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 Flow of S-matrix poles for elementary quantum potentials1This research was supported in part by an NSERC Undergraduate Summer Research Award (SGN) and an NSERC Discovery Grant (MAW).

2011 ◽  
Vol 89 (11) ◽  
pp. 1127-1140 ◽  
Author(s):  
B. Belchev ◽  
S.G. Neale ◽  
M.A. Walton
Keyword(s):  
Bound States ◽  
Scattering Matrix ◽  
Nucl Phys ◽  
Quantum Scattering ◽  
Research Award ◽  
Momentum Plane ◽  
S Matrix ◽  
Potential Depth ◽  
Square Well ◽  
Insight Into

The poles of the quantum scattering matrix (S-matrix) in the complex momentum plane have been studied extensively. Bound states give rise to S-matrix poles, and other poles correspond to non-normalizable antibound, resonance, and antiresonance states. They describe important physics but their locations can be difficult to determine. In pioneering work, Nussenzveig (Nucl. Phys. 11, 499 (1959)) performed the analysis for a square well (wall), and plotted the flow of the poles as the potential depth (height) varied. More than fifty years later, however, little has been done in the way of direct generalization of those results. We point out that today we can find such poles easily and efficiently using numerical techniques and widely available software. We study the poles of the scattering matrix for the simplest piecewise flat potentials, with one and two adjacent (nonzero) pieces. For the finite well (wall) the flow of the poles as a function of the depth (height) recovers the results of Nussenzveig. We then analyze the flow for a potential with two independent parts that can be attractive or repulsive, the two-piece potential. These examples provide some insight into the complicated behavior of the resonance, antiresonance, and antibound poles.

Quantum Mechanics

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Quantum Mechanics ◽  
Ground State ◽  
Variational Principles ◽  
Free Particle ◽  
Uncertainty Relations ◽  
Probabilistic Interpretation ◽  
Historical Account ◽  
Potential Wells ◽  
Harmonic Oscillator Potential ◽  
Ground State Energies

In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.

Scattering of a Gaussian beam by an anisotropic-coated eccentric conducting circular cylinder

2020 ◽  
Vol 12 (9) ◽  
pp. 900-905
Author(s):  
Shi-Chun Mao ◽  
Zhen-Sen Wu ◽  
Zhaohui Zhang ◽  
Jiansen Gao ◽  
Lijuan Yang
Keyword(s):  
Gaussian Beam ◽  
Plane Waves ◽  
Approximate Expression ◽  
Transverse Magnetic ◽  
Addition Theorem ◽  
Electric Polarization ◽  
Electric Conductor ◽  
Solution Convergence ◽  
Local Coordinates ◽  
Scattered Fields

AbstractA solution to the problem of Gaussian beam scattering by a circular perfect electric conductor coated with eccentrically anisotropic media is presented. The incident Gaussian beam source is expanded as an approximate expression in the simple form with Taylor's series. The transmitted field in the anisotropically coated region is expressed as an infinite summation of Eigen plane waves with different polar angles. The unknown coefficients of the scattered fields are obtained with the aid of the boundary conditions. The addition theorem for cylindrical functions is applied to transfer from the local coordinates to the global ones. The infinite series can be truncated under the prerequisite of achieving the solution convergence. Only the case of transverse-electric polarization is discussed. The similar formulation of transverse-magnetic polarization can be obtained by adopting a similar method. Some numerical results are presented and discussed. The result is in agreement with that available as expected when the eccentric geometry comes to the concentric one.

