Free Particle, Wave Packet and Dynamics, Quantum Dots, Defects and Traps

2021 ◽  
pp. 32-56
Author(s):  
Duncan G. Steel
Keyword(s):  
Wave Packet ◽  
Dimensional Space ◽  
Free Particle ◽  
Plane Waves ◽  
Higher Dimensions ◽  
Destructive Interference ◽  
Piecewise Constant ◽  
Wave Package ◽  
Quantum Behavior ◽  
The Waves

This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum well. A free particle can exist over all space or can be localized in a wave package. The wave packet is a coherent superposition of the plane waves that make up the wave function that localizes the particle because of constructive and destructive interference. The wave packet spreads out in time because the waves leading to constructive interference get out of phase. In Chapter 2, the particle was localized by a quadratic potential energy. Here, the potentials are described as piecewise constant. The approach is based on assuming a one-dimensional space, x, which is relevant to many problems in the laboratory. The solution is easily generalized to higher dimensions (x-y or x-y-z), but the physics remains the same. The objective is to understand the shape of the eigenfunctions in space and to be able to relate this to the probability density of locating the particle, as well as understanding the relevance of these systems to today’s technology.

scholarly journals HIDDEN SCALE IN QUANTUM MECHANICS

2009 ◽  
Vol 24 (27) ◽  
pp. 2203-2211 ◽  
Author(s):  
PULAK RANJAN GIRI
Keyword(s):  
Wave Packet ◽  
Quantum Particle ◽  
Dimensional Space ◽  
Free Particle ◽  
Three Dimensional ◽  
Particle Systems ◽  
Quantum Mechanical ◽  
Scale Invariant ◽  
Mechanical Models ◽  
Three Dimensional Space

We show that the intriguing localization of a free particle wave-packet is possible due to a hidden scale present in the system. Self-adjoint extensions (SAE) is responsible for introducing this scale in quantum mechanical models through the nontrivial boundary conditions. We discuss a couple of classically scale invariant free particle systems to illustrate the issue. In this context it has been shown that a free quantum particle moving on a full line may have localized wave-packet around the origin. As a generalization, it has also been shown that particles moving on a portion of a plane or on a portion of a three-dimensional space can have unusual localized wave-packet.

scholarly journals The relativity theory of plane waves

1926 ◽  
Vol 111 (757) ◽  
pp. 95-104 ◽  
Keyword(s):  
Gravitational Field ◽  
Electromagnetic Waves ◽  
Small Amplitude ◽  
Plane Waves ◽  
Cumulative Effect ◽  
Relativity Theory ◽  
Transverse Waves ◽  
Propagation Of Light ◽  
Cumulative Change ◽  
The Waves

Weyl has shown that any gravitational wave of small amplitude may be regarded as the result of the superposition of waves of three types, viz.: (i) longitudinal-longitudinal; (ii) longitudinal-transverse; (iii) transverse-transverse. Eddington carried the matter much further by showing that waves of the first two types are spurious; they are “merely sinuosities in the co­ordinate system,” and they disappear on the adoption of an appropriate co-ordinate system. The only physically significant waves are transverse-transverse waves, and these are propagated with the velocity of light. He further considers electromagnetic waves and identifies light with a particular type of transverse-transverse wave. There is, however, a difficulty about the solution as left by Eddington. In its gravitational aspect light is not periodic. The gravitational potentials contain, in addition to periodic terms, an aperiodic term which increases without limit and which seems to indicate that light cannot be propagated indefinitely either in space or time. This is, of course, explained by noting that the propagation of light implies a transfer of energy, and that the consequent change in the distribution of energy will be reflected in a cumulative change in the gravitational field. But, if light cannot be propagated indefinitely, the fact itself is important, whatever be its explana­tion, for the propagation of light over very great distances is one of the primary facts which the relativity theory or any like theory must meet. In endeavouring to throw further light on this question, it seemed desirable to avoid the assumption that the amplitudes of the waves are small; terms neglected on this ground might well have a cumulative effect. All the solu­tions discussed in this paper are exact.

scholarly journals Solutions of the Relativistic Two-Body Problem. II. Quantum Mechanics

1972 ◽  
Vol 25 (2) ◽  
pp. 141 ◽  
Author(s):  
JL Cook
Keyword(s):  
Bound States ◽  
Free Particle ◽  
Plane Waves ◽  
Addition Theorem ◽  
Partial Wave Analysis ◽  
Harmonic Oscillator Potential ◽  
Two Body Problem ◽  
Probability Densities ◽  
Square Well ◽  
Potential Harmonic

This paper discusses the formulation of a quantum mechanical equivalent of the relative time classical theory proposed in Part I. The relativistic wavefunction is derived and a covariant addition theorem is put forward which allows a covariant scattering theory to be established. The free particle eigenfunctions that are given are found not to be plane waves. A covariant partial wave analysis is also given. A means is described of converting wavefunctions that yield probability densities in 4-space to ones that yield the 3-space equivalents. Bound states are considered and covariant analogues of the Coulomb potential, harmonic oscillator potential, inverse cube law of force, square well potential, and two-body fermion interactions are discussed.

