes its phase winding:These are normalized with respect to $\mathrm{d}\varphi$. The corresponding complex Fourier series isUnlike the wrapped sine subset, this orthonormal basis includes the constant $k=0$ state and both sine and cosine combinations. If one retains arc-length normalization instead, replace $1/\sqrt{2\pi }$ by $1/\sqrt{a}$ and integrate with $r\mathrm{d}\varphi$.