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arXiv:2010.14930

Equivalence of quantum harmonic oscillators and classical oscillators subject to random forces

Can Gokler

Published Oct 26, 2020• 2 citations
arXivHTMLPDF

Abstract

We show that the Schrödinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals $O( m / \hbar)$, when $\hbar / m \ll 1$. Conversely, every solution to the Schrödinger equation, including all the superposition states, arises this way. In other words, the quantum harmonic oscillator is approximately nothing but the classical harmonic oscillator, with the same mass and frequency as the quantum harmonic oscillator, subject to a random force. We generalize the result to multiple non-interacting oscillators. We show that the Schrödinger equation for $n$ non-interacting quantum harmonic oscillators with masses $m_1, ..., m_n$ and frequencies $ω_1, ..., ω_n$ can be derived as an approximation to the Newtonian mechanics of $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This is valid for time intervals $O( \tilde{m} / \hbar)$, where $\tilde{m}$ is the mass of the minimum mass oscillator, when $\hbar/\tilde{m} \ll 1$. Conversely, every solution, including all the entangled states, to the Schrödinger equation arises this way. In other words, $n$ non-interacting quantum harmonic oscillators are approximately nothing but $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This provides a local Newtonian model of entanglement of non-interacting quantum oscillators. The correlations required by entangled states are embedded in the phase space probability density of the classical oscillators.

DOI
10.48550/arxiv.2010.14930

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We show that the Schrödinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals O(m/ℏ)O( m / \hbar)O(m/ℏ), when ℏ/m≪1\hbar / m \ll 1ℏ/m≪1. Conversely, every solution to the Schrödinger equation, including all the superposition states, arises this way. In other words, the quantum harmonic oscillator is approximately nothing but the classical harmonic oscillator, with the same mass and frequency as the quantum harmonic oscillator, subject to a random force. We generalize the result to multiple non-interacting oscillators. We show that the Schrödinger equation for nnn non-interacting quantum harmonic oscillators with masses m1,...,mnm_1, ..., m_nm1​,...,mn​ and frequencies ω1,...,ωnω_1, ..., ω_nω1​,...,ωn​ can be derived as an approximation to the Newtonian mechanics of nnn non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This is valid for time intervals O(m~/ℏ)O( \tilde{m} / \hbar)O(m~/ℏ), where m~\tilde{m}m~ is the mass of the minimum mass oscillator, when ℏ/m~≪1\hbar/\tilde{m} \ll 1ℏ/m~≪1. Conversely, every solution, including all the entangled states, to the Schrödinger equation arises this way. In other words, nnn non-interacting quantum harmonic oscillators are approximately nothing but nnn non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This provides a local Newtonian model of entanglement of non-interacting quantum oscillators. The correlations required by entangled states are embedded in the phase space probability density of the classical oscillators.
Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.
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.
An earlier study of quantum-mechanical time has suggested that, from the quantum-theoretic point of view, time is a random variable. The present work explores how such a description of time can be used to bring greater detail into the analysis of the one-dimensional harmonic oscillator. Specifically, the classical equations for harmonic motion are used as a point transformation between phase space and time-energy space, permitting probability distributions in the two spaces to be different representations of the same thing. The familiar energy-quantization hypothesis is supplemented with the hypothesis that quantized energy values are means (averages) of energy distributions characteristic of each energy state. With the probability distribution of time determined from the previously cited study, the desired quantized energy distributions follow from the requirements: (1) that they should be normalized, non-negative, and with means equal to the quantized energy values; (2) that the equivalent representation of distribution in position-momentum space should have marginal densities that are proportional (except near the origin) to the densities in position and momentum, ${|\ensuremath{\phi}(q)|}^{2}$ and ${|\ensuremath{\psi}(p)|}^{2}$, which are postulated in quantum theory. Requirement (2) is a weakening of the Born postulate. In the time-energy space, the probability distributions exclude energies smaller than the mode (energy value with maximum probability). In position-momentum space this exclusion requirement becomes the dynamical restriction: $\frac{{q}^{2}m{\ensuremath{\omega}}^{2}}{2}+\frac{{p}^{2}}{2m}\ensuremath{\ge}\frac{{\ensuremath{\Lambda}}_{n}\ensuremath{\hbar}\ensuremath{\omega}}{2}$, where, for the first three energy states, ${\ensuremath{\Lambda}}_{0}=0$, ${\ensuremath{\Lambda}}_{1}=\frac{3}{2}$, ${\ensuremath{\Lambda}}_{2}=2+\ensuremath{\surd}\frac{3}{2}$. In general terms it is thus shown that the quantized harmonic oscillator can be described as a statistical ensemble of classical oscillators. Some comparisons with work on phase-space formulations of quantum mechanics (especially with the work of Wigner) are made and some aspects of the theory that should submit to experimental verification are pointed out.
In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.
In physics, harmonic motion is among the most representative types of motion. A simple harmonic oscillator is often the source of any vibration with a restoring force proportional to Hooke’s law. Every minimum potential has a solution in the form of the harmonic oscillator potential. Little oscillations at the minimum are characteristic of almost all natural potentials and of many quanta mechanical systems. Harmonic motion is an essential building block for these more complex uses. The Schrödinger equation is a defining feature of the harmonic oscillator. Here, we demonstrate that the time-frequency plane is a useful tool for analyzing their dynamics. We numerically integrate several examples involving different input forces and demonstrate that the oscillations are clearly displayed and easily interpretable in the time-frequency plane.
The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable (x\mathit{x}x , v^\hat{v}v^) , with the commutation relation [x\mathit{x}x , v^\hat{v}v^] = iℏ\mathit{i}\hbariℏ/m\mathit{m}m , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion K\mathit{K}K(x,v,t\mathit{x, v, t}x,v,t) of the classical system, The comparison with the quantization in the space (x,p\mathit{x, p}x,p) is done with the usual Schrödinger’s equation for the Hamiltonian H\mathit{H}H(x,p,t\mathit{x, p, t}x,p,t), and with the commutation relation [x\mathit{x}x , p^\hat{p}p^​] = iℏ\mathit{i}\hbariℏ. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (x\mathit{x}x , v^\hat{v}v^) quantization has fewer oscillations than the (x\mathit{x}x , p^\hat{p}p^​) quantization, the average energy of the system is higher in (x\mathit{x}x , p^\hat{p}p^​) quantization than on the (x\mathit{x}x , v^\hat{v}v^) quantization, and the Boltzmann- Shannon entropy on the (x\mathit{x}x , p^\hat{p}p^​) quantization is higher than on the (x\mathit{x}x , v^\hat{v}v^) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.