REALIZATION OF DYNAMICAL GROUP FOR A SYMMETRIC WELL POTENTIAL tan2(πx/a) AND ITS CONTROLLABILITY

2006 ◽  
Vol 21 (28n29) ◽  
pp. 5833-5843
Author(s):  
SHI-HAI DONG ◽  
M. LOZADA-CASSOU ◽  
MARCO A. ARJONA L
Keyword(s):  
Exact Solutions ◽  
Quantum System ◽  
Ground State ◽  
Bound States ◽  
Factorization Method ◽  
Creation Operator ◽  
Ladder Operators ◽  
Dynamical Group ◽  
Commutation Relations ◽  
Square Well

The exact solutions of quantum system with a symmetric well potential V(x) = D tan 2(πx/a) are obtained. The ladder operators are constructed directly from the normalized eigenfunctions with the factorization method. It is shown that these ladder operators satisfy the commutation relations of the generators for an su(1, 1) algebra. The infinitely deep square well and harmonic limits of this potential are briefly studied. The controllability of this system is also investigated. It is demonstrated that this system with discrete bound states can be strongly completely controlled. This may be realized theoretically by acting the creation operator [Formula: see text] on the ground state.

EXACT SOLUTIONS, LADDER OPERATORS AND BARUT–GIRARDELLO COHERENT STATES FOR A HARMONIC OSCILLATOR PLUS AN INVERSE SQUARE POTENTIAL

2005 ◽  
Vol 19 (28) ◽  
pp. 4219-4227 ◽  
Author(s):  
SHI-HAI DONG ◽  
M. LOZADA-CASSOU
Keyword(s):  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Coherent States ◽  
Factorization Method ◽  
Ladder Operators ◽  
Dynamical Group ◽  
One Dimensional ◽  
Hidden Symmetry ◽  
The One ◽  
Analytical Expressions

We present exact solutions of the one-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square potential. The ladder operators are constructed by the factorization method. We find that these operators satisfy the commutation relations of the generators of the dynamical group SU(1, 1). Based on those ladder operators, we obtain the analytical expressions of matrix elements for some related functions ρ and [Formula: see text] with ρ=x2. Finally, we make some comments on the Barut–Girardello coherent states and the hidden symmetry between E(x) and E(ix) by substituting x→ix.

AN ALGEBRAIC APPROACH TO A HARMONIC OSCILLATOR PLUS AN INVERSE SQUARE POTENTIAL IN TWO DIMENSIONS

2005 ◽  
Vol 20 (24) ◽  
pp. 5663-5670 ◽  
Author(s):  
SHI-HAI DONG ◽  
GUO-HUA SUN ◽  
M. LOZADA-CASSOU
Keyword(s):  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Algebraic Approach ◽  
Two Dimensions ◽  
Wave Functions ◽  
Matrix Elements ◽  
Ladder Operators ◽  
Radial Wave ◽  
Commutation Relations ◽  
The Matrix

The exact solutions of the Schrödinger equation with a harmonic oscillator plus an inverse square potential are obtained in two dimensions. We construct the ladder operators directly from the radial wave functions and find that these operators satisfy the commutation relations of an SU (1, 1) group. We obtain the explicit expressions of the matrix elements for some related functions ρ and [Formula: see text] with ρ = r2. We also explore another symmetry between the eigenvalues E(r) and E(ir) by substituting r→ir.

scholarly journals Electromagnetic transition form factors of baryons in a relativistic Faddeev approach

2018 ◽  
Vol 181 ◽  
pp. 01013 ◽  
Author(s):  
Reinhard Alkofer ◽  
Christian S. Fischer ◽  
Hèlios Sanchis-Alepuz
Keyword(s):  
Ground State ◽  
Bound States ◽  
Body Wave ◽  
Form Factors ◽  
Wave Functions ◽  
Electromagnetic Form Factors ◽  
Transition Form ◽  
Electromagnetic Transition ◽  
Three Body ◽  
Gluon Interaction

The covariant Faddeev approach which describes baryons as relativistic three-quark bound states and is based on the Dyson-Schwinger and Bethe-Salpeter equations of QCD is briefly reviewed. All elements, including especially the baryons’ three-body-wave-functions, the quark propagators and the dressed quark-photon vertex, are calculated from a well-established approximation for the quark-gluon interaction. Selected previous results of this approach for the spectrum and elastic electromagnetic form factors of ground-state baryons and resonances are reported. The main focus of this talk is a presentation and discussion of results from a recent investigation of the electromagnetic transition form factors between ground-state octet and decuplet baryons as well as the octet-only Σ0 to Λ transition.

