# Two dimensional hydrogen atom

I was trying to solve for the wave function of an electron in a hydrogen atom confined to only two dimensions (in flatland). First of all I figured out the electrostatic potential energy in flatland. This came out to be $$U=-\frac{Ze^2}{2\pi\epsilon_0}\log{r}$$ Here I have chosen $$r=1$$ as the surface of zero potential.

The Schrödinger equation in two dimensions is $$-(\frac{1}{r}\partial_r(r\partial_{r}\Psi)+\frac{1}{r^2}\partial^{2}_{\theta}\Psi) + U\Psi=E\Psi$$ To avoid excessive typing I have decided to set $$\hbar/2\mu$$ to one. Using $$\Psi=R(r)Y(\theta)$$ the equation can be separated into radial and angular equations.

The angular equation is $$\partial^{2}_{\theta}Y=-m^2Y$$ and the radial equation is $$\frac{1}{r}\partial_r(r\partial_{r}R)+(E+\frac{Ze^2}{2\pi\epsilon_0}\log{r}-\frac{m^2}{r^2})R=0$$

I am able to easily solve and quantise the angular equation but I have been stuck on the radial equation for days now. I do not know how to solve the radial differential equation (I am not very good at solving second order DE). I am also not able to figure out how to quantise the total energy of the electron.

Questions:

1) How to solve the radial wave function equation and how does the energy quantise in such an atom?

2) When I looked for online sources for help I found that everyone considered the potential energy to be of the form $$1/r$$ instead of $$\log(r)$$. According to Gauss law the electrostatic potential will be logarithmic in two dimensions. Why do they consider the potential to be of the form $$1/r$$?

• If a differential equation is not analytically solvable, you can always solve it numerically. – G. Smith Apr 15 '20 at 16:40
• @G.Smith How exactly? – Manvendra Somvanshi Apr 15 '20 at 18:03
• Wikipedia’s Numerical methods for ODEs mentions several algorithms. – G. Smith Apr 15 '20 at 18:17
• Never think you're the first one to have thought about a problem: newton.ex.ac.uk/research/qsystems/portnoi/JMP_43_4681.pdf and web.math.ku.dk/~schlicht/DL/2013/HydrogenAtom.pdf – Gert Jun 4 '20 at 15:28
• @Gert I have seen both of the papers you have sited here already. Both of them are different from the problem that I am (or was) solving. The potential used in both papers is the coulomb potential, which not applicable in flat land, it is only applicable in confined two dimensions in 3D space ( as explained by @Vadium). Also I never said that I am the first one to think of this in my post, all I said that I was having a problem here and needed help. – Manvendra Somvanshi Jun 4 '20 at 16:03

I agree that this equation probably does not admit a solution in terms of elementary functions. However, with a little digging I found this paper: Atabek et. al. Phys. Rev. A 1974 covering your questions (they use a logarithmic 2D potential and they discuss the spectrum of energy eigenvalues).

One useful thing they do is substitute $$R_m = r^{-½}f_m$$ to eliminate the term proportional to $$R^\prime$$ in the equation and to generate an equation of the form: $$f^{\prime \prime}_m = g(r,m) f_m$$ which is nicer to work with.

Regarding a numerical solution:

What you have here is a "2-point boundary value problem (BVP)" (since you specify the value of $$R_m$$, or equivalently $$f_m$$, at $$r = 0$$ and you want it to vanish as $$r \rightarrow \infty$$) which are often treated with so-called "Shooting Codes".

If you are familiar with simpler (1-step, explicit) numerical methods such as RK4, one simple thing to do is iterate over values of $$E$$ with a guess for $$R^\prime(0)$$ and check the value of solution ($$R$$) at some large, final value of $$r$$. Depending on your units, even something like $$r_f = 10$$ seems to suffice. Then you can manually adjust your search-grid of $$E$$'s until this final value, $$R(r_f)$$, is approximately zero.

I coded-up this RK4-iterative scheme just to check the qualitative shape of the radial wave functions (with $$m = 0$$) reported in the linked paper from 1974 and they seem to agree (c.f. Fig. 4 of the paper).

One caveat: don't pay attention to the numerical values of $$E$$ or $$R_{m=0}$$ in the following figure, the units are probably a bit confused. Also, I normalized the peaks of $$R_m$$ to 1 for comparison across $$m$$'s.

Nevertheless, for qualitative inspection, here are three eigenfunctions of the equation, one for $$m = 0$$, and two for $$m = 1$$. These were found using the iterative RK4 method.

I doubt that this equation is solvable, although it would be wise to check in a book on special functions, such as Abramovitz&Stegun or Gradshtein &Ryzhik.

Your equation seems to be correct for a flatland. Many problems however deal with real 3D world where the motion is confined to two or even one dimensions. In this case the potential remains 3D: $$1/r$$. One-dimensional case is notable, since the binding energy diverges, which long posed a problem for analyzing excitons in carbon nanotubes.