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Physical Review

The Inverse-Cube Central Force Field in Quantum Mechanics

George H. Shortley

Published Jul 1, 1931• 53 citations
Publisher

Abstract

The problem of the motion of a particle in an inverse-cube central force field is fully treated by quantum mechanics and the results compared with the classical theory. Taking the effective radial potential energy as $\frac{S}{{r}^{2}}$, although the solutions for negative energy for $0\ensuremath{\geqq}S\ensuremath{\geqq}\frac{\ensuremath{-}{h}^{2}}{32{\ensuremath{\pi}}^{2}\ensuremath{\mu}}$ satisfy the usual boundary conditions, they can not be admitted because the Hamiltonian is not Hermitian in these solutions. This corresponds to taking ${(l+\frac{1}{2})}^{2}$ in place of $l(l+1)$ as the analogue of the square of the classical angular momentum. If we do this, we get a complete analogy between the classical and quantum mechanically allowed solutions, with no quantization. The solutions involve Bessel functions of both real and imaginary orders with both real and imaginary arguments.

DOI
10.1103/physrev.38.120

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The problem of the motion of a particle in an inverse-cube central force field is fully treated by quantum mechanics and the results compared with the classical theory. Taking the effective radial potential energy as Sr2\frac{S}{{r}^{2}}r2S​, although the solutions for negative energy for 0\ensuremath≧S\ensuremath≧\ensuremath−h232\ensuremathπ2\ensuremathμ0\ensuremath{\geqq}S\ensuremath{\geqq}\frac{\ensuremath{-}{h}^{2}}{32{\ensuremath{\pi}}^{2}\ensuremath{\mu}}0\ensuremath≧S\ensuremath≧32\ensuremathπ2\ensuremathμ\ensuremath−h2​ satisfy the usual boundary conditions, they can not be admitted because the Hamiltonian is not Hermitian in these solutions. This corresponds to taking (l+12)2{(l+\frac{1}{2})}^{2}(l+21​)2 in place of l(l+1)l(l+1)l(l+1) as the analogue of the square of the classical angular momentum. If we do this, we get a complete analogy between the classical and quantum mechanically allowed solutions, with no quantization. The solutions involve Bessel functions of both real and imaginary orders with both real and imaginary arguments.
Abstract Angular momentum in classical mechanics is given by a vector. The plane perpendicular to this vector, in accordance to central field theory, determines the space in which particle motion takes place. No such simple picture exists in quantum mechanics. The states of a particle in a central field are proportional to spherical harmonics which do not define any plane of motion. In the first part of this paper we discuss the angular distribution of particle position and compare it to the classical probabilistic approach. In the second part, the matter of addition of angular momenta is discussed. In classical mechanics this means addition of vectors, while in quantum mechanics Clebsch–Gordan coefficients have to be used. We have found classical approximations to quantum coefficients and the limit of their applicability. This analysis gives a basis for the so-called ‘vector addition model’ used in some elementary textbooks on atomic physics. It can help to better understand the addition of angular momenta in quantum mechanics.
Complex numbers were created by introducing the imaginary unit i to represent the square root of -1, allowing solutions to equations that involved square roots of negative numbers. Complex numbers were further extended to quaternionic numbers by introducing additional imaginary units, namely " j " and " k ". Quaternions are non-commutative and involve four components, with each component being a real number or a multiple of an imaginary unit. Usually complex numbers are used to represent wave functions in quantum mechanics. Solutions using quaternions to square well, spin and angular momentum, Dirac equations have been obtained by many researchers. In this article, we have made use of quaternions to study the generalization of adding angular momenta, the digital signal processing of a quaternionic function and circular potential of a particle in real Hilbert space and have obtained quaternionic solutions in terms of Bessel functions.
One of the most important problems in quantum mechanics concerns the motion of a particle in a potential of the form V ( r ) = V ( r ), where r ≡ | r | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203741481/4ab22e1a-c1c5-4151-9df7-a6d7ea2b05e0/content/eq763.tif"/> . This system has the simplifying feature of spherical symmetry, which allows the angular dependence of the motion to be reduced to purely geometrical considerations and hence solved once and for all, while the physics enters only through the radial motion. Treating this problem in detail also provides a good opportunity to illustrate the quantum mechanical formalism.
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.