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Journal of Modern Physics

The Classical Limit of the Quantum Kepler Problem

A. Martín-Ruiz, J. Bernal, Alejandro Frank, Adrián Carbajal-Domı́nguez

Published Jan 1, 2013• 17 citations
Publisher

Abstract

The classical limit of the quantum mechanical Kepler problem is derived by using a simple mathematical procedure recently proposed. The method is based both on Bohr’s correspondence principle and the local averages of the quantum probability distribution. We illustrate in a clear fashion the difference between Planck’s limit and Bohr’s correspondence principle. We discuss the confinement effect in macroscopic systems.

DOI
10.4236/jmp.2013.46112

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The classical limit of the quantum mechanical Kepler problem is derived by using a simple mathematical procedure recently proposed. The method is based both on Bohr’s correspondence principle and the local averages of the quantum probability distribution. We illustrate in a clear fashion the difference between Planck’s limit and Bohr’s correspondence principle. We discuss the confinement effect in macroscopic systems.
According to the correspondence principle, as formulated by Bohr, both in the old and the modern quantum theory, the classical limit should be recovered for large values of the quantum numbers in any quantum system. However, this classical limit of quantum theory is not so straightforward as in the interface of other generalizations of classical mechanics and other domains. In particular, relativistic kinematics and mechanics reduce to Newtonian equations by simple algebra in the case of bodies moving with small velocities compared to the speed of light in vacuum. In this paper we consider the correspondence limit to the two-body problem in gravitational physics, the limit in which both the principal and the angular quantum numbers, N, L are very large. In this limit, we compare with the classical elliptical orbits and we find that the macroscopic coherent quantum states correspond to the statistical average of every classical state compatible with conservation laws for the total energy and angular momentum. We also consider the perturbed Kepler problem with a central perturbation force proportional to the inverse of the cube of the distance to the central body. The exact solution for the quantum eigenstates shows that the first order perturbation to the energy eigenvalues are obtained classically as the temporal orbital average of the perturbation potential.
In this paper we suggest a simple mathematical procedure to derive the classical probability density of quantum systems via Bohr’s correspondence principle. Using Fourier expansions for the classical and quantum distributions, we assume that the Fourier coefficients coincide for the case of large quantum number. We illustrate the procedure by analyzing the classical limit for the quantum harmonic oscillator and the particle in a box, although the method is quite general. We find, in an analytical fashion, the classical distribution arising from the quantum one as the zeroth order term in an expansion in powers of Planck’s constant. We interpret the correction terms as residual quantum effects at the microscopic-macroscopic boundary.
Abstract Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck’s version of the correspondence principle, classical mechanics is recovered when the limit in which a dimensionless parameter containing Planck’s constant h goes to zero is taken, while Bohr’s version entails taking the limit of large quantum numbers. However, despite what is usually stated in textbooks, the relation between the two theories is much more complex to state and understand. Here we deal with this issue by analysing some key examples, in some of which also the analogously subtle relation between wave and geometric optics is considered. Implications for quantum mechanics teaching at undergraduate level are carefully discussed.
Quantum and classical mechanics are fundamentally different theories, but the correspondence principle states that quantum particles behave classically in the appropriate limit. For high-energy periodic quantum systems, the emergence of the classical description should be understood in a distributional sense, i.e., the quantum probability density approaches the classical distribution when the former is coarse-grained. Following a simple reformulation of this limit in the Fourier space, in this paper, we investigate the macroscopic behavior of freely falling quantum particles. To illustrate how the method works and to fix some ideas, we first successfully apply it to the case of a particle in a box. Next, we show that, for a particle bouncing under the gravity field, in the limit of a high quantum number, the leading term of the quantum distribution corresponds to the exact classical distribution plus sub-leading corrections, which we interpret as quantum corrections at the macroscopic level.
We consider the Bohr correspondence limit of the Schrödinger wave function for an atomic elliptic state. We analyze this limit in the context of Nelson’s stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler’s laws of planetary motion in a quantum mechanical setting. In this quantum mechanical setting, local mild instabilities occur in the Keplerian orbit for eccentricities greater than 12 which do not occur classically.