Wave Packet Model Research Articles

Wave packet with special relativity demonstrating quantum rules, Schrödinger's equation and propagator integral

The physics of special relativity is applied to a wave-packet model of a particle without the usual de Broglie-like assumption of Planck's rule, E = hv. The velocity of the particle is equated to the group-velocity of the wave-packet to associate the particle with its wave-packet. With such an identification it is demonstrated using special relativity that the momentum four-vector, P = [p, E/c2] of the particle is parallel and proportional to a central wave-number four-vector, K0 = [k0. ω0/c2], of the particle's wave-packet. The constant of proportionality, relating these two four-vectors, has the dimension of action and by diffraction experiments it has the value h/2π where h is Planck's constant. This refinement of the de Broglie wave-like nature of a particle suggests that special relativity underlies quantum physics more deeply than heretofore believed. The above quantum rules are used to find the wave-packet function in the one-dimensional case of a non-relativistic free particle and a particle in a conservative force field of potential V (x). This wave-packet function is shown to satisfy an integral time-evolution equation which relates the wave function at time t0 to the function at a slightly later time t. The resulting integral equation is equivalent to the propagator equation or path-integral hypothesised by Feynman. This fact also demonstrates, using the method of Feynman, that the wave-packet function satisfies Schrodinger's partial differential equation. The wave-packet function of a particle in a force field is directly shown to satisfy a linear integral equation which relates the wave-packet at one instant of time to its values at all later times. The kernel or propagator of this integral equation is also equivalent to Feynman's path integral. A computation demonstrates that the position of a moving particle in a simple quadratic potential well does not become more uncertain with the passage of time. This contrasts with the behaviour of a moving free particle which as a function of time becomes more and more difficult to accurately locate.

Resonant tunneling with electron-phonon interactions: an exactly solvable model applied to desorption

The problem of hot electron and/or laser induced desorption of atoms or molecules on surfaces is considered. The equivalence between a previous wavepacket model for this process and the problem of inelastic resonant electron tunneling through a quantum well with electron-phonon coupling is established. Numerical consequences of some exactly solved quantum well models are explored as they relate to desorption.

A method for deriving particle phase shift

Regarding a method for deriving the particle phase-shift based on the nonspreading wave packet model with a de Broglie faster-than-light phase wave for a particle, a slight difference between nonrelativistic quantum theory and this method is shown, to the second order in the value of the particle mass, as determined by the calculation of particle phase shift due to the gravity of the Earth. This difference will be examined by advanced experiments in neutron interferometry.

Evolution of the modern photon

The term ‘‘photon’’ represents at least four distinct models and carries different connotations for students and for practicing physicists. This reflects the long and complex historical evolution of the concept and its association with the largely misinterpreted principle of duality. The unsatisfactory nature of the corpuscular and wave packet models is discussed, and the pedagogical desirability urged of replacing them with a semiclassical approach in elementary presentations. Derivations of the photoelectric (PE) effect without photons are cited and a vector analysis is given, demonstrating that the PE effect cannot be considered as simply the interaction of a photon and electron.

Semi-quantal differential spectra for the process Cs + O 2(υ = 0) → Cs + + O 2−(υ *) → Cs + O 2(υ′)

Total differential cross sections and vibrational distributions as a function of the scattering angle are presented for neutral scattering in Cs + O 2 collisions. These results were calculated by integration in time of the time-dependent Schrödinger equation along a classical straight-line trajectory. The vibrational motion of the molecular ion during the collision proves to be an important factor in the collision dynamics. The agreement with available experimental data, and with the results of the classical surface hopping trajectory model and the semi-classical moving wave packet model, is satisfactory.

A semiclassical wave packet model for the investigation of elastic and inelastic gas–surface scattering

A semiclassical wave packet model that permits simultaneous study of surface diffraction, Debye–Waller effects, and inelastic energy transfer processes at the gas–surface interface is presented. The incident atomic beam is represented by a quantum mechanical wave packet whose momentum-space probability density corresponds to that employed in the molecular-beam experiments being simulated. A semiclassical forced-oscillator-type approximation couples the surface motion to that of the wave packet through a time-varying potential obtained from a solution of Hamilton’s equations of motion for the lattice. Explicit integration procedures are used to evolve the wave packet through this time-varying field. The average final kinetic energy of the beam, the angular distributions and diffraction patterns, Debye–Waller broadening effects, and the surface phonon spectrum are obtained from the scattered wave packet and its Fourier transform. The model has been applied to the case of an atomic hydrogen beam having a square distribution of incident momenta impinging upon a simple two-dimensional lattice. The results yield diffraction patterns that correlate with the known grating of the lattice. The average final kinetic energy of the beam is found to vary linearly with incident energy and with surface temperature in accord with the results of recent molecular beam experiments, and a surface phonon spectrum that exhibits one-, two-, and three-phonon processes is obtained.

Critical behaviour of a disordered wave packet model

Disordered systems with n isotropically coupled spin components are considered. For n = 1, 2, 3 in 3.99 dimensions, there is a new stable random fixed point of the renormalization. For n = 4, the exponents depend continuously on the disorder.

The Determinism of Quantum-Mechanical Probability Statements

A presentation showing how the statements which relate to microphysical objects as they are different from the statements of classical mechanics is made. The determinism of classical and of quantum-mechanical theories is qualified. A (crucial) distinction between causality and determinism is given. Detailed analyses of diffraction as a result of single and double-slit demonstrations point to paradoxes arising from the use of particle or wave models, respectively, for photons and electrons. The compromising wave-packet model is underscored. The meanings for the Psi-function and for |ψ|2 are presented, with an explanation of the limitations that arise from giving an interpretation of |ψ|2 as a probability. The meaning of ψ-waves is analyzed. A case is made for the ψ-function's describing a well-ordered list of betting odds, based on previous statistical experiences, for the different results of specific experiments of a microscopic object of study with macroscopic apparatus. The Schrödinger wave equation is evaluated for meaning and judged as a deterministic expression.