Page conjectured very recently [Phys. Rev. Lett. 71, 1291(1993] that if a quantum system of Hilbert space dimension mn is in a random pure state, the average entropy of a subsystem of dimension m & n is given by S "= P&" &1 -™ . We outline a proof of this elegant formula.PACS numbers: 05.30.Ch, 02.90. +p, 03.65.w, 05.90.+m Very recently Page [1] considered the problem of getting entropy out of a system in a pure quantum state. He took a quantum system AB of Hilbert dimension mn with normalized density matrix, and divided it into two coupled

No abstract

No abstract

The simplest random evolution is the motion of a particle on a straight line with constant velocity suffering random collisions, which reverse the velocity. The position x(t) of the particle at time t is the stochastic process that defines Kac's path-integral solution of the telegrapher equation. We view Kac's prescription as a path-dependent time reparametrization, which associates to the time t spent by the particle in going from xo to x along a path co, the time r(co) the particle would have taken to go from xo to x without reversal. We compute explicitly the probability distribution of the random variable r. It is then possible to compute by simple quadrature any path integral with an integrand function of rico).

No abstract

No abstract

Low Btu gas has been produced by gasifying coal in a spouted bed reactor. Coal of size 0.8‐3.6 mm is fed continuously to a 0.15 m diameter spouted bed of inerts and gasified using mixtures of steam and air. Tests of the process with Western Canadian coals of free swelling index 0, 4 and 7 are reported. Gases of heating value to 3.61 MJ/m3 were produced at coal throughputs of 0.188 kg/m2s with the reactor operating at atmospheric pressure and temperatures to 930°C. Characteristics of the spouted bed gasifier are presented and results compared to commercial moving and fluidized bed systems. A simple mathematical model based on the two‐region spouted bed model of Mathur and Lim is used to predict the effect on steam utilization of bed composition, bed height and diameter, and particle size.

Projectile motion is usually the first non-uniform two-dimensional motion that students will encounter in a pre-university physics course. In this article, we introduce a novel technique for capturing the trajectory of projectile motion on an inclined Perspex plane. This is achieved by coating the Perspex with a thin layer of fine water droplets that allows the projectile to leave a trail as it passes. The experiment was field-tested in Singapore and the response was very good.