Bohr’s molecular model and the melding of classical and quantum mechanics

Abstract

With considerable interest I read the article “Bohr’s molecular model, a century later” by Anatoly Svidzinsky, Marlan Scully, and Dudley Herschbach (Physics Today, January 2014, page 33). It is always interesting to see how our current understanding sheds new light on the revolutionary scientific ideas of the past. I agree with the authors that although quantum mechanics is the real basis for atomic- and molecular-physics computations, the old Bohr model that treats electrons in atoms like tiny planets moving around the sun-nucleus is intuitively clear and very attractive.As I see it, the aim of the article is to show how to use Bohr’s approach to treat not only simple atoms but molecules. An important point in that approach is reconciliation of quantum mechanics with Bohr’s ideas. The authors claim that in infinite dimensions, quantum mechanics “morphs into classical mechanics.”I cannot say, however, that the infinite-dimension system is a clarifying model to describe physical or chemical objects. I do not see that the reference to chromodynamics (and to Edward Witten’s article in Physics Today, July 1980, page 38) is a clarifying one. Far from convincing are statements like “Hence the large-D limit, where 1/D → 0, is closer to the real world (1/D = 1/3) than is the oft-used D = 1 regime. Indeed, results obtained at large D usually resemble those for D = 3.” I confess it sounds too lacking in rigor to be convincing.But the large-D limit per se is not what bothers me. Of great concern is the assumption that the radial part of the D-dimensional Schrödinger equation in Hartree units looks like this:(1)where Z is the nuclear charge and l is the angular momentum.Equation (1) has no sense for D = 1. Indeed, in one dimension, a finite angular momentum requires infinite speed of a rotating particle. Therefore, for D = 1, the correct equation is(2)which does not follow from equation (1) at D = 1.But most important is that equation (1) implies that the Coulomb potential does not depend on D. That could have been considered correct before it became clear that the Coulomb law follows from Maxwell’s equations when one considers a field generated by a point-like electric charge. So to obtain the Coulomb law for a two-dimensional space, one has to consider Maxwell equations in a two-dimensional world. For a point-like charge, those equations lead to a 1/(ln r) instead of 1/r dependence. If one takes into account the D-dependence of the Coulomb potential, derivations performed in the Physics Today article become meaningless. Indeed, when using an unrealistic equation (1), how can one believe that it illuminates Bohr’s quite realistic and physical postulates?© 2014 American Institute of Physics.