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American Journal of Physics

The Bohr-Sommerfeld Theory and the Uncertainty Principle

Yardley Beers

Published Aug 1, 1972• 8 citations
Publisher

Abstract

It is suggested that, if the Wilson-Sommerfeld quantization postulate is modified such that phase integrals for nonignorable coordinates are placed equal to half-integers times Planck's constant while those for ignorable coordinates are placed equal to integers times Planck's constant, eigenvalues that are in much better agreement with quantum mechanics are obtained. Also there results a picture of the radiation process that is conceptually consistant with classical theory. It avoids the picture of an electron making many revolutions in a closed orbit without radiating. Some of the discussion is based upon developments in classical theory that have evolved since the Bohr-Sommerfeld theory was formulated. In addition, some of the effects of the uncertainty principle are made more apparent.

DOI
10.1119/1.1986774

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arXiv:gr-qc/0605005•Apr 30, 2006•1 citation
It is suggested that, if the Wilson-Sommerfeld quantization postulate is modified such that phase integrals for nonignorable coordinates are placed equal to half-integers times Planck's constant while those for ignorable coordinates are placed equal to integers times Planck's constant, eigenvalues that are in much better agreement with quantum mechanics are obtained. Also there results a picture of the radiation process that is conceptually consistant with classical theory. It avoids the picture of an electron making many revolutions in a closed orbit without radiating. Some of the discussion is based upon developments in classical theory that have evolved since the Bohr-Sommerfeld theory was formulated. In addition, some of the effects of the uncertainty principle are made more apparent.
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.
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.
Classical mechanics and the quantum conditions of Planck, Bohr, Sommerfeld, Wilson and Einstein are presented. The virtues and defects of this “old quantum theory” are pointed out. Its replacement by quantum mechanics is described, leading to the Schrödinger equation for the wave function and the corresponding energy eigenvalues. For separable systems, the reduction of this equation to ordinary differential equations and their asymptotic solution by the WKB method are described, as well as the resulting corrected quantum conditions with integer or half-integer quantum numbers. For nonseparable systems, the analogous asymptotic solution constructed by the author is described, together with the corrected quantum conditions to which it leads. Examples of the use of these conditions in the solution of eigenvalue problems are presented. It is explained that difficulties arise in using this method when the classical motion is stochastic or chaotic. Suggestions for overcoming these difficulties are mentioned.
The development of quantum mechanics in the early twentieth century marked one of the most profound conceptual revolutions in physics. Beginning with Planck's quantization of radiation and Einstein's interpretation of the photoelectric effect, followed by Bohr's atomic model, and later formalized by Heisenberg and Schrödinger, the theory provided a new description of nature—one in which probability amplitudes replaced deterministic trajectories. Within this framework, microscopic systems such as atoms, electrons, and photons are characterized by wavefunctions and operators rather than by points in phase space.
The striking synthetic effort made by Bohr to unite radiation facts with atomic structure facts led to the formulation of his “quantum theory of atomic structure,” which with a large group of subsequent developments, is now usually referred to as the “classical quantum theory.” As a conceptual aid it takes over the idea of an “astronomical atom” consisting of a positive nucleus around which electrons revolve in definite orbits, more or less in the fashion of the solar system. More or less. The similarities and differences as a bit of description are interesting and impressive, but when the theory is reduced to its postulates—and it is this which interests us here—it sounds very much like a re-statement (for a purpose) of the Planckian decisions.
We consider the description of classical oscillatory motion in ZM theory, and explore the relationship of ZM theory to semi-classical Bohr-Sommerfeld quantization. The treatment illustrates some features of ZM theory, especially the inadequacies of classical and semi-classical treatments due to non-analyticity of the mapping of classical trajectories onto the ZM clock field. While the more complete ZM formalism is not developed here, the non-analyticities in the classical treatment resemble issues in the comparison of classical and quantum formalisms. We also show that semi-classical quantization is valid for a periodic manifold in ZM theory, though the quantum number n=0n=0n=0 is allowed, as it would be in quantum mechanics for a periodic manifold. Still, this suggests a connection to the first-order success of Bohr theory in describing the phenomenology of atomic quantum states. The approximate nature of the semi-classical treatment of three dimensional atomic orbits is, however, also apparent in relation to ZM theory. These observations are preliminary to a discussion of ZM theory in relation to quantum mechanics and quantum field theory in subsequent papers.