Representations of the n-dimensional rotation group

1961 ◽  
Vol 57 (3) ◽  
pp. 469-475 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Bound States ◽  
Differential Operators ◽  
Energy Levels ◽  
Radial Wave Function ◽  
Rotation Group ◽  
Wave Mechanics ◽  
Radial Wave ◽  
Shift Operators ◽  
Branching Rule ◽  
Infinitesimal Operators

ABSTRACTThe commutators of the infinitesimal operators of the n-dimensional rotation group Rn with vector operators under Rn are expressed in a vectorial notation. The infinitesimal operators for the representations (l 0…0) are treated in detail. Shift operators for l are constructed and are used to derive the branching rule for these representations. The energy levels and degeneracy of bound states of a particle under an inverse square force in n dimensions are found by wave mechanics and by expressing the Hamiltonian in terms of Casimir's operator for Rn+1. Differential operators which transform one radial wave function into another are obtained.

Analytical solution of the Shredinger equation for the linear combination of the Manning-Rosen and the class of Yukawa potential

2021 ◽  
pp. 157-169
Author(s):  
G.A. Bayramova ◽  
Keyword(s):  
Analytical Solution ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Yukawa Potential ◽  
Energy Levels ◽  
Energy Eigenvalue ◽  
Radial Wave ◽  
Rosen Potential ◽  
Analytical Expressions ◽  
Potential Parameters

In the present work, an analytical solution for bound states of the modified Schrödinger equation is found for the new supposed combined Manning-Rosen potential plus the Yukawa class. To overcome the difficulties arising in the case l ≠ 0 in the centrifugal part of the Manning-Rosen potential plus the Yukawa class for bound states, we applied the developed approximation. Analytical expressions for the energy eigenvalue and the corresponding radial wave functions for an arbitrary value l ≠ 0 of the orbital quantum number are obtained. And also obtained eigenfunctions expressed in terms of hypergeometric functions. It is shown that energy levels and eigenfunctions are very sensitive to the choice of potential parameters.

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.

DEFORMED SHELL CLOSURES FOR LIGHT ATOMIC CLUSTERS

2008 ◽  
Vol 22 (28) ◽  
pp. 4917-4935 ◽  
Author(s):  
R. A. GHERGHESCU ◽  
D. N. POENARU ◽  
A. SOLOVYOV ◽  
W. GREINER
Keyword(s):  
Input Data ◽  
Energy Levels ◽  
Radial Wave Function ◽  
Atomic Clusters ◽  
Shell Correction ◽  
Axially Symmetric ◽  
Cylindrical Coordinates ◽  
Radial Wave ◽  
Shell Effects ◽  
Shell Closures

The spheroidal shell model of the Nilsson type is used to describe the deformed states of atomic clusters. The Hamiltonian is analytically solved in cylindrical coordinates, where l2 term is treated as deformation-dependent. The usual asymptotic eigenfunctions are obtained for axially symmetric potentials without approximation, and the radial wave function usually employed for further computation is no longer needed. The energy levels obtained in such a way are used as input data for shell correction calculations. Minima due to shell effects are obtained as a function of the number of atoms in the atomic cluster as well as the δ-deformation-dependent. Calculations are performed for N up to 200, and spheroidally (oblate and prolate) deformed shell closures are predicted.

SUPERSYMMETRY APPROACHES TO THE BOUND STATES OF THE GENERALIZED WOODS–SAXON POTENTIAL

2004 ◽  
Vol 19 (08) ◽  
pp. 615-625 ◽  
Author(s):  
H. FAKHRI ◽  
J. SADEGHI
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Bound States ◽  
Differential Operators ◽  
Energy Levels ◽  
Negative Energy ◽  
Saxon Potential ◽  
First Order ◽  
Jacobi Differential Equation ◽  
Algebraic Solutions

Using the associated Jacobi differential equation, we obtain exactly bound states of the generalization of Woods–Saxon potential with the negative energy levels based on the analytic approach. According to the supersymmetry approaches in quantum mechanics, we show that these bound states by four pairs of the first-order differential operators, represent four types of the laddering equations. Two types of these supersymmetry structures, suggest the derivation of algebraic solutions by two different approaches for the bound states.

