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arXiv:1202.1876

Applying quantum mechanics to macroscopic and mesoscopic systems

N. Poveda T., N. Vera-Villamizar

Published Feb 9, 2012• 0 citations
arXivHTMLPDF

Abstract

There exists a paradigm in which Quantum Mechanics is an exclusively developed theory to explain phenomena on a microscopic scale. As the Planck's constant is extremely small, $h\sim10^{-34}{J.s}$, and as in the relation of de Broglie the wavelength is inversely proportional to the momentum; for a mesoscopic or macroscopic object the Broglie wavelength is very small, and consequently the undulatory behavior of this object is undetectable. In this paper we show that with a particle oscillating around its classical trajectory, the action is an integer multiple of a quantum of action, $S = nh_{o}$. The quantum of action, $h_{o}$, which plays a role equivalent to Planck's constant, is a free parameter that must be determined and depends on the physical system considered. For a mesoscopic and macroscopic system: $h_{o}\gg h$, this allows us to describe these systems with the formalism of quantum mechanics.

DOI
10.48550/arxiv.1202.1876

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There exists a paradigm in which Quantum Mechanics is an exclusively developed theory to explain phenomena on a microscopic scale. As the Planck's constant is extremely small, h∼10−34J.sh\sim10^{-34}{J.s}h∼10−34J.s, and as in the relation of de Broglie the wavelength is inversely proportional to the momentum; for a mesoscopic or macroscopic object the Broglie wavelength is very small, and consequently the undulatory behavior of this object is undetectable. In this paper we show that with a particle oscillating around its classical trajectory, the action is an integer multiple of a quantum of action, S=nhoS = nh_{o}S=nho​. The quantum of action, hoh_{o}ho​, which plays a role equivalent to Planck's constant, is a free parameter that must be determined and depends on the physical system considered. For a mesoscopic and macroscopic system: ho≫hh_{o}\gg hho​≫h, this allows us to describe these systems with the formalism of quantum mechanics.
An explicit model allowing a unified description of microscopic and macroscopic systems is exhibited. First, a modified quantum dynamics for the description of macroscopic objects is constructed and it is shown that it forbids the occurrence of linear superpositions of states localized in far-away spatial regions and induces an evolution agreeing with classical mechanics. This dynamics also allows a description of the evolution in terms of trajectories. To set up a unified description of all physical phenomena, a modification of the dynamics, with respect to the standard Hamiltonian one, is then postulated also for microscopic systems. It is shown that one can consistently deduce from it the previously considered dynamics for the center of mass of macroscopic systems. Choosing in an appropriate way the parameters of the so-obtained model one can show that both the standard quantum theory for microscopic objects and the classical behavior for macroscopic objects can all be derived in a consistent way. In the case of a macroscopic system one can obtain, by means of appropriate approximations, a description of the evolution in terms of a phase-space density distribution obeying a Fokker-Planck diffusion equation. The model also provides the basis for a conceptually appealing description of quantum measurement.
In the 17th century, it was thought that motion of macroscopic and microscopic objects could be well-expressed by using the laws of classical mechanics. But towards the end of 19th century, experimental studies showed that classical mechanics failed when applied to microscopic objects. This was mainly because classical mechanics did not take into account the concept of de-Broglie’s dual nature of matter and Heisenberg’s uncertainty principle. Also, classical mechanics does not put any restrictions on the values of dynamical properties (likes position, momentum, energy etc.) calculated for microscopic object. According to classical mechanics, microscopic objects can have any value of dynamical properties, ( exact , large or small is permissible). Also any number of properties can be calculated simultaneously for a system.
The principle of action plays a central role in both classical and quantum mechanics, yet its physical meaning is often introduced in a purely formal manner. In this work, the concept of action is examined from a scale-based perspective that clarifies its role in the quantum–classical correspondence. By emphasizing the dimensionless ratio S/ℏ, we show that the emergence of classical dynamics can be understood as a consequence of quantum phase coherence rather than as an independent postulate. Within the path integral formulation, classical trajectories arise naturally from stationary-phase conditions when the action associated with a physical process becomes large compared to Planck’s constant. This interpretation provides a unified conceptual framework in which variational principles, quantum interference, and classical determinism are linked through the scale of action. The analysis highlights the role of Planck’s constant as a universal parameter governing the physical relevance of alternative histories, analogous to the function of Boltzmann’s constant in statistical physics. Beyond its theoretical implications, the proposed perspective offers pedagogical value by providing a coherent bridge between classical and quantum mechanics suitable for advanced undergraduate and early graduate physics education.
In the 17th century, it was thought that motion of macroscopic and microscopic objects could be well-expressed by using the laws of classical mechanics. But towards the end of 19th century, experimental studies showed that classical mechanics failed when applied to microscopic objects. This was mainly because classical mechanics did not take into account the concept of de-Broglie’s dual nature of matter and Heisenberg’s uncertainty principle. Also, classical mechanics does not put any restrictions on the values of dynamical properties (likes position, momentum, energy etc.) calculated for microscopic object. According to classical mechanics, microscopic objects can have any value of dynamical properties, ( exact, large or small is permissible). Also any number of properties can be calculated simultaneously for a system.
Quantum mechanics is the theoretical framework that describes physical phenomena on the microscopic scale. In particular, it is used in those situations in which Planck's constant h , approximately equal to 6.62 × 10 -34 Js, cannot be assumed to be negligible, despite its apparently tiny magnitude. Macroscopic phenomena, on the other hand, involve magnitudes that are so large that h can be set equal to zero without appreciable error. As an illustration of this in the macroscopic context, consider the following example: a ladybug ( Hippodamia convergens ) of mass 1 g moves radially at a speed of 0.01 ms -1 on a turntable rotating at 10 rpm. In the rotating frame of reference, the ladybug has a kinetic energy of about an erg, and, in the laboratory frame, its angular momentum is roughly 10 -5 Js when it is at a distance of 0.1 m from the center of the rotating turntable. The latter angular momentum is of the order of 10 28 h . Thus, relative to this value, h is well approximated by zero and, indeed, only classical mechanics is needed to describe the ladybug's motion. In contrast to the preceding example, an electron in the “2p” level of hydrogen has an energy of about 10 -25 J and an angular momentum whose magnitude is approximately h /(2π) = ħ (pronounced “h-bar”). Here, h cannot be ignored and only a quantum description can accurately account for the properties of this system.
Since the discovery of quantum dynamics in the 1900s, we have observed many of its effects on the macroscopic world. Max Planck gave the first quantum hypothesis in 1900, explaining how electromagnetic waves transmit energy in “quanta.” Building upon that, Louis De Broglie stated the theory of wave-particle duality, and Erwin Schrodinger postulated the equation to interpret this wave nature. Physicists expanded the field as the theories of Quantum Electrodynamics and Quantum Chromodynamics were put forward, explaining the Electromagnetic Force and the Strong Nuclear Force in terms of quanta. Theories such as the Many Worlds Interpretation were also proposed to explain quantum mechanical phenomena. Although there were many discoveries, there were also problems such as the Hierarchy Problem that arose because of the unusual mass of the Higgs Boson. Nevertheless, This unusual mass of the Higgs needs to be “unusual” for life to exist. If it were something else, the discovery of Quantum Mechanics would not have been possible, and everything we have learned about it till now would not exist; we would not exist. Thus, we can look at how quantum dynamics affects the macroscopic world even though it governs the world of the “very small.”