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American Journal of Physics

The right way to introduce complex numbers in damped harmonic oscillators

Jason D. Tran, Leanne Doughty, James K. Freericks

Published Apr 22, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.1119/5.0220797

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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.
Abstract By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator X^ \hat X X^ in x representation. The evolution of ⟨x^⟩ \left\langle {\hat x} \right\rangle ⟨x^⟩ is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.
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.
We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.
We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.
The motion of a particle subject to an external force and coupled to a large number of harmonic oscillators is reducible to a nonlinear integro-differential equation for the position operator. This is achieved by eliminating the degrees of freedom corresponding to the motion of the oscillators. Upon this elimination the commutator of the position and the mechanical momentum of the particle becomes time dependent and a c number for a quadratic potential, but in general a q number for other fields of force. For a harmonically bound particle interacting with the heat bath the time evolution of the position and momentum operators are given in terms of the solutions of an integro-differential equation. Certain forms of the coupling of the motion of the particle to the heat bath allow for further reduction of these equations to quartic differential equations with constant coefficients. For these cases the time dependence of the position-momentum commutator, the energy of the particle as well as the heat bath are studied.