= Solution
Solving $-i\hbar u'(\phi)=\lambda u(\phi)$ gives an exponential. The <periodic boundary condition> quantizes its phase winding:
$$
\boxed{u_k(\phi)=\frac{e^{ik\phi}}{\sqrt{2\pi}},\quad
\hat l_z u_k=\hbar k u_k,\quad k\in\mathbb Z.}
$$
These are normalized with respect to $\mathrm d\phi$. The corresponding complex <Fourier series> is
$$
\boxed{f(\phi)=\sum_{k=-\infty}^{\infty}c_k u_k(\phi),\qquad
c_k=\int_0^{2\pi}u_k(\phi)^*f(\phi)\,\mathrm d\phi.}
$$
Unlike 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 $r\,\mathrm d\phi$.
Back to article page
