Schrödinger’s radial equation

The August 2014 issue of Physics Today contains a letter (page 8 ) in which M. Y. Amusia comments on the article “Bohr’s molecular model, a century later” by Anatoly Svidzinsky, Marlan Scully, and Dudley Herschbach (Physics Today, January 2014, page 33 ). Amusia points out that the radial part of the article’s D-dimensional Schrödinger equation in Hartree units is

and he criticizes the implication that the Coulomb potential does not depend on D, which it surely does. Half a century ago, following up on the work of Paul Ehrenfest, ¹ who had studied the hydrogen atom in n spatial dimensions using Bohr orbit theory, I reformulated the problem ² using Schrödinger’s equation extended to n dimensions, in which I had the Coulomb potential for n ≥ 3. I did not include the Coulomb potential for n = 1,2, as discussed by Amusia, as I was only interested in the stability of the higher-dimensional atom for n > 3. (For the dimensionality of space I used “n” rather than “D.”) In that work I gave the radial equation equivalent to the one above: I wrote, “If we now transform to n-dimensional polar coordinates, introduce n-dimensional spherical harmonics, and factor out the angular dependence, ³ the resulting radial equation takes the form

In my paper, I did not eliminate the dR/dr term to arrive at a form equivalent to the first equation. However, that is readily done by setting R = r^{−(n − 1)/2}ϕ and generalizing to nuclear charge Z, and setting n → D. The reason I have the D-dependent Coulomb potential for D > 3 and the article authors do not is that they have a different goal—to gain new insight into the real, atomic-molecular 3D world using the limiting behavior of the D-dimensional kinetic energy. In contrast, as indicated above, I was only interested in whether the D-dimensional Schrödinger “hydrogen atom” would have stable bound states for D > 3.