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.

scholarly journals BOUNDARY ONE-POINT FUNCTIONS, SCATTERING, AND BACKGROUND VACUUM SOLUTIONS IN TODA THEORIES

2003 ◽  
Vol 18 (06) ◽  
pp. 879-899 ◽  
Author(s):  
V. A. FATEEV ◽  
E. ONOFRI
Keyword(s):  
Exact Solutions ◽  
Ground State ◽  
S Matrix ◽  
Parametric Families ◽  
Vacuum Solutions ◽  
Ground State Energies

The parametric families of integrable boundary affine Toda theories are considered. We calculate boundary one-point functions and propose boundary S-matrices in these theories. We use boundary one-point functions and S-matrix amplitudes to derive boundary ground state energies and exact solutions describing classical vacuum configurations.

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 Spin(p + 1, p + 1) COVARIANT Dp-BRANE BOUND STATES

2001 ◽  
Vol 16 (17) ◽  
pp. 3025-3040 ◽  
Author(s):  
P. SUNDELL
Keyword(s):  
Field Theory ◽  
Ground State ◽  
Bound States ◽  
Type Ii ◽  
Duality Group ◽  
Zero Force ◽  
Force Condition ◽  
The Stability

We construct Spin (p + 1, p + 1) covariant D p-brane bound states by using the fact that the potentials in the RR sector of toroidically compactified type II supergravity transform as a chiral spinor of the T duality group. As an application, we show the invariance of the zero-force condition for a probe D-brane under noncommutative deformations of the background, which gives a holographic proof of the stability of the corresponding field theory ground state under noncommutative deformations. We also identify the Spin (p + 1, p + 1) transformation laws by examining the covariance of the D-brane Lagrangians.

scholarly journals Fundamental dissipation due to bound fermions in the zero-temperature limit

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
S. Autti ◽  
S. L. Ahlstrom ◽  
R. P. Haley ◽  
A. Jennings ◽  
G. R. Pickett ◽  
...  
Keyword(s):  
Critical Velocity ◽  
Ground State ◽  
Bound States ◽  
Bound State ◽  
Temperature Limit ◽  
Pair Breaking ◽  
Zero Temperature ◽  
Andreev Bound States ◽  
Macroscopic Object

Abstract The ground state of a fermionic condensate is well protected against perturbations in the presence of an isotropic gap. Regions of gap suppression, surfaces and vortex cores which host Andreev-bound states, seemingly lift that strict protection. Here we show that in superfluid 3He the role of bound states is more subtle: when a macroscopic object moves in the superfluid at velocities exceeding the Landau critical velocity, little to no bulk pair breaking takes place, while the damping observed originates from the bound states covering the moving object. We identify two separate timescales that govern the bound state dynamics, one of them much longer than theoretically anticipated, and show that the bound states do not interact with bulk excitations.

Exact solutions of a quartic potential

2019 ◽  
Vol 34 (26) ◽  
pp. 1950208 ◽  
Author(s):  
Qian Dong ◽  
Guo-Hua Sun ◽  
M. Avila Aoki ◽  
Chang-Yuan Chen ◽  
Shi-Hai Dong
Keyword(s):  
Exact Solutions ◽  
Quantum System ◽  
Analytical Solutions ◽  
Wave Functions ◽  
Negative Potential ◽  
Potential Parameter ◽  
Real Numbers ◽  
Quartic Potential ◽  
Heun Functions ◽  
Potential Parameters

We find that the analytical solutions to quantum system with a quartic potential [Formula: see text] (arbitrary [Formula: see text] and [Formula: see text] are real numbers) are given by the triconfluent Heun functions [Formula: see text]. The properties of the wave functions, which are strongly relevant for the potential parameters [Formula: see text] and [Formula: see text], are illustrated. It is shown that the wave functions are shrunk to the origin for a given [Formula: see text] when the potential parameter [Formula: see text] increases, while the wave peak of wave functions is concaved to the origin when the negative potential parameter [Formula: see text] increases or parameter [Formula: see text] decreases for a given negative potential parameter [Formula: see text]. The minimum value of the double well case ([Formula: see text]) is given by [Formula: see text] at [Formula: see text].

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