scholarly journals Rational Trigonometry in Higher Dimensions and a Diagonal Rule for $2$-planes in Four-dimensional Space

KoG ◽  
2017 ◽  
pp. 47-54
Author(s):  
Norman Wildberger
Keyword(s):  
Dimensional Space ◽  
Higher Dimensions ◽  
Natural Generalization ◽  
Principal Angles ◽  
Rational Trigonometry ◽  
Pythagoras Theorem ◽  
The Cross

We extend rational trigonometry to higher dimensions by introducing rational invariants between $k$-subspaces of $n$-dimensional space to give an alternative to the canonical or principal angles studied by Jordan and many others, and their angular variants. We study in particular the cross, spread and det-cross of $2$-subspaces of four-dimensional space, and show that Pythagoras theorem, or the Diagonal Rule, has a natural generalization forsuch $2$-subspaces.

Initial phase and free-particle wave packet evolution

2009 ◽  
Vol 77 (6) ◽  
pp. 538-545 ◽  
Author(s):  
Theodore L. Beach
Keyword(s):  
Wave Packet ◽  
Initial Phase ◽  
Free Particle

scholarly journals Electromagnetically induced transparency in plasma

2001 ◽  
Vol 19 (2) ◽  
pp. 175-179 ◽  
Author(s):  
B. ERSFELD ◽  
D.A. JAROSZYNSKI
Keyword(s):  
Electromagnetic Waves ◽  
Plasma Frequency ◽  
Electromagnetically Induced Transparency ◽  
Plane Waves ◽  
Electron Mass ◽  
Number Density ◽  
Relativistic Fluid ◽  
Longitudinal Plasma ◽  
Induced Transparency ◽  
The Waves

The coupled propagation of two electromagnetic waves in plasma is studied to establish the conditions for induced transparency. Induced transparency refers to the situation where both waves propagate unattenuated, although the frequency of one (or both) of them is below the plasma frequency so that it could not propagate in the absence of the other. The effect is due to the interaction of the waves through their beat, which modulates both the electron mass and, by exciting longitudinal plasma oscillations, their number density, and thus the plasma frequency. Starting from a relativistic fluid description, a dispersion relation for plane waves of weakly relativistic intensities is derived, which takes into account the polarization of the waves and the nonlinearities with respect to both their amplitudes. This serves as a basis for the exploration of the conditions for induced transparency and the modes of propagation.

Gravity Waves in a Horizontal Shear Flow. Part I: Growth Mechanisms in the Absence of Potential Vorticity Perturbations

2009 ◽  
Vol 39 (3) ◽  
pp. 481-496 ◽  
Author(s):  
Nikolaos A. Bakas ◽  
Brian F. Farrell
Keyword(s):  
Shear Flow ◽  
Wave Packet ◽  
Gravity Waves ◽  
Static Stability ◽  
Horizontal Shear ◽  
Growth Mechanisms ◽  
Trapping Level ◽  
Energy Growth ◽  
Zonal Velocity ◽  
The Waves

Abstract Interaction of internal gravity waves with a horizontal shear flow in the absence of potential vorticity perturbations is investigated making use of closed-form solutions. Localized wave packet trajectories are obtained, the energy growth mechanisms occurring are identified, and the potential role of perturbation growth in wave breaking is assessed. Regarding meridional propagation, the wave packet motion is limited by turning levels where the waves are reflected and trapping levels where the waves stagnate. Regarding perturbation energy amplification, two growth mechanisms can be distinguished: growth due to advection of zonal velocity and growth due to downgradient Reynolds stresses. The three-dimensional perturbations producing optimal energy growth reveal that these two mechanisms produce large and robust amplification of zonal velocity and/or density and vertical velocity, potentially leading to shear or convective instability. For large static stability, amplification of density perturbations in conjunction with vertical orientation of the constant phase lines close to the trapping level potentially leads to a convective collapse of the wave packet near the trapping level, in agreement with existing direct numerical simulation studies. For lower static stability and for waves with phase lines oriented horizontally, growth due to advection of zonal velocity dominates, leading to rapid growth of streamwise streaks within the localized wave packet and potentially to shear instability.

Effect of weak nonlinearities on the plane waves in a plasma stream

1976 ◽  
Vol 54 (1) ◽  
pp. 39-47 ◽  
Author(s):  
S. R. Seshadri
Keyword(s):  
Plane Waves ◽  
Equation Of Motion ◽  
Wave Number ◽  
Plasma Stream ◽  
Electromagnetic Mode ◽  
Correct Equation ◽  
Long Time ◽  
Modulational Stability ◽  
The Waves ◽  
Slow Modulation

The effect of weak nonlinearities on the monochromatic plane waves in a cold infinite plasma stream is investigated for the case in which the waves are progressing parallel to the drift velocity. The fast and the slow space-charge waves undergo amplitude-dependent frequency and wave number shifts. There is a long time slow modulation of the amplitude of the electromagnetic mode which becomes unstable to this nonlinear wave modulation. The importance of using the relativistically correct equation of motion for predicting correctly the modulational stability of the electromagnetic mode is pointed out.

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