Phillip R. Dukes

A GeoWall with Physics and Astronomy Applications

A GeoWall is a passive stereoscopic projection system that can be used by students, teachers, and researchers for visualization of the structure and dynamics of three-dimensional systems and data. The type of system described here adequately provides 3-D visualization in natural color for large or small groups of viewers. The name “GeoWall” derives from its initial development to visualize data in the geosciences.1 An early GeoWall system was developed by Paul Morin at the electronic visualization laboratory at the University of Minnesota and was applied in an introductory geology course in spring of 2001. Since that time, several stereoscopic media, which are applicable to introductory-level physics and astronomy classes, have been developed and released into the public domain. In addition to the GeoWalls application in the classroom, there is considerable value in its use as part of a general science outreach program. In this paper we briefly describe the theory of operation of stereoscopic projection ...

Quantum state revivals in quantum walks on cycles

Abstract Recurrence in the classical random walk is well known and described by the Polya number. For quantum walks, recurrence is similarly understood in terms of the probability of a localized quantum walker to return to its origin. Under certain circumstances the quantum walker may also return to an arbitrary initial quantum state in a finite number of steps. Quantum state revivals in quantum walks on cycles using coin operators which are constant in time and uniform across the path have been described before but only incompletely. In this paper we find the general conditions for which full-quantum state revival will occur.

The Evolution of our Probability Image for the Spin Orientation of a Spin — 1/2 — Ensemble as Measurements are Made on Several Members of the Ensemble — Connections with Information Theory and Bayesian Statistics

An analysis is made of the information acquired when measurements are made, of the component of spin along an arbitrarily chosen axis, (Θ, Φ), of each of several or many spin 1/2 systems, each prepared in the same way. Since the probability p↑ of the observed value of the spin being “up” is a unique function of the relative orientation of the axis of measurement, (Θ, Φ), and the quantization axis, (θ, φ), of the state into which the system was prepared, this probability function, p↑(θ, φ; Θ, Φ), may be interpreted as a likelihood function for the orientation, (θ, φ), of the state into which the system is prepared, conditioned by its having been measured with “up” spin along the axis with orientation (Θ, Φ). If one assumes that the preparation procedure prepares each system into the same unknown pure state, and further assumes that, a priori, each equal infinitisimal solid angle is equally likely to contain the orientation of this pure state, the likelihood function for the orientation, via Bayes’ theorem, becomes the (unnormalized) probability density function for the orientation of the pure state. Thus, from an ensemble of spin 1/2 systems, each prepared in the pure state with quantization axis (θ, φ), the probability for realizing n “up” values and m = N—n “down” values, in some specified order, in a sequence of N measurement events for the spin component along the (Θ, Φ) axis, is given by Π(θ, φ; Θ, Φ) = p ↑ n(1-p ↑)m = p ↑ n(1-p ↑]) N-n , (as in the Bernoulli coin-tossing problem). By the above arguments Π(θ, φ; Θ, Φ) dΩ, becomes, for fixed (Θ, Φ,n,N), the (unnormalized) probability for the system being prepared in the spin state whose direction is aligned to within dΩ of (θ, φ), conditioned by the measurement events which determined Π. If we take the density operator corresponding to this distribution function to be ρ = [∫dΩ,ρ0(θ, φ)Π(θ, φ; Θ, Φ)]/[∫dΩΠ(θ, φ; Θ, Φ)], where ρ0(θ, φ) is the density operator corresponding to the pure state aligned along (θ, φ), one finds that the probability of measuring “up” spin along (Θ, Φ), conditioned by a measurement of n “up” values in N trials is given by (n+1)/(N+2), which is Laplace’s rule of succession. Also presented is a geometrical parametrization of the density operator for a spin 1/2 system. The density operator for the pure state with spin orientation (θ, φ) is represented as the corresponding point on the unit sphere. Each point in the interior of the sphere represents a non-idempotent density operator, diagonal in the coordinate system in which the radial line to the representative point lies on the +2 axis, and such that the radius is equal to (σz) = λ↑ — λ↓. A measurement of n “up” values in TV trials, with a prior probability distribution that is uniform over the volume of this sphere, leads to (n+2)/(N+4) for the probability of measuring “up” spin; a more “cautious” rule of succession than that of Laplace.

A Lognormal State of Knowledge

The Lognormal distribution is derived as the representation of a particular state of knowledge using a combination of maximum entropy and group invariance arguments.

On the Assignment of Prior Expectation Values and a Geometric Means of Maximizing — Trρln ρ Constrained by Measured Expectation Values

An alternative method for applying the maximum entropy criterion to a simple two state system is presented. It evolves only a few rudiments of geometry and group theory. The advantages of this method are a vivid geometric depiction of the maximum entropy state as well as its analytic solution. The relationship between the maximum entropy state and the assignment of a prior expectation value made with it is also made apparent.