The motion of a particle in a periodic field of force

1928 ◽  
Vol 24 (3) ◽  
pp. 447-450
Author(s):  
S. L. Malurkar ◽  
J. Hargreaves
Keyword(s):  
Wave Packet ◽  
Uncertainty Relation ◽  
Classical Dynamics ◽  
Electron Theory ◽  
Periodic Field ◽  
Periodic Term ◽  
One Dimensional ◽  
De Broglie Waves ◽  
Suitable Combination

The electron theory of metals revived by Sommerfeld assumes that an electron moves in a metal as though this were an equipotential medium. Considering the nuclei fixed and regularly spaced we obtain a potential periodic in space coordinates. To study the effect of such fields we may simplify the problem so as to contain only one periodic term for each coordinate in its expression for potential. This problem can be reduced further to a one-dimensional one, of which the simplest example is the motion of an electron in a field with potential cos x or sin x. Darwin has shown that a suitable combination or packet of elementary de Broglie waves is capable of moving coherently in several instances. The motion of such a packet is found to be equivalent to that of a particle in classical dynamics with the Heissenberg uncertainty relation. The wave packet is used here for the motion of the electron in a periodic field. The result obtained is equivalent to that of classical dynamics. The wave packet again moves as a particle with an uncertainty relation.

CLASSICAL DYNAMICS OF ESCAPE IN A PERIODICALLY DRIVEN UNBOUND HAMILTONIAN SYSTEM: INITIAL PHASE EFFECTS

2011 ◽  
Vol 21 (09) ◽  
pp. 2587-2596 ◽  
Author(s):  
LEONIDAS KONSTANTINIDIS ◽  
VASSILIOS CONSTANTOUDIS ◽  
CLEANTHES A. NICOLAIDES
Keyword(s):  
External Field ◽  
Initial Phase ◽  
Field Amplitude ◽  
Classical Dynamics ◽  
Morse Oscillator ◽  
Molecular Dissociation ◽  
Escape Probability ◽  
Periodic Field ◽  
Periodically Driven ◽  
Energy Exchanges

We consider the problem of a classical Morse oscillator driven by an external periodic field of controllable characteristics and study systematically the effects of the initial phase of the external field on the probability of escape upon molecular dissociation at the end of the interaction time. First, it is shown that such effects indeed exist and may become important depending on the field amplitude and frequency. Then, an explanation is given in terms of the energy exchanges between the basic periodic orbits and the external field and of the associated modifications of the shape of regular islands in phase space. Finally, we exploit this explanation in order to make predictions regarding the behavior of the escape probability as a function of the initial phase and frequency of the external field.

Asymptotic estimate for the counting problems corresponding to the dynamical system on some decorated graphs

2017 ◽  
Vol 38 (5) ◽  
pp. 1697-1708 ◽  
Author(s):  
V. L. CHERNYSHEV ◽  
A. A. TOLCHENNIKOV
Keyword(s):  
Dynamical System ◽  
Wave Packet ◽  
General Theorem ◽  
Three Dimensional ◽  
Initial Time ◽  
Dimensional Torus ◽  
Counting Problems ◽  
One Dimensional ◽  
Moving Points ◽  
Narrow Wave

We study a topological space obtained from a graph via replacing vertices with smooth Riemannian manifolds, that is, a decorated graph. We consider the following dynamical system on decorated graphs. Suppose that, at the initial time, we have a narrow wave packet on a one-dimensional edge. It can be thought of as a point moving along the edge. When a packet arrives at the point of gluing, the expanding wavefront begins to spread on the Riemannian manifold. At the same time, there is a partial reflection of the wave packet. When the wavefront that propagates on the surface comes to another point of gluing, it generates a reflected wavefront and a wave packet on an edge. We study the number of Gaussian packets, that is, moving points on one-dimensional edges as time goes to infinity. We prove the asymptotic estimations for this number for the following decorated graphs: a cylinder with an interval, a two-dimensional torus with an interval and a three-dimensional torus with an interval. Also we prove a general theorem about a manifold with an interval and apply it to the case of a uniformly secure manifold.

