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Angle–Action Variables

Mark Sheard Child

Published Jan 3, 1991• 2 citations
Publisher

Abstract

Abstract The Bohr–Sommerfeld quantization rule, 12π∮pdq =(n+δ)ℏ, arose in Section 3.1 from the boundary conditions on the JWKB wave function. Here it is set in the context of classical angle–action theory from which it evolved (Bohr 1913). The idea is that any bound periodic motion can be simply described in special action-angle variables, termed here (1, α), such that the action, defined in Cartesian variables by the equation

DOI
10.1093/oso/9780198556541.003.0004

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Abstract The Bohr–Sommerfeld quantization rule, 12π∮pdq =(n+δ)ℏ, arose in Section 3.1 from the boundary conditions on the JWKB wave function. Here it is set in the context of classical angle–action theory from which it evolved (Bohr 1913). The idea is that any bound periodic motion can be simply described in special action-angle variables, termed here (1, α), such that the action, defined in Cartesian variables by the equation
Abstract Questions of quantization turn on a proper treatment of normally isolated regions where the JWKB description, expressed by eqn (2.13), breaks down. The most obvious of such special regions are those around the classical turning points, at each of which the analysis in Sections 2.2 and 2.3 yields an associated so-called Maslov (1972) phase correction of π/4; Delos (1986) and Littlejohn and Robbins (1987) discuss this Maslov correction from a more fundamental standpoint and Appendix A and parts of Section 11.2 bear on its choice. Other special regions around potential singularities are most conveniently handled by introducing a Langer (1932) correction term to the physical potential function (see Section 2.3). It is shown in Section 3.1 that systems with no other complications may be quantized by the Bohr-Sommerfeld rule (3.1)∮p(q)dq=(n+δ)h, n=0,1,2,3⋯ where n is the quantum number and b is a Maslov index, which takes the values ½, 0, ½ for vibrational, plane rotational, and orbital motion respectively.
For a single degree of freedom confined mechanical system with given energy, we know that the motion is always periodic and action-angle variables are convenient choice as conjugate phase-space variables. We construct action-angle coherent states in view to provide a quantization scheme that yields precisely a given observed energy spectrum { E n } for such a system. This construction is based on a Bayesian approach: each family corresponds to a choice of probability distributions such that the classical energy averaged with respect to this probability distribution is precisely E n up to a constant shift. The formalism is viewed as a natural extension of the Bohr-Sommerfeld rule and an alternative to the canonical quantization. In particular, it also yields a satisfactory angle operator as a bounded self-adjoint operator.
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.
A quantum mechanics is constructed which is patterned on classical Hamilton-Jacobi theory. The dynamical basis of the theory is a quantum Hamilton-Jacobi equation with accompanying physical boundary conditions. Basic objects of the formulation are quantum Hamilton's principal and characteristic functions. As an application of the formalism a theory of quantum action-angle variables is set up. The central feature of this theory is the definition of the quantum action variable which permits the determination of the bound-state energy levels without solving the dynamical equation.
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.