The concept of quantum ratio has emerged from recent efforts to understand how Newton’s equations appear for the center of mass (CM) of an isolated macroscopic body at finite body temperatures as a first approximation of quantum mechanical equations. It is defined as Q≡Rq/L0, where the quantum fluctuation range Rq is the spatial extension of the pure-state CM wave function, whereas L0 stands for the body’s linear size (the space support of the internal bound-state wave function). The two cases Rq/L0≲1 and Rq/L0≫1 roughly correspond to the body’s CM behaving classically or quantum mechanically, respectively. In the present note, we elaborate on this concept and illustrate it through several examples. An important notion following from introduction of the quantum ratio is that the elementary particles (thus, the electron and the photon) are quantum mechanical even when environment-induced decoherence places them into a mixed state. Thus, decoherence and classical state should not be identified. This simple observation, further illustrated by consideration of a few atomic and molecular processes, may have significant implications for the way that quantum mechanics works in biological systems.
The concept of quantum ratio has emerged from recent efforts to understand how Newton’s equations appear for the center of mass (CM) of an isolated macroscopic body at finite body temperatures as a first approximation of quantum mechanical equations. It is defined as Q≡Rq/L0, where the quantum fluctuation range Rq is the spatial extension of the pure-state CM wave function, whereas L0 stands for the body’s linear size (the space support of the internal bound-state wave function). The two cases Rq/L0≲1 and Rq/L0≫1 roughly correspond to the body’s CM behaving classically or quantum mechanically, respectively. In the present note, we elaborate on this concept and illustrate it through several examples. An important notion following from introduction of the quantum ratio is that the elementary particles (thus, the electron and the photon) are quantum mechanical even when environment-induced decoherence places them into a mixed state. Thus, decoherence and classical state should not be identified. This simple observation, further illustrated by consideration of a few atomic and molecular processes, may have significant implications for the way that quantum mechanics works in biological systems.
The concept of the Quantum Ratio was born out of the efforts to find a simple but universal criterion if the center of mass (CM) of an isolated (microscopic or macroscopic) body behaves quantum mechanically or classically, and under which conditions. It is defined as the ratio between the quantum fluctuation range, which is the spatial extension of the pure-state CM wave function, and the linear size of the body (the space support of the internal, bound-state wave function). The two cases where the ratio is smaller than unity or much larger than unity, roughly correspond to the body's CM behaving classically or quantum mechanically, respectively. An important notion following from the introduction of quantum ratio is that the elementary particles (thus the electron and the photon) are quantum mechanical. This is so even when the environment-induced decoherence turns them into a mixed state. Decoherence (mixed state) and classical state should not be identified. This simple observation is further elaborated, by analyzing some atomic or molecular processes. It may have far-reaching implications on the way quantum mechanics works, e.g., in biological systems.
Abstract The concept of the Quantum Ratio was born out of the efforts to find a simple but universal criterion if the center of mass (CM) of an isolated (microscopic or macroscopic) body behaves quantum mechanically or classically, and under which conditions. It is defined as the ratio between the quantum fluctuation range, which is the spatial extension of the pure-state CM wave function, and the linear size of the body (the space support of the internal, bound-state wave function). The two cases where the ratio is smaller than unity or much larger than unity, roughly correspond to the body’s CM behaving classically or quantum mechanically, respectively. An important notion following from the introduction of quantum ratio is that the elementary particles (thus the electron and the photon) are quantum mechanical. This is so even when the environment-induced decoherence turns them into a mixed state. Decoherence (mixed state) and classical state should not be identified. This simple observation is further elaborated, by analyzing some atomic or molecular processes. It may have far-reaching implications on the way quantum mechanics works, e.g., in biological systems.
Newton’s force law [Formula: see text] is derived from the Schrödinger equation for isolated macroscopic bodies, composite states of e.g. [Formula: see text] atoms and molecules, at finite body temperatures. We first review three aspects of quantum mechanics (QM) in this context: (i) Heisenberg’s uncertainty relations for their center of mass (CM), (ii) the diffusion of the CM wave packet, and (iii) a finite body temperature which implies a metastable (mixed-) state of the body: photon emissions and self-decoherence. They explain the origin of the classical trajectory for a macroscopic body. The ratio between the range [Formula: see text] over which the quantum fluctuations of its CM are effective, and the body’s (linear) size [Formula: see text], [Formula: see text] or [Formula: see text], tells whether the body’s CM behaves classically or quantum mechanically, respectively. In the first case, Newton’s force law for its CM follows from the Ehrenfest theorem. We illustrate this for weak gravitational forces, a harmonic-oscillator potential, and for constant external electromagnetic fields slowly varying in space. The derivation of the canonical Hamilton equations for many-body systems is also discussed. Effects due to the body’s finite size such as the gravitational tidal forces appear in perturbation theory. Our work is consistent with the well-known idea that the emergence of classical physics in QM is due to the environment-induced decoherence, but complements and completes it, by clarifying the conditions under which Newton’s equations follow from QM, and by deriving them explicitly.
Introduction Key features of quantum mechanics are the uncertainty principle, wave–particle duality, quantization of energies and the modification of classical probability laws. Biology is concerned with how natural systems function – from understanding how genetically coded information is replicated, to attaining a mechanistic model for complex multistep reactions. Recently researchers have been asking whether quantum mechanics, normally the domain of physics, is also needed to understand some biological processes. This field includes fascinating developments in theory and experiment, as well as multidisciplinary discussion, and the state-of-the-art is documented in this book. Erwin Schrödinger, in his famous book What is Life ? (Schrödinger, 1944), noted that quantum mechanics accounts for the stability of living things and their cellular processes because of our understanding, via quantum mechanics, of the stability and structure of molecules. The fact that quantum effects create, sometimes large, energy gaps between different states of a chemical system is also important. Such energy gaps, between electronic energy levels, enable living organisms to capture and store the energy carried from the sun by photons, and to visualize the world around them via optically induced chemical reactions. Davydov's view in Biology and Quantum Mechanics (Davydov, 1982) was that quantum mechanics is most relevant for isolated systems in pure states and therefore is of little importance for biological systems that are in statistical states at thermal equilibrium.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
The classical limit of quantum mechanics is investigated, by focusing on the study of the center of mass of a many-body system where each particle is described by quantum mechanics. We study how, in the limit when the number of particles diverges and under quite general assumptions, the center of mass of the system is not anymore described by quantum mechanics but by classical mechanics: the center of mass of the system becomes with a well-defined position and momentum, the state of the center of mass is fully determined by its position and by its momentum, and its dynamics is given by the classical law of dynamics. In order to get this result, three assumptions on the many-body system are necessary: the total mass of the system must be much larger than the mass of each particle composing the system; at most a finite number of particles has non-zero correlation in position with an infinite number of particles; finally, when the number of particles composing the system diverge, the variance in position of each particle converges to a finite value. These assumptions are commonly realized for a macroscopic solid and liquid systems. The results obtained in this paper show how the classical limit of quantum mechanics can be naturally achieved without the need to modify quantum mechanics, without the need to have an external environment and without the need of changing the interpretations of quantum mechanics.