Full solution of the quantum particle on a ring

In quantum particle in a ring problem, the general solution for the wavefunction, with $k = R sqrt{2 m E / hbar^2}$, $R$ being the ring radius, $c_{+, -}$ being constants, $E$ the energy, and $m$ the particle mass, is

$$psi(theta) = c_{+} e^{i k theta} + c_{-} e^{-i k theta}$$

Periodic boundary conditions give the eigenvalues $k = n$, with $n=0, pm 1, pm 2, ldots$.

At first glance the problem looks “underdetermined” – it seems that there are two unknowns ($c_{+}$ and $c_{-}$) but only one condition to find them, normalization of the wavefunction $langle psi lvert psirangle = 1$. But in my naive understanding, the problem is actually not underdetermined because one may set $c_{-} = 0$, as is seemingly done on Wikipedia in order to normalize the wavefunction.

Why may one set $c_{-} = 0$? Or, why do Wikipedia and textbooks (below) seemingly set $c_{-} = 0$? And how can we treat the general case with $c_{-} ne 0$?

Even if we do not trust Wikipedia, most textbooks on introductory quantum look at the ring problem and do the same thing. For example:

• “Introduction to Quantum Mechanics: in Chemistry, Materials Science, and Biology” By Sy M. Blinder
• “Atkins’ Physical Chemistry” By Peter Atkins, Julio de Paula
• “Quantum Mechanics for Chemists” David O. Hayward
• “Case studies in atomic physics 4” edited by E. McDaniel
• “Quantum Chemistry” By John P. Lowe