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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 $1/r$?

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  • $\begingroup$ If a differential equation is not analytically solvable, you can always solve it numerically. $\endgroup$ – G. Smith Apr 15 '20 at 16:40
  • $\begingroup$ @G.Smith How exactly? $\endgroup$ – Manvendra Somvanshi Apr 15 '20 at 18:03
  • $\begingroup$ Wikipedia’s Numerical methods for ODEs mentions several algorithms. $\endgroup$ – G. Smith Apr 15 '20 at 18:17
  • $\begingroup$ 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 $\endgroup$ – Gert Jun 4 '20 at 15:28
  • $\begingroup$ @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. $\endgroup$ – Manvendra Somvanshi Jun 4 '20 at 16:03
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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.

enter image description here

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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.

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