Daniel F. Styer

Nine formulations of quantum mechanics

Nine formulations of nonrelativistic quantum mechanics are reviewed. These are the wavefunction, matrix, path integral, phase space, density matrix, second quantization, variational, pilot wave, and Hamilton–Jacobi formulations. Also mentioned are the many-worlds and transactional interpretations. The various formulations differ dramatically in mathematical and conceptual overview, yet each one makes identical predictions for all experimental results.

Common misconceptions regarding quantum mechanics

This paper lists 15 commonly held misconceptions concerning quantum mechanics, such as ‘‘Energy eigenstates are the only allowed states’’ and ‘‘The wave function is dimensionless.’’ A few suggestions are offered to help combat these misconceptions in teaching.

Insight into entropy

What is the qualitative character of entropy? Several examples from statistical mechanics (including liquid crystal reentrant phases, two different lattice gas models, and the game of poker) demonstrate facets of this difficult question and point toward an answer. The common answer of “entropy as disorder” is regarded here as inadequate. An alternative but equally problematic analogy is “entropy as freedom.” Neither simile is perfect, but if both are used cautiously and not too literally, then the combination provides considerable insight.

Quantum revivals versus classical periodicity in the infinite square well

A particle of mass M moves in an infinite square well of width L (the “particle in a box”). Classically, the motion has period L2M/E, which depends on the initial condition through the energy E. Quantum mechanically, any wave function repeats exactly with period 4ML2/πℏ, independent of the initial condition. Given this qualitative difference, how can the classical motion possibly be the limit of the quantal time development? The resolution of this paradox involves the difference between the exact revival (recurrence) of the wave function and the approximate periodicity of expectation values such as 〈x(t)〉. (The latter may recur an odd integral number of times before the full wave function recurs.) The period of the expectation values does depend on the initial condition and can possess the expected classical limit. [An Appendix demonstrates that, under suitably quasiclassical conditions, the quantal time evolution of 〈x(t)〉 passes over to the classical result not only in period, but also in its exact func...

The motion of wave packets through their expectation values and uncertainties

The Ehrenfest equations govern the time evolution of quantal expectation values, and they have a widely known generalization that governs the time evolution of quantal uncertainties. Unfortunately, these results are usually applied only to free particles. Here, these methods are reviewed and applied to the simple harmonic oscillator, revealing a simple and beautiful picture of wave‐packet motion with an obvious classical limit.

Entropy and evolution

Quantitative estimates of the entropy involved in biological evolution demonstrate that there is no conflict between evolution and the second law of thermodynamics. The calculations are elementary and could be used to enliven the thermodynamics portion of a high school or introductory college physics course.

The Dobrushin−Shlosman phase uniqueness criterion and applications to hard squares

The rigorous Dobrushin-Shlosman phase uniqueness criterion is reviewed, then applied to the hard square model to prove that only a single phase exists at activityz=1.185. The criterion is violated (for a five-site by five-site lattice cell) atz=1.35557, but this does not imply phase nonuniqueness. This work complements that of Dobrushin, Kolafa, and Shlosman, who proved phase uniqueness for allz≤1. Certain “experimentally” discovered regularities are presented as conjectures: one for a more general problem and two for the application to hard squares. Even with these regularities, however, substantial further improvements in the algorithmic implementation of the criterion will be required before it can become a practical tool for locating phase transitions.

How do two moving clocks fall out of sync? A tale of trucks, threads, and twins

In special relativity, a pair of clocks synchronized in their own reference frame are not synchronized in another. How do two clocks, initially synchronized and at rest in the laboratory frame, fall out of sync as their speed relative to the lab gradually increases? The answer lies in general-relativistic time dilation. The path to the answer sheds light on the thread-between-spaceships paradox (also called the Bell spaceship paradox), on the twin paradox, and on the character of length contraction.

What good is the thermodynamic limit

Statistical mechanics applies to large systems: technically, its results are exact only for infinitely large systems in “the thermodynamic limit.” The importance of this proviso is often minimized in undergraduate courses. This paper presents six paradoxes in statistical mechanics that can be resolved only by acknowledging the thermodynamic limit. For example, it demonstrates that the widely used microcanonical “thin phase space limit” must be taken after taking the thermodynamic limit.

Vasserstein distances in two-state systems

We present formulas for the Vasserstein distance between two statistical mechanical states of a two-state system. For example, in a ferromagnetic spin-1/2 Ising model the Vasserstein distance is half the difference in the magnetizations.