Maximum counts of distinct states define important physical quantities, the most famous being entropy. Here we show that energy, momentum and other basic quantities of mechanics are also maximum counts. This generalizes a property familiar in communication with classical waves: frequency-width equals maximum number of distinct values that can be sent, per unit time. Basic mechanical quantities are frequency-widths of wavefunction evolution, and similarly bound number of distinct states. Evolutions that achieve maximum distinctness are effectively discrete. These properties refine our understanding of uncertainty, measurement, classical mechanics, and relativity.
Maximum counts of distinct states define important physical quantities, the most famous being entropy. Here we show that energy, momentum and other basic quantities of mechanics are also maximum counts. This generalizes a property familiar in communication with classical waves: frequency-width equals maximum number of distinct values that can be sent, per unit time. Basic mechanical quantities are frequency-widths of wavefunction evolution, and similarly bound number of distinct states. Evolutions that achieve maximum distinctness are effectively discrete. These properties refine our understanding of uncertainty, measurement, classical mechanics, and relativity.
Finite physical systems have only a finite amount of distinct state. This finiteness is fundamental in statistical mechanics, where the maximum number of distinct states compatible with macroscopic constraints defines entropy. Here we show that finiteness of distinct state is similarly fundamental in ordinary mechanics: energy and momentum are defined by the maximum number of distinct states possible in a given time or distance. More generally, any moment of energy or momentum bounds distinct states in time or space. These results generalize both the Nyquist bandwidth-bound on distinct values in classical signals, and quantum uncertainty bounds. The new certainty bounds are achieved by finite-bandwidth evolutions in which time and space are effectively discrete, including quantum evolutions that are effectively classical. Since energy and momentum count distinct states, they are defined in finite-state dynamics, and they relate classical mechanics to finite-state evolution.
The limited distinctness of physical systems is roughly expressed by uncertainty relations. Here we show distinctness is a finite resource we can exactly count to define basic physical quantities, limits to the resolution of space and time, and informational foundations for classical mechanics. Our analysis generalizes quantum speed limits: we count the distinct (orthogonal) states that can occur in a finite length of unitary change. As in Nyquist's bound on distinct signal values in classical waves, widths of superpositions bound the distinct states per unit length -- and basic conserved quantities are widths. Maximally distinct unitary evolution is effectively discrete -- and this characterizes classical systems. [see also Popular Summary in arxiv ancillary files]
For an isolated quantum system, average energy above the ground state and average momentum determine the minimum time and distance needed to transition through N distinct states. Other moments of energy and momentum provide similar bounds. For N≫1, we can equate average energy and momentum with the maximum average rates of distinct state change in time and space. Thus at least vestiges of finite-state dynamics persist in the classical world, and in its interpretation.
Reversible lattice dynamics embody basic features of physics that govern the time evolution of classical information. They have finite resolution in space and time, don't allow information to be erased, and easily accommodate other structural properties of microscopic physics, such as finite distinct state and locality of interaction. In an ideal quantum realization of a reversible lattice dynamics, finite classical rates of state-change at lattice sites determine average energies and momenta. This is very different than traditional continuous models of classical dynamics, where the number of distinct states is infinite, the rate of change between distinct states is infinite, and energies and momenta are not tied to rates of distinct state change. Here we discuss a family of classical mechanical models that have the informational and energetic realism of reversible lattice dynamics, while retaining the continuity and mathematical framework of classical mechanics. These models may help to clarify the informational foundations of mechanics.
Introduction In this chapter we discuss the basic physical concepts and equations related to the behavior of particles in the nanoworld. We introduce the Schrödinger wave equation for particles and determine the ways to calculate observable physical quantities. We find that, in wave mechanics, the motion of a particle confined to a finite volume is always characterized by discrete values of the energy and standing-wave-like wavefunctions, i.e., such a motion is quantized. While motion in an infinite space (i.e., free motion) is not quantized and is described by propagating waves, the energy of the particle is characterized by a continuous range of values. Keeping in mind the diverse variants of nanostructures, by using wave mechanics we analyze some particular examples, which highlight important quantum properties of particles. Many of the examples analyzed can serve as the simplest models of nanostructures and will be exploited in following chapters to understand the fundamentals of processes in nanoelectronics. The Schrödinger wave equation From the previous chapter, we conclude that nanosize physical systems are quantum-mechanical systems, inasmuch as their sizes are comparable to typical de Broglie wavelengths of the particles composing these systems. In dealing with quantum-mechanical systems, one aims at determining the wavefunction of a single particle or of the whole system. As we will demonstrate in the subsequent discussion, knowledge of the wavefunction in quantum mechanics is sufficient to describe completely a particle or even a system of particles.
The quantum measurement problem, i.e., the apparent conflict between unitary quantum evolution and non-unitary, stochastic wave-function collapse, remains unresolved a century after the formulation of quantum mechanics. We first review the standard picture, from the Copenhagen prescription with its Heisenberg cut, through von Neumann's movable cut, to environment-induced decoherence, which explains the emergence of stable classical records but still presupposes the Born rule and therefore cannot by itself replace the measurement postulates. We then connect this problem to the Maximum Entropy Principle in two complementary ways. First, decoherence drives the measurement apparatus toward the least biased state compatible with the dynamically protected pointer distribution, so that the emergent collapse may be viewed as a thermodynamic relaxation toward constrained maximum entropy. Second, we propose a Timeless Histories formulation of quantum mechanics in which the primitive objects are ordered sequences of events rather than evolving wave functions. Their conditional probabilities are assigned, relative to specified refinements, by a single Born-Boltzmann rule that combines the quadratic structure of Born's rule with Boltzmann's counting of equally probable microstates. The textbook form of Born's rule, the projection postulate, density operators, and wave functions are then recovered as derived informational constructs, while collapse becomes Bayesian conditioning on recorded events. Time and space enter only subsequently through unitary translation symmetry. Since the probability rule is postulated without reference to any physical collapse or classical observer, the usual circularity of the decoherence program is avoided. Decoherence can then be invoked to explain the stability and approximate additivity of macroscopic records, i.e., the emergence of classicality.