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arXiv:1108.2682 · American Journal of Physics

The uncertainty product of position and momentum in classical dynamics

A. R. Usha Devi, H. S. Karthik

Published Jul 17, 2012• 6 citations
arXivHTMLPDF

Abstract

It is generally believed that the classical regime emerges as a limiting case of quantum theory. Exploring such quantum-classical correspondences provides a deeper understanding of foundational aspects and has attracted a great deal of attention since the early days of quantum theory. It has been proposed that since a quantum mechanical wave function describes an intrinsic statistical behavior, its classical limit must correspond to a classical ensemble—not to an individual particle. This idea leads us to ask how the uncertainty product of canonical observables in the quantum realm compares with the corresponding dispersions in the classical realm. In this paper, we explore parallels between the uncertainty product of position and momentum in stationary states of quantum systems and the corresponding fluctuations of these observables in the associated classical ensemble. We confine ourselves to one-dimensional conservative systems and show, with the help of suitably defined dimensionless physical quantities, that first and second moments of the canonical observables match with each other in the classical and quantum descriptions—resulting in identical structures for the uncertainty relations in both realms.

DOI
10.1119/1.4720101

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It is generally believed that the classical regime emerges as a limiting case of quantum theory. Exploring such quantum-classical correspondences provides a deeper understanding of foundational aspects and has attracted a great deal of attention since the early days of quantum theory. It has been proposed that since a quantum mechanical wave function describes an intrinsic statistical behavior, its classical limit must correspond to a classical ensemble—not to an individual particle. This idea leads us to ask how the uncertainty product of canonical observables in the quantum realm compares with the corresponding dispersions in the classical realm. In this paper, we explore parallels between the uncertainty product of position and momentum in stationary states of quantum systems and the corresponding fluctuations of these observables in the associated classical ensemble. We confine ourselves to one-dimensional conservative systems and show, with the help of suitably defined dimensionless physical quantities, that first and second moments of the canonical observables match with each other in the classical and quantum descriptions—resulting in identical structures for the uncertainty relations in both realms.
Abstract Heisenberg’s uncertainty principle is often cited as an example of a ‘purely quantum’ relation with no analogue in the classical limit where ℏ → 0. However, this formulation of the classical limit is problematic for many reasons, one of which is dimensional analysis. Since ℏ is a dimensionful constant, we may always work in natural units in which ℏ = 1. Dimensional analysis teaches us that all physical laws can be expressed purely in terms of dimensionless quantities. This indicates that the existence of a dimensionally consistent constraint on ΔxΔp requires the existence of a dimensionful parameter with units of action, and that any definition of the classical limit must be formulated in terms of dimensionless quantities (such as quantum numbers). Therefore, bounds on classical uncertainty (formulated in terms of statistical ensembles) can only be written in terms of dimensionful scales of the system under consideration, and can be readily compared to their quantum counterparts after being non-dimensionalized. We compare the uncertainty of certain coupled classical systems and their quantum counterparts (such as harmonic oscillators and particles in a box), and show that they converge in the classical limit. We find that since these systems feature additional dimensionful scales, the uncertainty bounds are dependent on multiple dimensionless parameters, in accordance with dimensional considerations.
The focus of the present investigation is uncertainty relations for quantum particles, which quantify the fundamental limitations on some of their properties due to their incompatibility. In the first, longer part, we are concerned with preparational uncertainty \nrelations, while in the second we touch upon measurement inequalities through a generalisation of a model of joint measurement of position and momentum. \nSpecifically, starting from a triple of canonical operators, we prove product and sum inequalities for their variances with bounds larger than those following from combining the pairwise ones. We extend these results to N observables for a quantum particle and prove uncertainty relations for the sums and products of their variances \nin terms of the commutators. \nFurthermore, we present a general theory of preparational uncertainty relations for a quantum particle in one dimension and derive conditions for a smooth function of the second moments to assume a lower bound. The Robertson-Schroedinger inequality \nis found to be of special significance and we geometrically study the space of second moments. We prove new uncertainty relations for various functions of the variances and covariance of position and momentum of a quantum particle. Some of our findings are shown to extend to more than one spatial degree of freedom and we derive \nvarious types of inequalities. \nFinally, we propose a generalisation of the Arthurs-Kelly model of joint measurement of position and momentum to incorporate the case of more than two observables and derive joint-measurement inequalities for the statistics of the probes. For the case of \nthree canonical observables and suitable definitions of error and disturbance we obtain a number of error-error-error and error-error-disturbance inequalities and show that the lower bound is identical to the preparational one.
Physical observables which are measured in a particular state of a dynamical system can be conveniently represented in classical mechanics by mathematical variables, on the grounds that all have definite values in any classical state. In contrast, the statistical interpretation of wave functions describing the quantum states implies that there are uncertainties in the measurement of any two observables which can be represented by canonically conjugate variables, such as position q and momentum p q . This requires any measured dynamical observable A in a quantum state to be associated with an expectation value 〈 A 〉.
Abstract The correspondence principle provides a prescription to connect quantum physics to classical. It asserts that the physical quantities evaluated quantum mechanically approach their respective classical values for large quantum numbers. This has been shown for the pedagogically important cases of the particle in a box and a harmonic oscillator. However, a particle in a constant field has a wave function related to the Airy function and has at best been treated numerically. Employing energy eigenstates we obtain the expectation values of the position, the momentum and their moments upto fourth order, rigorously and without resorting to numerical or graphical techniques. We compare them with the corresponding classical values. We also examine the uncertainty product for the system.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.