The relativistic bound and scattering states of the Manning-Rosen potential with an improved new approximate scheme to the centrifugal term

Open Physics ◽  
2009 ◽  
Vol 7 (1) ◽  
Author(s):  
Gao-Feng Wei ◽  
Zhi-Zhong Zhen ◽  
Shi-Hai Dong
Keyword(s):  
Bound States ◽  
Gordon Equation ◽  
Scattering Amplitude ◽  
Energy Levels ◽  
Centrifugal Term ◽  
Radial Wave ◽  
Scattering States ◽  
Scattering State ◽  
Klein Gordon ◽  
Klein Gordon Equation

AbstractThe approximately analytical bound and scattering state solutions of the arbitrary l-wave Klein-Gordon equation for the mixed Manning-Rosen potentials are carried out by an improved new approximation to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Manning-Rosen potentials are presented and the corresponding energy equations for bound states and phase shifts for scattering states are derived. It is shown that the energy levels of the continuum states, reduce to the bound states of those at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and the special case for wave is studied briefly.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

THE ANALOGUE OF THE AHARONOV–BOHM EFFECT FOR BOUND STATES FOR NEUTRAL PARTICLES

2011 ◽  
Vol 26 (18) ◽  
pp. 1331-1341 ◽  
Author(s):  
KNUT BAKKE ◽  
C. FURTADO
Keyword(s):  
Bound States ◽  
Neutral Particle ◽  
Energy Levels ◽  
Quantum Ring ◽  
Persistent Currents ◽  
One Dimensional ◽  
Neutral Particles ◽  
Quantum Flux ◽  
Bohm Effect ◽  
Aharonov Bohm

We study the analogue of the Aharonov–Bohm effect for bound states for a neutral particle with a permanent magnetic dipole moment interacting with an external field. We consider a neutral particle confined to moving between two coaxial cylinders and show the dependence of the energy levels on the Aharonov-Casher quantum flux. Moreover, we show that the same flux dependence of the bound states can be found when the neutral particle is confined to a one-dimensional quantum ring and a quantum dot, and we also calculate the persistent currents in each case.

Approximate analytical solution of continuous states for the l-wave Schrödinger equation with a diatomic molecule potential

Open Physics ◽  
2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Gao-Feng Wei ◽  
Wen-Chao Qiang ◽  
Wen-Li Chen
Keyword(s):  
Schrödinger Equation ◽  
Diatomic Molecule ◽  
Schrodinger Equation ◽  
Scattering Amplitude ◽  
Energy Levels ◽  
Bound State ◽  
Radial Wave ◽  
Continuous States ◽  
Analytical Properties ◽  
Function Potential

AbstractThe continuous states of the l-wave Schrödinger equation for the diatomic molecule represented by the hyperbolical function potential are carried out by a proper approximation scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Schrödinger equation for the hyperbolical function potential are presented and the corresponding calculation formula of phase shifts is derived. Also, we interestingly obtain the corresponding bound state energy levels by analyzing analytical properties of scattering amplitude.

1.37 - 2.90 Micron Intersubband Transitions in GaN/AlN Superlattices

2006 ◽  
Vol 955 ◽  
Author(s):  
Eric Anthony DeCuir ◽  
Emil Fred ◽  
Omar Manasreh ◽  
Jinqiao Xie ◽  
Hadis Morkoc ◽  
...  
Keyword(s):  
Quantum Wells ◽  
Bound States ◽  
Energy Levels ◽  
Three Dimensional ◽  
Blue Shift ◽  
Bound State ◽  
Peak Position ◽  
Multiple Quantum ◽  
Intersubband Transitions ◽  
Photovoltaic Mode

ABSTRACTIntersubband transitions in the spectral range of 1.37-2.90 °Cm is observed in molecular beam epitaxy grown Si-doped GaN/AlN multiple quantum wells using a Fourier-transform spectroscopy technique. A blue shift in the peak position of the intersubband transition is observed as the well width is decreased. A sample with a well width in the order of 2.4 nm exhibited the presence of three bound states in the GaN well. The bound state energy levels are calculated using a transfer matrix method. An electrochemical capacitance voltage technique is used to obtain the three dimensional carrier concentrations in these samples which further enable the calculation of the Fermi energy level position. Devices fabricated from these GaN/AlN quantum wells are found to operate in the photovoltaic mode.

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