Spontaneous symmetry breakdown in non-relativistic quantum mechanics Am. J. Phys. 80, 891 (2012) Understanding the damping of a quantum harmonic oscillator coupled to a two-level system using analogies to classical friction Am. J. Phys. 80, 810 (2012) Relation between Poisson and Schrödinger equations Am. J. Phys. 80, 715 (2012) Comment on "Exactly solvable models to illustrate supersymmetry and test approximation methods in quantum mechanics," Am. J. Phys. 79, 755-761 (2011) Am. J. Phys. 80, 734 (2012) The uncertainty product of position and momentum in classical dynamics Am.Wigner functions and Weyl transforms of operators offer a formulation of quantum mechanics that is equivalent to the standard approach given by the Schrödinger equation. We give a short introduction and emphasize features that give insight into the nature of quantum mechanics and its relation to classical physics. A careful discussion of the classical limit and its difficulties is also given. The discussion is self-contained and includes complete derivations of the results presented.

Talbot and Talbot-Lau effects are frequently used in lensless imaging applications with light, ultrasound, x-rays, atoms and molecules--generally in situations where refractive optical elements are non-existent or not suitable. We here show an experimental visualization of the intriguing wave patterns that are associated with near-field interferometry behind a single periodic diffraction grating under plane wave illumination and which are often referred to as Talbot carpets or quantum carpets. We also show the patterns behind two separated diffraction gratings under nearly-monochromatic but spatially incoherent illumination that illustrate the nature of Talbot-Lau carpets.

We examine the free time evolution of a rectangular one dimensional Schrödinger wave packet of constant phase during the early stage which in the paraxial wave approximation is identical to the diffraction of a scalar field from a single slit. Our analysis, based on numerics and the Cornu spiral reveals considerable intricate detail behavior in the density and phase of the wave. We also point out a concentration of the intensity that occurs on axis and propose a new measure of width that expresses this concentration.

The pumping of a playground swing from the seated position is modeled using a dumbbell consisting of three masses. The equation of motion of the system when the dumbbell is forced to rock back and forth is obtained. When reasonable approximations for amplitudes are made, the equation of a linear harmonic oscillator with a driving and three parametric terms is obtained. Based on an analysis of this equation using numerical values of the parameters, it is concluded that the pumped swing is a driven harmonic oscillator for small-to-moderate amplitudes. Growth rates and relative phase of pumping and swing motion are determined and discussed. The source of the driving term is traced to the conservation of total angular momentum (motion of the center of mass and motion about the center of mass) about the point of support of the swing.

The pumping of a swing from a standing position is modeled as a rigid object forced to rotate back and forth at the lower ends of supporting ropes. This model after some approximations leads to a harmonic oscillator with driving and parametric terms. It is then argued that in the regime of the common playground swing the driving terms dominate and the pumping of a swing in the standing position is best characterized as a driven oscillator. Examination of the relative phase of the swinger and the swing also supports this conclusion. This model is compared with earlier work which claimed that the swing pumped by a standing swinger is characterized as a parametric oscillator. Simple demonstrations of both mechanisms are described. A comparison of pumping while standing and seated is also made.