Comment about particle on a circle

The wave function of a particle on a circle is a solution of the Schrödinger equation \begin{equation}\label{eq:20170129a} i \frac{\partial \psi}{\partial t} = - \frac{1}{2 m} \frac{\partial^2 \psi}{\partial x^2} \end{equation} with $x \in [0 , 2 \pi]$ and $\hbar = 1$. When \eqref{eq:20170129a} is solved in physics books, it is usually imposed that the wave function should be periodic [1]. I used to be puzzled why one has to impose the periodicity. After all, I thought, only the probability density function $|\psi|^2$ has physical meaning, so one could as well impose that \begin{equation}\label{eq:20170129b} \psi(2 \pi) = e^{ i \alpha} \psi(0) \quad\text{with}\quad \alpha\in\mathbb{R} \end{equation}