Jeffrey J. Prentis

Renormalisation theory of the self-avoiding Levy flight

The self-avoiding Levy flight (SALF) in d dimensions with Levy exponent mu is formulated as a geometrical equilibrium statistical mechanical problem. A direct renormalisation theory, based on modern field theoretic techniques, is used to derive the critical exponents and the end-to-end distance probability function through first order in epsilon =2 mu -d. The non-perturbative structure of the probability function is characterised by a universal scaling function. The SALF represents a simple many-body system that can assume a continuum of values of epsilon near zero.

Quantum jumps and classical harmonics

We present an introduction to quantum mechanics based on the formal correspondence between the atomic properties of quantum jumps and the classical harmonics of the electron’s periodic motion. By adding a simple quantum condition to the classical Fourier analysis, we readily find the energies of the stationary states, calculate the transition probabilities between the states, and construct the line spectrum of the emitted light. We provide examples to illustrate the asymptotic, and sometimes exact, agreement between the classical-quantum results (Fourier harmonics) and the exact quantum results (Heisenberg harmonics).

Experiments in statistical mechanics

We present experiments designed to illustrate the basic concepts of statistical mechanics using a gas of “motorized molecules.” Two molecular motion machines are constructed. The pressure fluctuation machine (mechanical interaction simulator) is a working model of two gases separated by a movable piston. The Boltzmann machine (canonical simulator) is a working model of a two-level quantum system in a temperature bath. Dynamical probabilities (fraction of time) are measured using mechanical devices, such as stop watches and motion sensors. Statistical probabilities (fraction of states) are calculated using physical statistics, such as microcanonical and canonical statistics. The experiments enable one to quantitatively test the fundamental principles of statistical mechanics, including the fundamental postulate, the ergodic hypothesis, and the statistics of Boltzmann.

The 1925 Born and Jordan paper “On quantum mechanics”

The 1925 paper “On quantum mechanics” by M. Born and P. Jordan, and the sequel “On quantum mechanics II” by M. Born, W. Heisenberg, and P. Jordan, developed Heisenberg’s pioneering theory into the first complete formulation of quantum mechanics. The Born and Jordan paper is the subject of the present article. This paper introduced matrices to physicists. We discuss the original postulates of quantum mechanics, present the two-part discovery of the law of commutation, and clarify the origin of Heisenberg’s equation. We show how the 1925 proof of energy conservation and Bohr’s frequency condition served as the gold standard with which to measure the validity of the new quantum mechanics.

Crossover from the exact factor to the Boltzmann factor

We derive the exact physical statistics of a system in thermal equilibrium with different environments—a gas, a spin, and a Boltzmann environment. The crossover from the exact factor to the Boltzmann factor is parametrized by scaled energy variables. The precise conditions that generate the Boltzmann factor emerge naturally. Such an analytic representation of the exact factor, the crossover, and the Boltzmann limit are not found in the standard model-free derivations of the Boltzmann factor.

Energy conservation in quantum mechanics

In the classical mechanics of conservative systems, the position and momentum evolve deterministically such that the sum of the kinetic energy and potential energy remains constant in time. This canonical trademark of energy conservation is absent in the standard presentations of quantum mechanics based on the Schrodinger picture. We present a purely canonical proof of energy conservation that focuses exclusively on the time-dependent position x(t) and momentum p(t) operators. This treatment of energy conservation serves as an introduction to the Heisenberg picture and illuminates the classical-quantum connection. We derive a quantum-mechanical work-energy theorem and show explicitly how the time dependence of x and p and the noncommutivity of x and p conspire to bring about a perfect temporal balance between the evolving kinetic and potential parts of the total energy operator.

Poincaré’s proof of the quantum discontinuity of nature

In his last memoir on mathematical physics, Henri Poincare presented one of the most profound and compelling proofs of the hypothesis of quanta. This highly original proof, which is actually three separate proofs, is based on first principles and is full of physical insight, mathematical rigor, and elegant simplicity. The memoir is refreshingly uncluttered by some of the conventional, and more abstract concepts, such as temperature and entropy, that Planck and others relied on in their work. Poincare’s analysis is based on an ingenious physical model consisting of long‐period resonators interacting with short‐period resonators. A unique formulation of statistical mechanics, based on the calculus of probabilities, Fourier’s integral, and complex analysis, logically unfolds throughout the memoir. Poincare invents an ‘‘inverse statistical mechanics’’ that allows him to prove a crucial result that no one had proved before: The hypothesis of quanta is both a sufficient and a necessary condition to account for ...

Elliptical Orbit ⇒ 1/r2 Force

Newtons proof of the connection between elliptical orbits and inverse-square forces ranks among the “top ten” calculations in the history of science. This time-honored calculation is a highlight in an upper-level mechanics course. It would be worthwhile if students in introductory physics could prove the relation elliptical orbit ⇒ 1/r2 force without having to rely on upper-level mathematics. We introduce a simple procedure—Newtons Recipe—that allows students to readily and accurately deduce the algebraic form of force laws from a geometric analysis of orbit shapes.

The art of statistical mechanics: Looking at microscopic spectra and seeing macroscopic phenomena

Simple graphical spectra are presented as visual paradigms for the basic ideas of statistical mechanics. Each spectrum is designed so that the mechanical information can be readily converted into thermal and statistical properties.

Power law approximations for radioactive decay chains

Consider a radioactive decay chain X 1 ? ? ? X n ? and let N n ( t ) be the amount of X n at time t. This paper establishes error bounds for small and intermediate time approximations to N n ( t ) including the power-law approximation N n ( t ) ? Ct m - 1 for ? m + 1 < < t < < ? m where ? j is the jth largest half-life. The approximations shed light on the qualitative behavior of N n ( t ) and are useful for reducing the roundoff error when computing N n ( t ) for small t which is a problem with the usual formula. The error bounds allow one to find the range of t for which these approximations can be used with a given degree of precision.