BOUND STATES FOR SCHRÖDINGER HAMILTONIANS: PHASE SPACE METHODS AND APPLICATIONS

1996 ◽  
Vol 08 (04) ◽  
pp. 503-547 ◽  
Author(s):  
PH. BLANCHARD ◽  
J. STUBBE
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Dynamical Systems ◽  
Bound States ◽  
Energy Levels ◽  
Schrödinger Operators ◽  
Particle Systems ◽  
Existence And Nonexistence ◽  
Phase Space Methods ◽  
Special Case

Properties of bound states for Schrödinger operators are reviewed. These include: bounds on the number of bound states and on the moments of the energy levels, existence and nonexistence of bound states, phase space bounds and semi-classical results, the special case of central potentials, and applications of these bounds in quantum mechanics of many particle systems and dynamical systems. For the phase space bounds relevant to these applications we improve the explicit constants.

Invariance Groups in Classical and Quantum Mechanics

1973 ◽  
Vol 28 (3-4) ◽  
pp. 538-540 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Energy Levels ◽  
Classical Mechanics ◽  
Symmetry Groups ◽  
Casimir Invariants ◽  
Classical Phase Space ◽  
Invariance Groups ◽  
Classical And Quantum Mechanics

AbstractThis is a report on some new relations and analogies between classical mechanics and quantum mechanics which arise out of the work of Kostant and Souriau. Topics treated are i) the role of symmetry groups; ii) the notion of elementary system and the role of Casimir invariants; iii) energy levels; iv) quantisation in terms of geometric data on the classical phase space. Some applications are described.

scholarly journals Discrete spectrum of many body Schrödinger operators with non-constant magnetic fields II

1997 ◽  
Vol 145 ◽  
pp. 69-98
Author(s):  
Tetsuya Hattori
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Discrete Spectrum ◽  
Bound States ◽  
Atomic System ◽  
Energy Levels ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Many Body ◽  
Discrete Energy

This paper is continuation from [10], in which we studied the discrete spectrum of atomic Hamiltonians with non-constant magnetic fields and, more precisely, we showed that any atomic system has only finitely many bound states, corresponding to the discrete energy levels, in a suitable magnetic field. In this paper we show another phenomenon in non-constant magnetic fields that any atomic system has infinitely many bound states in a suitable magnetic field.

SUPERSYMMETRIC QUANTUM MECHANICS APPLIED TO NONRELATIVISTIC QUARK MODELS

1993 ◽  
Vol 08 (08) ◽  
pp. 1437-1455 ◽  
Author(s):  
E.J.O. GAVIN ◽  
H. FIEDELDEY ◽  
H. LEEB ◽  
S.A. SOFIANOS
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Approximation Scheme ◽  
Energy Levels ◽  
Supersymmetric Quantum Mechanics ◽  
S Wave ◽  
Nucleon Spectrum ◽  
Quark Potential ◽  
Quark Models ◽  
Original Potential

We examine the effect of changing the energy levels and normalization constants of bound states corresponding to baryons and mesons in nonrelativistic quark models. We do this by applying the transformations of supersymmetric quantum mechanics (SUSYQM) to the potentials used in these models. In particular, we fit the spectra and leptonic decay widths of [Formula: see text] and [Formula: see text] mesons by modifying several existing [Formula: see text] potentials by means of supersymmetric transformations. It is found that the potentials are unchanged beyond 2 fm, and that fitting the widths induces greater oscillations in the potentials than those generated by adjusting the energy levels only. Transformations of SUSYQM are applied to the hypercentral potential in order to accommodate the Roper resonance in the s-wave nucleon spectrum. The quark-quark potential found by inverting the transformed hypercentral potential via a new exact Abel transform differs significantly from the original potential up to 5 fm from the origin and violates the concavity requirement. The [Formula: see text] potential related to this potential by Lipkin’s rule does not reproduce the meson spectrum. As the Hall-Post lower bound is also accurate for baryons, the results of the application of supersymmetric transformations in this approximation scheme are also considered and compared to the upper bound of the hypercentral approximation.

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.

scholarly journals Modes and Noise Propagation in Phase Space

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
J. S. Ben-Benjamin ◽  
L. Cohen
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Dispersion Relation ◽  
Wigner Distribution ◽  
Hermitian Operator ◽  
Type Equation ◽  
Hamiltonian Operator ◽  
Dispersive Media ◽  
Phase Space Methods ◽  
Complex Dispersion

AbstractWe show that phase space methods developed for quantum mechanics, such as the Wigner distribution, can be effectively used to study the evolution of nonstationary noise in dispersive media. We formulate the issue in terms of modes and show how modes evolve and how they are effected by sources.We show that each mode satisfies a Schrödinger type equation where the “Hamiltonian” may not be Hermitian. The Hamiltonian operator corresponds to dispersion relationwhere thewavenumber is replaced by the wavenumber operator. A complex dispersion relation corresponds to a non Hermitian operator and indicates that we have attenuation. A number of examples are given.

scholarly journals INVERSE SQUARE PROBLEM AND so(2, 1) SYMMETRY IN NONCOMMUTATIVE SPACE

2009 ◽  
Vol 24 (14) ◽  
pp. 2655-2663 ◽  
Author(s):  
PULAK RANJAN GIRI
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Energy Levels ◽  
Noncommutative Space ◽  
Noncommutative Phase Space

We study the quantum mechanics of a system with inverse square potential in noncommutative space. Both the coordinates and momenta are considered to be noncommutative, which breaks the original so(2, 1) symmetry. The energy levels and eigenfunctions are obtained. The generators of the so(2, 1) algebra are also studied in noncommutative phase space and the commutators are calculated, which shows that the commutators obtained in noncommutative space is not closed. However the commutative limit Θ, [Formula: see text] for the commutators smoothly go to the standard so(2, 1) algebra.

The Agmon spectral function for molecular hamiltonians with symmetry restrictions

1993 ◽  
Vol 440 (1910) ◽  
pp. 621-638 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Spectral Function ◽  
Schrödinger Operator ◽  
Spectral Properties ◽  
Bound States ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Polyatomic Systems

In this paper we introduce symmetry considerations into our earlier work, which was concerned with geometric spectral properties of Schrödinger operators including the N -body operators of quantum mechanics. The point of emphasis is a function introduced by Shmuel Agmon which we have named the Agmon spectral function. We show that this function is symmetric for an N -body Schrödinger operator restricted to a subspace of prescribed symmetry. We then show how it can be used to obtain criteria for the finiteness and infiniteness of bound states of polyatomic systems.

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