Consider a particle of mass \(M\) moving in the \(xy\) plane in the presence of the Coulomb potential \[V = -\frac{\alpha}{\sqrt{x^2 + y^2}} \qquad (\alpha > 0) \,.\] Bound states \(\psi\) and energy lelvels \(E < 0\) are determined by the time-independent Schrödinger equation \[-\frac{\hbar^2}{2 M} \nabla^2 \psi + V \psi = E \psi \,.\] In polar coordinates (\(x = r \cos \theta\), \(y = r \sin \theta\)), the equation reads \[-\frac{\hbar^2}{2 M} \left( \frac{\partial^2}{\partial r^2} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} \right) \psi - \frac{\alpha}{r} \psi = E \psi \,.\] Using separation of variables, along with the condition that \(\psi\) is single-valued, we get \[\psi(r,\theta) = R(r) e^{i m \theta} \qquad (m \in \mathbb{Z}) \,,\] where the radial wave function \(R(r)\) satisfies \[\frac{d^2 R}{d r^2} + \frac{1}{r} \frac{d R}{d r} + \left( \frac{2 M E}{\hbar^2} + \frac{2 M \alpha}{\hbar^2 r} - \frac{m^2}{r^2}

Nuts and bolts of quantum mechanics