scholarly journals Discretized representations of harmonic variables by bilateral Jacobi operators

2000 ◽  
Vol 4 (4) ◽  
pp. 297-308
Author(s):  
Andreas Ruffing
Keyword(s):  
Quantum Optics ◽  
Harmonic Oscillator ◽  
Sequence Space ◽  
Heisenberg Algebra ◽  
Harmonic Oscillators ◽  
Quantum Oscillator ◽  
Oscillator Algebra ◽  
Jacobi Operators ◽  
Quantum Properties ◽  
Weighted Sequence Space

Starting from a discrete Heisenberg algebra we solve several representation problems for a discretized quantum oscillator in a weighted sequence space. The Schrödinger operator for a discrete harmonic oscillator is derived. The representation problem for aq-oscillator algebra is studied in detail. The main result of the article is the fact that the energy representation for the discretized momentum operator can be interpreted as follows: It allows to calculate quantum properties of a large number of non-interacting harmonic oscillators at the same time. The results can be directly related to current research on squeezed laser states in quantum optics. They reveal and confirm the observation that discrete versions of continuum Schrodinger operators allow more structural freedom than their continuum analogs do.

Rosen-Chambers Variation Theory of Linearly-Damped Classic and Quantum Oscillator

2014 ◽  
Vol 4 (1) ◽  
pp. 404-426
Author(s):  
Vincze Gy. Szasz A.
Keyword(s):  
Harmonic Oscillator ◽  
Classical Mechanics ◽  
Universal Time ◽  
Variation Theory ◽  
Quantum Oscillator ◽  
Variation Principle ◽  
Poisson Brackets ◽  
Mathematical Meaning ◽  
Damped Harmonic Oscillator ◽  
Damped Oscillator

Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.

Harmonic Oscillators with Nonlinear Damping

2017 ◽  
Vol 27 (11) ◽  
pp. 1730037 ◽  
Author(s):  
J. C. Sprott ◽  
W. G. Hoover
Keyword(s):  
Dynamical Systems ◽  
Harmonic Oscillator ◽  
Strange Attractor ◽  
Nonlinear Damping ◽  
Harmonic Oscillators ◽  
Coexisting Attractors ◽  
Simple Harmonic Oscillator ◽  
Interesting Behavior

Dynamical systems with special properties are continually being proposed and studied. Many of these systems are variants of the simple harmonic oscillator with nonlinear damping. This paper characterizes these systems as a hierarchy of increasingly complicated equations with correspondingly interesting behavior, including coexisting attractors, chaos in the absence of equilibria, and strange attractor/repellor pairs.

scholarly journals Integrals of the motion and green functions for time-dependent mass harmonic oscillators

2018 ◽  
Vol 64 (1) ◽  
pp. 30
Author(s):  
Surarit Pepore
Keyword(s):  
Harmonic Oscillator ◽  
Harmonic Oscillators ◽  
Time Dependent ◽  
Green Functions ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Damped Harmonic Oscillator ◽  
System Trajectory ◽  
Dependent Mass ◽  
Motion Method

The application of the integrals of the motion of a quantum system in deriving Green function or propagator is established. The Greenfunction is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phasespace. The explicit expressions for the Green functions of the damped harmonic oscillator, the harmonic oscillator with strongly pulsatingmass, and the harmonic oscillator with mass growing with time are obtained in co-ordinate representations. The connection between theintegrals of the motion method and other method such as Feynman path integral and Schwinger method are also discussed.

Approach of the continuous q-Hermite polynomials to x-representation of q-oscillator algebra and its coherent states

2019 ◽  
Vol 17 (02) ◽  
pp. 2050021
Author(s):  
H. Fakhri ◽  
S. E. Mousavi Gharalari
Keyword(s):  
Harmonic Oscillator ◽  
Generating Function ◽  
Coherent States ◽  
Finite Interval ◽  
Hermite Polynomials ◽  
Oscillator Algebra ◽  
Text Representation ◽  
Ladder Operators ◽  
Recursion Relations ◽  
Wilson Operator

We use the recursion relations of the continuous [Formula: see text]-Hermite polynomials and obtain the [Formula: see text]-difference realizations of the ladder operators of a [Formula: see text]-oscillator algebra in terms of the Askey–Wilson operator. For [Formula: see text]-deformed coherent states associated with a disc in the radius [Formula: see text], we obtain a compact form in [Formula: see text]-representation by using the generating function of the continuous [Formula: see text]-Hermite polynomials, too. In this way, we obtain a [Formula: see text]-difference realization for the [Formula: see text]-oscillator algebra in the finite interval [Formula: see text] as a [Formula: see text]-generalization of known differential formalism with respect to [Formula: see text] in the interval [Formula: see text] of the simple harmonic oscillator.

Quantum algebras and parity-dependent spectra

2007 ◽  
Vol 463 (2086) ◽  
pp. 2415-2427 ◽  
Author(s):  
C.V Sukumar ◽  
Andrew Hodges
Keyword(s):  
Quantum Optics ◽  
Harmonic Oscillator ◽  
Combinatorial Theory ◽  
Squeezed States ◽  
Quantum Algebras ◽  
Matrix Elements ◽  
Euler Numbers ◽  
Simple Harmonic Oscillator ◽  
Non Gaussian ◽  
Special Case

We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.

COHERENT STATES AND SCHWINGER MODELS FOR PSEUDO GENERALIZATION OF THE HEISENBERG ALGEBRA

2009 ◽  
Vol 24 (25) ◽  
pp. 2039-2051 ◽  
Author(s):  
H. FAKHRI ◽  
B. MOJAVERI ◽  
A. DEHGHANI
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Heisenberg Algebra ◽  
Creation And Annihilation Operators ◽  
Number Of Generators ◽  
Simple Harmonic Oscillator

We show that the non-Hermitian Hamiltonians of the simple harmonic oscillator with [Formula: see text] and [Formula: see text] symmetries involve a pseudo generalization of the Heisenberg algebra via two pairs of creation and annihilation operators which are [Formula: see text]-pseudo-Hermiticity and [Formula: see text]-anti-pseudo-Hermiticity of each other. The non-unitary Heisenberg algebra is represented by each of the pair of the operators in two different ways. Consequently, the coherent and the squeezed coherent states are calculated in two different approaches. Moreover, it is shown that the approach of Schwinger to construct the su(2), su(1, 1) and sp(4, ℝ) unitary algebras is promoted so that unitary algebras with more linearly dependent number of generators are made.

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