Quantum and classical dynamics of H + CaCl(X 2Σ+) → HCl + Ca(1S) reaction and vibrational energy levels of the HCaCl complex

2016 ◽  
Vol 18 (23) ◽  
pp. 15673-15685 ◽  
Author(s):  
Rui Shan Tan ◽  
Huan Chen Zhai ◽  
Feng Gao ◽  
Dianmin Tong ◽  
Shi Ying Lin
Keyword(s):  
Wave Packet ◽  
Cross Sections ◽  
Ground State ◽  
Energy Surface ◽  
Vibrational Energy ◽  
Energy Levels ◽  
Classical Trajectory ◽  
Classical Dynamics ◽  
Quantum Wave ◽  
Vibrational Energy Levels

We carried out accurate quantum wave packet as well as quasi-classical trajectory (QCT) calculations for H + CaCl (νi = 0, ji = 0) reaction occurring on an adiabatic ground state. Recent ab initio potential energy surface is employed to calculate the quantum and QCT reaction probabilities for several partial waves (J = 0, 10, and 20) as well as state resolved QCT integral and differential cross sections.

The W. K. B. approximation for Bloch electrons with an application to interband tunnelling

1964 ◽  
Vol 280 (1381) ◽  
pp. 185-197 ◽  
Keyword(s):  
Wave Function ◽  
Wave Equation ◽  
Effective Mass ◽  
Periodic Term ◽  
One Dimensional ◽  
Auxiliary Functions ◽  
Varying Coefficients ◽  
Exponential Behaviour ◽  
The One ◽  
Bloch Electrons

The W. K. B. technique for solving the one-dimensional wave equation is extended to the case when the potential field includes a rapidly varying periodic term as well as a slowly varying term. A pair of auxiliary functions are introduced which are identical to the wave function and its derivative respectively at the edges of the periodic cells, but which have a simple exponential behaviour within the cells. The auxiliary functions satisfy a pair of auxiliary (related) differential equations, with slowly varying coefficients, which are valid for all energy values. Solution of the auxiliary equations by the well-known W. K. B. technique yields approximations to the wave function. These approximations break down in the neighbourhood of the band edges, which are the turning points of the problem. Connexion formulae are established across the band edges and employed to calculate the interband tunnelling probability. In the immediate neighbourhood of a band edge the analysis yields an effective-mass wave equation and a closed form for the wave function. The auxiliary functions are closely related to the effective-mass modulating wave function and the results of this paper may be regarded as an extension of effective-mass theory for the one-diinensional case, throughout the whole of the energy ranges of allowed bands and forbidden gaps.

scholarly journals Non-Markovianity induced by a single-photon wave packet in a one-dimensional waveguide

2016 ◽  
Vol 41 (13) ◽  
pp. 3126 ◽  
Author(s):  
D. Valente ◽  
M. F. Z. Arruda ◽  
T. Werlang
Keyword(s):  
Wave Packet ◽  
Single Photon ◽  
One Dimensional

QUANTUM EFFECTS OF A PARTICLE IN AN INFINITE SQUARE WELL WITH A MOVING WALL

2000 ◽  
Vol 14 (10) ◽  
pp. 1059-1065 ◽  
Author(s):  
JIAN ZOU ◽  
BIN SHAO
Keyword(s):  
Wave Packet ◽  
Boundary Conditions ◽  
Coherent State ◽  
Dynamical Behavior ◽  
Squeezed State ◽  
Initial State ◽  
One Dimensional ◽  
Moving Wall ◽  
Quantum Behavior ◽  
Square Well

The quantum behavior of a particle in a one-dimensional infinite square well potential with a moving wall is studied. The particle is assumed to be initially prepared in the coherent state (Gaussian wave packet) and although the boundary is far from the particle, it is shown that the changing of the boundary conditions can instantaneously affect the dynamical behavior of the particle. It is also shown that the initial state can evolve into a squeezed state, and in some cases the spreading of the wavepacket could be suppressed. Finally the Pancharatnam phase is also discussed.

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