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.

The S-Matrix and Its Analysis

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Acoustic Waves ◽  
Bound States ◽  
Scattering Matrix ◽  
Infinite Dimensional ◽  
Levinson’S Theorem ◽  
S Matrix ◽  
Radial Schrödinger Equation ◽  
Square Well ◽  
Watson Transform ◽  
Remote Past

This chapter focuses on the scattering matrix, or S-matrix, an infinite-dimensional matrix or operator that expresses the state of a scattering system consisting of waves or particles or both in the far future in terms of its state in the remote past. In the case of electromagnetic (or acoustic) waves, the S-matrix connects the intensity, phase, and polarization of the outgoing waves in the far field at various angles to the direction and polarization of the beam pointed toward an obstacle. The chapter first considers the problem of scattering by a square well, located symmetrically with respect to the origin, before discussing bound states and a heuristic derivation of the Breit-Wigner formula. It als describes the Watson transform and Regge poles before concluding with an analysis of the time-independent radial Schrödinger equation and Levinson's theorem.

Subwavelength Defect Characterization Using Guided Wave Scattering Matrix

2013 ◽  
Vol 330 ◽  
pp. 504-509
Author(s):  
Yang Zheng ◽  
Jin Jie Zhou ◽  
Hui Zheng
Keyword(s):  
Wave Scattering ◽  
Scattering Matrix ◽  
Guided Wave ◽  
Defect Characterization ◽  
Challenging Problem ◽  
Recognition Method ◽  
Quantitative Characterization ◽  
S Matrix ◽  
Imaging Algorithms

Although many imaging algorithms such as ellipse and hyperbola algorithm can roughly locate defects in large plate-like structures with sparse guided wave arrays, quantitative characterization of them is still a challenging problem, especially for those small defects known as subwavelength defects. Scattering signals of defects contain abundant information so that can be used to evaluate defects. A defects recognition method using the S-matrix (scattering matrix) was presented. S-matrices of hole and crack with S0 mode incident were experimentally measured. The results show that defects can be recognized from the morphology of 2D S-matrix chart. This method has great potential to achieve more specific parameters of small defects with sparse guided wave arrays.

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.

One-Dimensional Jost Solutions: The S-Matrix Revisited

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Boundary Conditions ◽  
Transmission Coefficient ◽  
Radial Equation ◽  
Scattering Problems ◽  
One Dimensional ◽  
S Matrix ◽  
Jost Functions ◽  
Jost Solutions ◽  
Square Well ◽  
Matrix Problems

This chapter examines the properties of one-dimensional Jost solutions for S-matrix problems. It first considers how the left–right transmission and reflections coefficients can be expressed in terms of the elements of the S-matrix for one-dimensional scattering problems on, focusing on poles of the transmission coefficient. It then uses the radial equation to revisit the problem of the square-well potential from the perspective of the Jost solution, with Jost boundary conditions at r = 0 and as r approaches infinity. It also presents the notations for the Jost functions and the S-matrix before discussing the problem of scattering from a constant spherical inhomogeneity.

The scattering matrix and the Green functions

2021 ◽  
pp. 99-116
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Scattering Matrix ◽  
Quantum Scattering ◽  
Green Functions ◽  
Particle Interactions ◽  
Quantum Fields ◽  
Spinor Fields ◽  
Quantized Fields

In this chapter, the book begins to develop a perturbative formalism to describe the interactions of quantized fields and, in particular, the interactions of particles in terms of their quantum fields. Quantum scattering requires the description of particle interactions and asymptotic states, which are introduced in detail. The n-point Green functions are defined. An essential part of the chapter is devoted to deriving the reduction of the S-matrix in terms of the Green functions. The compact description of these notions is achieved by introducing the generating functionals of Green functions and the S-matrix. The same constructions are also introduced for the spinor fields.

EXACT SOLUTION OF THE CONTINUOUS STATES FOR GENERALIZED ASYMMETRICAL HARTMANN POTENTIALS UNDER THE CONDITION OF PSEUDOSPIN SYMMETRY

2007 ◽  
Vol 22 (26) ◽  
pp. 4825-4832 ◽  
Author(s):  
JIAN-YOU GUO ◽  
FANG ZHOU ◽  
FENG-LIANG GUO ◽  
JIAN-HONG ZHOU
Keyword(s):  
Exact Solution ◽  
Bound States ◽  
Continuation Method ◽  
Radial Wave Function ◽  
Pseudospin Symmetry ◽  
Radial Wave ◽  
Confluent Hypergeometric Functions ◽  
Momentum Plane ◽  
Scattering Phase ◽  
Hartmann Potential

Under the condition of pseudospin symmetry, the exact solution of Dirac equation is studied and that no bound solutions are observed for generalized asymmetrical Hartmann potential, which is in agreement with that for Coulomb potential. With the analytic continuation method, the unbound solutions are presented by mapping the wave functions of bound states in the complex momentum plane. Furthermore, the scattering phase shifts are obtained from the radial wave function by analyzing the asymptotic behavior of the confluent hypergeometric functions.

scholarly journals BOUND STATE BOUNDARY S MATRIX OF THE SINE-GORDON MODEL

1994 ◽  
Vol 09 (27) ◽  
pp. 4801-4810 ◽  
Author(s):  
SUBIR GHOSHAL
Keyword(s):  
Field Theory ◽  
Bound States ◽  
Bound State ◽  
Two Dimensional ◽  
Integrable Field Theory ◽  
S Matrix ◽  
State Boundary ◽  
Sine Gordon Model ◽  
Integrable Field

We study the boundary S matrix for the reflection of bound states of the two-dimensional sine-Gordon integrable field theory in the presence of a boundary.

A new variational expression for the scattering matrix

1988 ◽  
Vol 53 (9) ◽  
pp. 1873-1880 ◽  
Author(s):  
William H. Miller
Keyword(s):  
Variational Method ◽  
Scattering Matrix ◽  
Matrix Version ◽  
S Matrix ◽  
Concise Expression ◽  
Variational Expression

The S-matrix version of the Kohn variational method is used to obtain a new, more concise expression for the scattering matrix, one that has both esthetic and practical advantages over earlier ones that have been used.

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