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arXiv:quant-ph/0611147

How often is the coordinate of a harmonic oscillator positive?

Boris S. Tsirelson

Published Nov 14, 2006• 4 citations
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Abstract

The coordinate of a harmonic oscillator is measured at a time chosen at random among three equiprobable instants: now, after one third of the period, or after two thirds. The (total) probability that the outcome is positive depends on the state of the oscillator. In the classical case the probability varies between 1/3 and 2/3, but in the quantum case -- between 0.29 and 0.71.

DOI
10.48550/arxiv.quant-ph/0611147

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The coordinate of a harmonic oscillator is measured at a time chosen at random among three equiprobable instants: now, after one third of the period, or after two thirds. The (total) probability that the outcome is positive depends on the state of the oscillator. In the classical case the probability varies between 1/3 and 2/3, but in the quantum case -- between 0.29 and 0.71.
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.
Harmonic oscillators are important in physical applications and in demonstrating the working of the postulates of quantum formalism. A classical harmonic oscillator has a state space, i.e., its phase space, coordinated by the oscillator’s position x and momentum p . Observables correspond to real-valued functions on the state space. The evolution of the state is determined by Hamilton’s equations, which reduce to Eq. (27.12) . The time dependence of the state automatically determines the time dependence of observables.
The evaluation of the average value of the position coordinate, 〈x〉, of a particle moving in a harmonic oscillator potential (V(x)=kx2/2) with a small anharmonic piece (V(x)=−λkx3) is a standard calculation in classical Newtonian mechanics and statistical mechanics where the problem has relevance to thermal expansion. In each case, the calculation is most easily done using a perturbative expansion. In this note, we perform the same computation of 〈x〉 in quantum mechanics using time-independent perturbation theory and the ladder operator formalism to show how similar results are obtained. We also indicate how a semiclassical calculation using a classical probability distribution can also be used to obtain the same result.
The time evolution orbits of Gaussian vectors under a harmonic oscillator dynamics are computed and discussed. The generic case is shown to be this: a minimum uncertainty vector which is coherent with respect to a harmonic oscillator Hamiltonian with smaller frequency than the one of the driving Hamiltonian, evolves into a minimum uncertainty vector with reduced position variance for exactly two discrete instants of time per period. The results are applied to the case of a perturbing frequency jump in the driving harmonic oscillator Hamiltonian. The total transition probability to the excited (unperturbed) oscillator levels from the ground-state vector is computed.