I'm trying to solve the Schrödinger equation for a free particle in a helix, but I have found it difficult to understand the normalization of the wave function. Well, let $\alpha(\phi) = b\cos(\phi)\hat{i} + b\sin(\phi)\hat{j} + a\phi \hat{k}$ the parametrization of a circular helix, for $a$ and $b$ constants and $\phi \in \mathbb{R}$ an angular variable. We get the Laplacian $ \nabla^{2} = \dfrac{1}{a^{2} + b^{2}}\dfrac{d^{2}}{d\phi^{2}}.$
It follows that the Schrödinger equation is $$ -\dfrac{\hbar^{2}}{2m}\dfrac{1}{a^{2} + b^{2}}\dfrac{d^{2}\psi}{d\phi^{2}} = E\psi$$ or $$ \dfrac{d^{2}\psi}{d\phi^{2}} + k^{2}\psi = 0, $$ for $k^{2} = \dfrac{2m(a^{2} + b^{2})}{\hbar^{2}}E$. Consequently, $\psi(\phi) = Ae^{ik\phi}$, where $A$ is a normalization constant.
Looking at the geometry of the curve, I believe that the wave function must be periodic, that is, $\psi(\phi) = \psi(\phi + 2\pi)$. Thus, $k$ must be an integer: $k_{n} = n = 0, \pm 1, \pm 2, \cdots. $ With this, the possible values for energy are obtained: $$ E_{n} = \dfrac{\hbar^{2}n^{2}}{2m(a^{2} + b^{2})}.$$
My biggest problem is finding the normalization constant. If I consider $0 < \phi < 2\pi$ (because the wave function is periodic at $2\pi$), then $$ 1 = \int_{0}^{2\pi}|\psi|^{2}d\phi = |A|^{2}2\pi$$ or $A = \dfrac{1}{\sqrt{2\pi}}$.
But, in fact, $\phi$ can take on any real value in the helix. In that case, we get that $\int_{-\infty}^{\infty}|\psi|^{2}d\phi$ diverges! Which of the two cases is correct? In the first, the particle behaves similarly to a particle in a ring, in the second there are no quantized energies.
