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Journal of Physics Conference Series

Novel connection between classical and quantum mechanics for the isotropic harmonic oscillator

James K. Freericks

Published Mar 1, 2026• 0 citations
Publisher

Abstract

Abstract In 1930, Born and Jordan wrote the textbook Elementare Quantenmechanik , which focused on treating quantum theory from a matrix mechanics perspective. When they treated the harmonic oscillator, they showed a way to decouple the classical equations of motion, which naturally leads to the quantum treatment via the factorization method. This work shows how this approach extends to the isotropic harmonic oscillator, treated in spherical coordinates. This type of reasoning provides a new way to connect classical to quantum mechanics, which appears more natural than other ways used to connect the two. Hopefully, this will assist students with being able to better relate classical and quantum mechanics to each other rather than seeing them as completely separate and unrelated fields.

DOI
10.1088/1742-6596/3203/1/012034

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Abstract In 1930, Born and Jordan wrote the textbook Elementare Quantenmechanik , which focused on treating quantum theory from a matrix mechanics perspective. When they treated the harmonic oscillator, they showed a way to decouple the classical equations of motion, which naturally leads to the quantum treatment via the factorization method. This work shows how this approach extends to the isotropic harmonic oscillator, treated in spherical coordinates. This type of reasoning provides a new way to connect classical to quantum mechanics, which appears more natural than other ways used to connect the two. Hopefully, this will assist students with being able to better relate classical and quantum mechanics to each other rather than seeing them as completely separate and unrelated fields.
The introduction of complex numbers to solve the damped harmonic oscillator in classical mechanics can seem mysterious to students. We show how an approach based on Born and Jordan's 1930 quantum mechanics textbook both demystifies the use of complex numbers in classical mechanics and makes a strong connection to quantum mechanics. In that work, they converted the harmonic oscillator equations of motion into two uncoupled first-order equations. In classical mechanics, this mapping explicitly introduces complex numbers into the motion of the oscillator and directly shows how to solve for the position and momentum observables. Here, we explain how this mapping works and show how to demystify the use of complex numbers in damped-driven harmonic oscillators. As an added bonus, this approach also shows how to determine the energy as a function of time and makes a strong connection to quantum mechanics.
Chapter 3 introduces the student to the matrix formulation of quantum mechanics, pioneered by Heisenberg in collaboration with Born and Jordan. It provides the basic formal elements of the Heisenberg description in terms of operators and their matrix representation, derives the Heisenberg inequality, and applies the formalism to the harmonic oscillator as a central and particularly simple quantum problem. It introduces the state vectors and their Hilbert space, and concludes with a brief discussion hinting at the physical significance of the replacement of classical continuous dynamical variables with operators as a manifestation of the coupling of the particle to the radiation field.
There are significant differences in the structures of course presentations of classical mechanics and quantum mechanics, but also essential links that connect the two approaches. A large part of the differences in approach do not involve quantization, but rather the traditional use of instantaneous values to describe macroscopic objects and position probability densities to describe microscopic objects. A pedagogic reformulation of the classical problem to bridge this gap is presented here. The method is numerical in nature, but for illustrative purposes it is herein applied to two specific cases for which analytic solutions also exist, the Kepler–Coulomb potential and the isotropic harmonic oscillator. Using this approach, classical systems can be examined in the presence of perturbations in the same way as is used in quantum mechanics courses. Semiclassical quantization can also be introduced to extend the connection to a quantum-mechanical treatment.
In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.
Abstract In the present work we use a point transformation defined in the extended configuration space to show that, in both classical and quantum formalisms, the evolution of a particle under the influence of a constant gravitational field can be related to the evolution of the harmonic oscillator. That is, in the classical mechanics regime, we show that this transformation allows us to establish the connection in the Newtonian, Lagrangian, Hamilton-Jacobi and Hamiltonian formalisms. In the quantum mechanics regime, the transformation determines a relationship between the wave functions, solutions to the corresponding Schrödinger equations, the propagators and the Wigner functions. Furthermore, by using the quantum potential approach to quantum mechanics, we show that the classical point transformation, in a natural manner, induces a quantum point transformation in the corresponding extended quantum configuration space, that determines the connection between the corresponding Madelung-Bohm trajectories. We find that the quantum transformation has the same functional form as the classical one even though the quantum or Madelung-Bohm trajectories are in general totally different from the classical ones. Then, we show that the general results can also be applied to the corresponding paraxial wave equations, and by using the stationary solutions of the quantum Harmonic oscillator and the Airy solution obtained by Berry and Balazs, as two applications, we determine the evolution of the Hermite-Gaussian and Airy beams in free space, in a linear and in a quadratic optical media. Finally, by using the point transformation, we construct two metrics whose geodesics correspond precisely to the classical trajectories for the harmonic oscillator and a particle in a uniform gravitational field.