## Posts

### Two-dimensional hydrogen atom

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}