A non-Gaussian single-mode squeezed state of the simple harmonic oscillator

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.

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Harmonic Oscillators with Nonlinear Damping

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417300373 ◽

2017 ◽

Vol 27 (11) ◽

pp. 1730037 ◽

Cited By ~ 8

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.

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Propagator for the simple harmonic oscillator

American Journal of Physics ◽

10.1119/1.19003 ◽

1998 ◽

Vol 66 (11) ◽

pp. 1022-1024 ◽

Cited By ~ 14

Author(s):

Nora S. Thornber ◽

Edwin F. Taylor

Keyword(s):

Harmonic Oscillator ◽

Simple Harmonic Oscillator

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(ANTI)COMMUTATIVITY OF THE HARMONIC CREATION AND ANNIHILATION OPERATORS

International Journal of Modern Physics B ◽

10.1142/s0217979206034327 ◽

2006 ◽

Vol 20 (11n13) ◽

pp. 1808-1818

Author(s):

S. KUWATA ◽

A. MARUMOTO

Keyword(s):

Irreducible Representation ◽

Harmonic Oscillator ◽

Linear Algebra ◽

Commutation Relation ◽

Complex Number ◽

Single Mode ◽

Equation Of Motion ◽

Sufficient Condition ◽

Special Linear ◽

Creation And Annihilation Operators

It is known that para-particles, together with fermions and bosons, of a single mode can be described as an irreducible representation of the Lie (super) algebra 𝔰𝔩2(ℂ) (2-dimensional special linear algebra over the complex number ℂ), that is, they satisfy the equation of motion of a harmonic oscillator. Under the equation of motion of a harmonic oscillator, we obtain the set of the commutation relations which is isomorphic to the irreducible representation, to find that the equation of motion, conversely, can be derived from the commutation relation only for the case of either fermion or boson. If Nature admits of the existence of such a sufficient condition for the equation of motion of a harmonic oscillator, no para-particle can be allowed.

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