Bound states, plane waves, and orthogonalization

1978 ◽  
Vol 18 (6) ◽  
pp. 2412-2420 ◽  
Author(s):  
A. Y. Sakakura ◽  
W. E. Brittin ◽  
M. D. Girardeau
Keyword(s):  
Bound States ◽  
Plane Waves

scholarly journals SINGULARITY-FREE BREIT EQUATION FROM CONSTRAINT TWO-BODY DIRAC EQUATIONS

1996 ◽  
Vol 05 (04) ◽  
pp. 589-615 ◽  
Author(s):  
HORACE W. CRATER ◽  
CHUN WA WONG ◽  
CHEUK-YIN WONG
Keyword(s):  
Quantum Electrodynamics ◽  
Bound States ◽  
Two Body Problem ◽  
Dirac Equations ◽  
Coupled Equations ◽  
Interaction Terms ◽  
Potential Form ◽  
Orthogonality Constraint ◽  
Hyperbolic Form ◽  
Breit Equation

We examine the relation between two approaches to the quantum relativistic two-body problem: (1) the Breit equation, and (2) the two-body Dirac equations derived from constraint dynamics. In applications to quantum electrodynamics, the former equation becomes pathological if certain interaction terms are not treated as perturbations. The difficulty comes from singularities which appear at finite separations r in the reduced set of coupled equations for attractive potentials even when the potentials themselves are not singular there. They are known to give rise to unphysical bound states and resonances. In contrast, the two-body Dirac equations of constraint dynamics do not have these pathologies in many nonperturbative treatments. To understand these marked differences we first express these contraint equations, which have an “external potential” form, similar to coupled one-body Dirac equations, in a hyperbolic form. These coupled equations are then recast into two equivalent equations: (1) a covariant Breit-like equation with potentials that are exponential functions of certain “generator” functions, and (2) a covariant orthogonality constraint on the relative momentum. This reduction enables us to show in a transparent way that finite-r singularities do not appear as long as the exponential structure is not tampered with and the exponential generators of the interaction are themselves nonsingular for finite r. These Dirac or Breit equations, free of the structural singularities which plague the usual Breit equation, can then be used safely under all circumstances, encompassing numerous applications in the fields of particle, nuclear, and atomic physics which involve highly relativistic and strong binding configurations.

scholarly journals Leaky Modes of Waveguides as a Classical Optics Analogy of Quantum Resonances

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Sara Cruz y Cruz ◽  
Oscar Rosas-Ortiz
Keyword(s):  
Quantum Mechanics ◽  
Short Range ◽  
Bound States ◽  
Waveguide Structure ◽  
One Dimensional ◽  
Leaky Modes ◽  
Scattering States ◽  
Quantum Resonances ◽  
Zone Of Influence ◽  
Square Well

A classical optics waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics. The analogy is based on the well-known resemblance between the guided and radiation modes of a waveguide with the bound and scattering states of a quantum well. As resonances are scattering states that spend some time in the zone of influence of the scatterer, we associate them with the leaky modes of a waveguide, the latter characterized by suffering attenuation in the direction of propagation but increasing exponentially in the transverse directions. The resemblance is complete because resonances (leaky modes) can be interpreted as bound states (guided modes) with definite lifetime (longitudinal shift). As an immediate application we calculate the leaky modes (resonances) associated with a dielectric homogeneous slab (square well potential) and show that these modes are attenuated as they propagate.

scholarly journals Scattering and Bound States of Duffin-Kemmer-Petiau Particles forq-Parameter Hyperbolic Pöschl-Teller Potential

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Hilmi Yanar ◽  
Ali Havare ◽  
Kenan Sogut
Keyword(s):  
Bound States ◽  
Scalar Potential ◽  
Spatial Dimension ◽  
Shape Parameters ◽  
Energy Eigenvalues ◽  
Vector Potentials ◽  
Scattering States ◽  
Potential Shape ◽  
Probability Densities ◽  
Interaction Approach

The Duffin-Kemmer-Petiau (DKP) equation in the presence of a scalar potential is solved in one spatial dimension for the vectorq-parameter Hyperbolic Pöschl-Teller (qHPT) potential. In obtaining complete solutions we used the weak interaction approach and took the scalar and vector potentials in a correlated form. By looking at the asymptotic behaviors of the solutions, we identify the bound and scattering states. We calculate transmission (T) and reflection (R) probability densities and analyze their dependence on the potential shape parameters. Also we investigate the dependence of energy eigenvalues of the bound states on the potential shape parameters.

Sign in / Sign up

Export Citation Format

Share Document