Particle Descriptions

In an inertial reference frame of relatively small but constant velocity the classic Newtonian description of particle motion is a very good approximation, provided the particle velocity and its space-time variability are small. If either of these conditions are not met, relativistic or quantum theoretic descriptions are necessary since the concept of a classic particle becomes questionable. A review of classical particle dynamics is first presented, followed at large particle speeds by Einstein's amazing relativistic linkage of space and time as well as of mass and energy. In regions of small space-time dimensions the need for a probabilistic description of particle behavior leads to a non-standard derivation of the Schroedinger wave description of single particle motion and the quantum phenomena resulting therefrom.

The classical equations of single particle motion describe the position q and momentum p at time t of a particle in p,q phase space. For motion in a gravitational or other potential field an energy Hamiltonian H(p,q) provides a convenient means for deriving these classical equations. Classical particle motion may also be described by a wave function f(q,t) that satisfies a wave equation, obtained by setting -(¶/¶ t)f = H(p,q) and (¶/¶ x)f = p, the physical significance of which is not clear classically. For particle speeds less than the speed of light, Einstein’s relativistic energy Hamiltonian is used.

In the quantum range of space-time scales, the motion of a single particle requires a statistical description. The probable average position <q(t)> of a particle at time t is defined in terms of a complex wave function y(q,t), whose amplitude squared represents the probability density of finding the particle. Derivation of a defining equation for y(q,t) follows from the requirement that averages of particle dynamic observables calculated from the wave function y(q,t) are identical to corresponding classical averages which on the average obey the classical equations of motion . For relatively weak space-time variability, the quantum wave description reduces approximately to the classic dynamical description of the particle. This wave-particle duality in appropriate ranges, well known in quantum mechanics, is not unfamiliar in classical wave dynamics where it is usually expressed in ray optic language.

For a localized system of classical point particles, each having an energy H(p,q), an ensemble averaged kinetic distribution function f(p,q,t) provides a means for determination of the average position and momentum of the localized system. A kinetic equation for f(p,q,t) may be derived from a knowledge of H(p,q). The overall system may be described in terms of variables that are either kinetic (momentum, space, and time dependent), fluid-dynamic (space and time dependent), or macro-particle (time dependent). The different descriptions are derivable from the kinetic via momentum, or velocity and spatial moments, of the kinetic distribution function.

 Wavepacket Descriptions

The equations of propagation of a classical wavepacket y(x,t) may be described in terms of a frequency ("energy") operator function H(P,Q) where P and Q are wavepacket momentum and position operators in x-space. The functional form of H(P,Q) determines the equations of wavepacket propagation in space-time. If the space-time scales of the wavepacket and of the propagating medium are appropriately large, the wavepacket may also be represented approximately as a system of quasiparticles, each of momentum k and position x, whose dynamics is governed by a function H(k,x) indicative of the frequency ("energy") of a propagating wavepacket type. The functional form of H(k,x) determines the overall quasiparticle dynamics and suggests a physical picture of how a wavepacket evolves in space-time. At the kinetic level of description, the quasiparticle system at time t is expressed in terms of a quasiparticle distribution function F(k,x,t) in k,x phase space. An x,t dependent "fluid dynamic" level of description is derived from k moments of F(k,x,t). An average t-dependent description, indicative of the average position x(t), wavenumber (momentum) k(t) and frequency w(t) of the wave packet is recovered from k and x integrals of F(k,x,t).

Classical and Quantum Description of Single Particle and Wavepacket Systems

Kinetic Quasiparticle Description of a Classical Wavepacket

Macro-Particle Description of a Wavepacket