Solution of 1D Schrodinger equation with the potential $V(x) = -\frac{1}{|x|}$

Question

Consider a potential, $$V(x) = -\frac{1}{|x|}$$ and, if we apply this to a one-dimensional Schrodinger's equation, I'd like to know the solution for the wave function in 1D. Does it have a simple analytical solution? Does it have any oscillatory behavior like $$\psi(x,t) = P(x) e^{ikx}e^{i\omega t}$$ I mean will there be a factor like $e^{ikx}$ ? From the internet search, looking at one-dimensional hydrogen atom, first of all I am not sure whether there is any analytical solution, but I guess it was suggested that an exponential decay, something like $$P(x) = e^{-\alpha x}$$ is present. But I am not sure about presence of oscillations like $e^{ikx}$. Hence I'd appreciate some suggestions and clarification.

PS : I am not interested in Hydrogen atom, but in this specific 1D potential.

Answer 1

With a potential $V(x) = -\frac{\alpha}{|x|}$, with the notation $a = \frac{\hbar^2}{m \alpha}$, solutions are $$u^+_n(x,t) \sim x e^{-\frac{x}{na}} L_{n -1}^1 \left(\frac{2x}{na}\right) e^{-\frac{1}{\hbar}E_nt} \;\text{for}\; x\gt0 \\ u^+_n(x,t) = 0 \;\text{for}\; x\le0$$ and $$u^-_n(x,t) \sim x e^{+\frac{x}{na}} L_{n -1}^1 \left(\frac{2x}{na} \right) e^{-\frac{1}{\hbar}E_nt} \;\text{for}\; x\lt 0 \\ u^-_n(x,t) = 0 \;\text{for}\; x\ge0$$ whose energy is $$E_n = - \frac{1}{n^2} \left(\frac{m \alpha^2}{2 \hbar^2}\right)$$ where $L_n^\gamma$ is the Generalized Laguerre Polynomial.

[EDIT] There are 2 different set of basis functions, see this reference, page 192, formulae 20a and 20b.

Answer 2

One-dimensional Schrödonger equation with Coulomb potential is known to be problematic, as the ground state energy diverges. It can however be solved by introducing an appropriate cutoff. Here is the abstract from the classical Loudon's paper One-dimensional hydrogen atom, which solves this problem:

The quantum-mechanical system which consists of a particle in one dimension subjected to a Coulomb attraction (the one-dimensional hydrogen atom) is shown to have a ground state of infinite binding energy, all the excited bound states of the system having a twofold degeneracy. The breakdown of the theorem that a one-dimensional system cannot have degeneracy is examined. The treatment illustrates a number of properties common to the quantum mechanics of one-dimensional systems.

Although initially published in American Journal of Physics, as being of only academical interest, the paper has become a rather cited one in the field of carbon nanotubes, which are effectively one-dimensional systems, and where the exciton energies were predicted to be anomalously large (of course, in the nanotubes there is a natural cutoff parameter — the nanotube diameter). The problem is less manifest in other one-dimensional structures, such, e.g., as semiconductor quantum wires, since there is screening of the Coulomb potential by the electrons present in the material outside of the wire.