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What happened to the Bohr-Sommerfeld elliptic orbits in Schrodinger's wave mechanics?

Michael Nauenberg

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

According to Heisenberg and Pauli, two of the great pioneers of quantum mechanics, in the domain of atoms and molecules the Bohr-Sommerfeld elliptic orbits disappear. But Bohr's correspondence principle requires that for large quantum numbers, quantum mechanics corresponds to classical mechanics. It is shown here how this correspondence, generally ignored in physics textbooks, takes place.

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According to Heisenberg and Pauli, two of the great pioneers of quantum mechanics, in the domain of atoms and molecules the Bohr-Sommerfeld elliptic orbits disappear. But Bohr's correspondence principle requires that for large quantum numbers, quantum mechanics corresponds to classical mechanics. It is shown here how this correspondence, generally ignored in physics textbooks, takes place.
Abstract We review the Bohr atomic model, where certain conditions (quantum conditions) were used to select certain “allowed” circular orbits for the electrons in an atom. Sommerfeld realized that a more generalized condition, applied to the action variables, would be consistent with the Bohr condition and also allow elliptical orbits. This condition amounted to the quantization of the phase space occupied by an orbit. This had independently been suggested by Planck. We also discuss the similarities between classical mechanics and the geometric optics limit of a wave theory.
The results of Bohr model can be used to explain the motion of macroscopic objects or big objects (like orbiting planets and sun, falling stone etc.). These macroscopic objects have essentially a particle nature and Newtonian laws of motion can be applied on them. But it fails when applied to explain the motion of microscopic objects or small objects (like electrons, atoms, molecules etc.). This is because: According to Bohr model, an orbit is a well-defined path and this path can completely be defined only if both the position and velocity of the electron are known exactly at the same time. But this is not possible according to the Heisenberg’s uncertainty principle. Thus, Bohr model contradicts the Heisenberg’s uncertainty principle . According to Bohr model, an electron is regarded as a charged particle moving in well-defined circular orbits concentric about the nucleus. The wave character of the electron is not considered. Thus, Bohr model ignores the de-Broglie dual nature of matter .
The results of Bohr model can be used to explain the motion of macroscopic objects or big objects (like orbiting planets and sun, falling stone etc.). These macroscopic objects have essentially a particle nature and Newtonian laws of motion can be applied on them. But it fails when applied to explain the motion of microscopic objects or small objects (like electrons, atoms, molecules etc.). This is because: According to Bohr model, an orbit is a well-defined path and this path can completely be defined only if both the position and velocity of the electron are known exactly at the same time. But this is not possible according to the Heisenberg’s uncertainty principle. Thus, Bohr model contradicts the Heisenberg’s uncertainty principle . According to Bohr model, an electron is regarded as a charged particle moving in well-defined circular orbits concentric about the nucleus. The wave character of the electron is not considered. Thus, Bohr model ignores the de-Broglie dual nature of matter .
Before the Heisenberg-Schrödinger formulation of quantum mechanics, the semiclassical Bohr-Sommerfeld theory successfully accounted for quantized properties such as the energy levels in the hydrogen atom. However, the forcing of closed orbits for particle motion ran afoul of the uncertainty principle. Recently, the use of Dscaling, in which the motion of each particle is described by a vector in D dimensions, was used to reintroduce the uncertainty principle to this earlier theory. When properly done, such equations reduce to the correct Schrödinger form for D = 3 but can still be solved in the more tractable D→ ∞ limit. This D scaling approach was applied successfully to atoms but did not yield bound states for molecules.