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arXiv:2102.10857 · European Journal of Physics

Dimensional analysis and the correspondence between classical and quantum uncertainty

Viola Gattus, Sotirios Karamitsos

Published Jul 16, 2020• 6 citations

Abstract

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.

DOI: 10.1088/1361-6404/aba6bc

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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.

Abstract As an undergraduate exercise, in an article (2012 Am. J. Phys. 80 780–14 ), quantum and classical uncertainties for dimensionless variables of position and momentum were evaluated in three potentials: infinite well, bouncing ball, and harmonic oscillator. While original quantum uncertainty products depend on ℏ and the number of states ( n ), a dimensionless approach makes the comparison between quantum uncertainty and classical dispersion possible by excluding ℏ . But the question is whether the uncertainty still remains dependent on quantum number n . In the above-mentioned article, there lies this contrast; on the one hand, the dimensionless quantum uncertainty of the potential box approaches classical dispersion only in the limit of large quantum numbers ( n → ∞ )—consistent with the correspondence principle. On the other hand, similar evaluations for bouncing ball and harmonic oscillator potentials are equal to their classical counterparts independent of n . This equality may hide the quantum feature of low energy levels. In the current study, we change the potential intervals in order to make them symmetric for the linear potential and non-symmetric for the quadratic potential. As a result, it is shown in this paper that the dimensionless quantum uncertainty of these potentials in the new potential intervals is expressed in terms of quantum number n . In other words, the uncertainty requires the correspondence principle in order to approach the classical limit. Therefore, it can be concluded that the dimensionless analysis, as a useful pedagogical method, does not take away the quantum feature of the n -dependence of quantum uncertainty in general. Moreover, our numerical calculations include the higher powers of the position for the potentials.

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 Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck’s version of the correspondence principle, classical mechanics is recovered when the limit in which a dimensionless parameter containing Planck’s constant h goes to zero is taken, while Bohr’s version entails taking the limit of large quantum numbers. However, despite what is usually stated in textbooks, the relation between the two theories is much more complex to state and understand. Here we deal with this issue by analysing some key examples, in some of which also the analogously subtle relation between wave and geometric optics is considered. Implications for quantum mechanics teaching at undergraduate level are carefully discussed.

The uncertainty principle, formulated by Heisenberg in 1927, is a fundamental principle in quantum mechanics. It forces us to review key concepts of classical physics, and from this analysis are born the concepts of quantum physics. Consequences of the Uncertainty principle also introduces the concept of incompatible sizes, a completely new concept compared to classical physics. A state of the system, which we call a quantum state, will be determined from the knowledge of a set maximum of certain quantities that are independent and compatible with each other. A measurable physical quantity is called an observable. If at a given moment a measurement is performed by one or more compatible observables it is said they are performing an observation of the system. The basic principles of quantum mechanics are expressed by a number of postulates. Their numbers and can vary from one text to another.

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.