Textbook Problem

A particle on a ring has a wavefunction $\Psi =e^{im\varphi }$, where $\varphi =0\text{to}2\pi$ and $m$ is a constant.

(a) Normalize the wavefunction, where $d\tau$ is $d\varphi$. How does the normalization constant depend on the constant $m$?

(b) What is the probability that the particle is in the ring indicated by the angular range $\varphi =0\text{to}2\pi /3$? Does this answer make sense? How does the probability depend on constant $m$?