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The Journal of Physical Chemistry B

Revisiting Bohr's Semiclassical Quantum Theory

Dor Ben‐Amotz

Published Jul 19, 2006• 1 citation

Abstract

Bohr's atomic theory is widely viewed as remarkable, both for its accuracy in predicting the observed optical transitions of one-electron atoms and for its failure to fully correspond with current electronic structure theory. What is not generally appreciated is that Bohr's original semiclassical conception differed significantly from the Bohr-Sommerfeld theory and offers an alternative semiclassical approximation scheme with remarkable attributes. More specifically, Bohr's original method did not impose action quantization constraints but rather obtained these as predictions by simply matching photon and classical orbital frequencies. In other words, the hydrogen atom was treated entirely classically and orbital quantized emerged directly from the Planck-Einstein photon quantization condition, E = h nu. Here, we revisit this early history of quantum theory and demonstrate the application of Bohr's original strategy to the three quintessential quantum systems: an electron in a box, an electron in a ring, and a dipolar harmonic oscillator. The usual energy-level spectra, and optical selection rules, emerge by solving an algebraic (quadratic) equation, rather than a Bohr-Sommerfeld integral (or Schroedinger) equation. However, the new predictions include a frozen (zero-kinetic-energy) state which in some (but not all) cases lies below the usual zero-point energy. In addition to raising provocative questions concerning the origin of quantum-chemical phenomena, the results may prove to be of pedagogical value in introducing students to quantum mechanics.

DOI: 10.1021/jp061993b

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We review Bohr's atomic model and its extension by Sommerfeld from a mathematical perspective of wave mechanics.The derivation of quantization rules and energy levels is revisited using semiclassical methods.Sommerfeld-type integrals are evaluated by elementary techniques, and connections with the Schr odinger and Dirac equations are established.Historical developments and key transitions from classical to quantum theory are discussed to clarify the structure and significance of the old quantum mechanics.

The old quantum theory of Bohr and Sommerfeld was abandonned for the wrong reason. Its contradictions were caused not by the orbit concept but by a mental barrier—the inconceivability that an electron might collide with the atomic nucleus. Removing that barrier resolves the theory’s main failures—incorrect orbital momenta, He atom, H + 2 molecule ion. The inclusion of electron oscillations through the nucleus—a concept called “Coulomb oscillator”—renders the old quantum theory consistent with quantum mechanics (although devoid of wave character). The triple success of the Bohr-Sommerfeld model is its correct description of the H atom (and one-electron ions) concerning (1) the energy levels Enl, (2) the orbital angular momenta Lnl—if corrected as L 2 = l(l + 1)~ 2 and with the Coulomb oscillator included—and (3) the orbits’ space quantization—with (Lnl)z = ml~. These achievements are succinctly represented by the principal, angular and magnetic quantum numbers (n, l, ml) and visualized by orbital ellipse geometry—major axis, vertex curvature, and tilt angle, respectively. Orbit geometry also accounts for the average orbit size. Moreover, the Coulomb oscillator provides a natural explanation of (isotropic) hyperfine interaction. The shortcomings of the old quantum theory lie in its neglect of three properties of particles—their spin, their wave nature and their quantum statistics. These deficiencies notwithstanding, the visual appeal of the Bohr-Sommerfeld model remains a pedagogical asset to complement the abstract character of quantum mechanics.

Abstract The development of the complex of assumptions and methods now referred to as the “old quantum theory” mainly took place in the first five years following the introduction of the Bohr atomic model in 1913. Three guiding principles emerged that were used, sometimes in overlapping ways, to explain the flood of spectroscopic data that needed to be explained. First, quantization rules (or conditions) were proposed to single out the allowed orbital motions of electrons in atoms. These rules were derived in various forms by Planck, Sommerfeld, and Wilson, but were put into their most general form by Schwarzschild, who recognized the underlying principle as the quantization of the action variables of a multiply periodic classical system. Second, the special role of the action variables in quantization was given convincing support by the transfer of the adiabatic principle of mechanics to quantum theory (work primarily due to Paul Ehrenfest). Third, the correspondence principle, or statement of asymptotic coincidence of quantum and classical theory in the limit of large quantum numbers, originally introduced by Bohr in 1913 as a supporting argument for his quantization of angular momentum in his theory of the hydrogen atom, was extended by Bohr and Kramers to provide selection rules and approximate intensity predictions evening the regime low quantum numbers.

The old quantum theory of Bohr and Sommerfeld was abandonned for the wrong reason. Its contradictions were caused not by the orbit concept but by a mental barrier--the inconceivability that an electron might collide with the atomic nucleus. Removing that barrier resolves the theory's main failures--incorrect orbital momenta, He atom, H2+ molecule ion. The inclusion of electron oscillations through the nucleus--a concept called "Coulomb oscillator"--renders the old quantum theory consistent with quantum mechanics (although devoid of wave character). The triple success of the Bohr-Sommerfeld model is its correct description of the H atom (and one-electron ions) concerning (1) the energy levels Enl, (2) the orbital angular momenta Lnl--if corrected as Lnl^2 = l(l+1) hbar^2 and with the Coulomb oscillator included--and (3) the orbits' space quantization--with (Lnl)z = ml hbar. These achievements are succinctly represented by the principal, angular and magnetic quantum numbers (n, l, ml) and visualized by orbital ellipse geometry--major axis, vertex curvature, and tilt angle, respectively. Orbit geometry also accounts for the average orbit size. Moreover, the Coulomb oscillator provides a natural explanation of (isotropic) hyperfine interaction. The shortcomings of the old quantum theory lie in its neglect of three properties of particles--their spin, their wave nature and their quantum statistics. These deficiencies notwithstanding, the visual appeal of the Bohr-Sommerfeld model remains a pedagogical asset to complement the abstract character of quantum mechanics.

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.