Trajectories and the perception of classical motion in the free propagation of wave packets

Abstract

The free propagation in time of a normalisable wave packet is the oldest problem of continuum quantum mechanics. Its motion from microscopic to macroscopic distance is the way in which most quantum systems are detected experimentally. Although much studied and analysed since 1927 and presented in many textbooks, here the problem is re-appraised from the standpoint of semi-classical mechanics. Particular aspects are the emergence of deterministic trajectories of particles emanating from a region of atomic dimension and the interpretation of the wave function as describing a single particle or an ensemble of identical particles. Of possible wave packets, that of Gaussian form is most studied due to the simple exact form of the time-dependent solution in real and in momentum space. Furthermore, this form is important in laser optics. Here the equivalence of the time-dependent Schrödinger equation to the paraxial equation for the propagation of light is demonstrated explicitly. This parallel helps to understand the relevance of trajectory concepts and the conditions necessary for the perception of quantum motion as classical.

The propagation of Hermite–Gauss wave packets in optics and quantum mechanics

“…This generalises the result given in Ref. [6] for the ground state. It gives a new interpretation of the Gouy phase as the 'adiabatic' energy phase arising from the continuous propagation of the HO HG wave function.…”

“…However, as shown in Ref. [6], the PE and TDSE are identical in that the approximation steps to derive the PE from the three‐dimensional Helmholtz equation are exactly those leading from the three‐dimensional time‐independent Schrödinger equation to the two‐dimensional TDSE.…”

“…In particular, the fact that the coordinates $$x,t$$ are connected by CM asymptotically has not been accorded the importance it deserves. Recently, attention has been called increasingly to the fact that the asymptotic form of the wave function represents the quantum to classical transition in a very general way 6,21,22 . Of course, this is the direct equivalent of the wave optics to ray optics transition.…”

A century ago the Stern-Gerlach experiment ruled unequivocally in favor of Quantum Mechanics

“…Most recently, John Briggs applied the 1937 imaging theorem of Edwin Kemble [76] to the motion of atoms over macroscopic dis- tances such as those encountered in the SGE and concluded that the perception of classical (trajectories) or quantal (wavepackets) behavior depends on the accuracy of detecting the atoms' motion. Briggs' treatment is based on the assumption that the wavefunction always describes a statistical ensemble of identically-prepared particles and that no meaning can be ascribed to the wavefunction of a single particle [77,78]; it leads to the conclusion that the quantum-to-classical transition occurs due to a unitary evolution of the wavefunction. In contrast, decoherence theory [79][80][81][82] assumes a singleparticle wavefunction whose evolution is non-unitary due to environmental effects and whose collapse leads to a particular outcome of a measurement.…”

A Century Ago the Stern–Gerlach Experiment Ruled Unequivocally in Favor of Quantum Mechanics

“…[73] Most recently, John Briggs applied the 1937 imaging theorem of Edwin Kemble [74] to the motion of atoms over macroscopic distances such as those encountered in the SGE and concluded that the perception of classical (trajectories) or quantal (wavepackets) behavior depends on the accuracy of detecting the atoms' motion. Briggs' treatment is based on the assumption that the wavefunction always describes a statistical ensemble of identically-prepared particles and that no meaning can be ascribed to the wavefunction of a single particle; [75,76] it leads to the conclusion that the quantum-to-classical transition occurs due to a unitary evolution of the wavefunction. In contrast, decoherence theory [77][78][79][80] assumes a single-particle wavefunction whose evolution is non-unitary due to environmental effects and whose collapse leads to a particular outcome of a measurement.…”