TWO-VARIABLE HERMITE POLYNOMIALS AS TIME-EVOLUTIONAL TRANSITION AMPLITUDE FOR DRIVEN HARMONIC OSCILLATOR

For the two-variable Hermite polynomials Hm,n(β,β*) we find its new physical explanation in the dynamics of a linear forced quantum harmonic oscillator (or a dispaced oscillator), i.e. Hm,n(β,β*) can be explained as the time-evolutional transition amplitude from an initial number state |m〉 at t0 to a final state |n〉 at t of the dispaced oscillator. Two new properties of the time-evolutional operator for driven oscillator are revealed.

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Deformed u(2) Algebra as the Symmetry Algebra of the Planar Anisotropic Quantum Harmonic Oscillator with Rational Ratio of Frequencies

International Journal of Modern Physics A ◽

10.1142/s0217751x97001742 ◽

1997 ◽

Vol 12 (19) ◽

pp. 3335-3346 ◽

Cited By ~ 3

Author(s):

Dennis Bonatsos ◽

P. Kolokotronis ◽

D. Lenis ◽

C. Daskaloyannis

Keyword(s):

Harmonic Oscillator ◽

Hermite Polynomials ◽

Symmetry Algebra ◽

Quantum Harmonic Oscillator ◽

General Applicability ◽

Finite Dimensional Representation ◽

Finite Dimensional ◽

Superintegrable Systems ◽

Frequency Ratios ◽

Special Case

The symmetry algebra of the two-dimensional anisotropic quantum harmonic oscillator with rational ratio of frequencies is identified as a deformation of the u(2) algebra. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are determined using algebraic methods of general applicability to quantum superintegrable systems. For labelling the degenerate states an "angular momentum" operator is introduced, the eigenvalues of which are roots of appropriate generalized Hermite polynomials. The cases with frequency ratios 1:n correspond to generalized parafermionic oscillators, while in the special case with frequency ratio 2:1 the resulting algebra corresponds to the finite W algebra [Formula: see text].

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Feed-Forward Neural Networks Based on the Eigenstates of the Quantum Harmonic Oscillator

Journal of Advanced Computational Intelligence and Intelligent Informatics ◽

10.20965/jaciii.2006.p0567 ◽

2006 ◽

Vol 10 (4) ◽

pp. 567-577

Author(s):

Gerasimos Rigatos ◽

Keyword(s):

Neural Networks ◽

Harmonic Oscillator ◽

Hermite Polynomials ◽

Basis Functions ◽

System Modelling ◽

General Category ◽

Quantum Harmonic Oscillator ◽

Feed Forward ◽

Change Of Scale ◽

Feed Forward Neural Networks

The paper introduces feed-forward neural networks where the hidden units employ orthogonal Hermite polynomials for their activation functions. The proposed neural networks have some interesting properties: (i) the basis functions are invariant under the Fourier transform, subject only to a change of scale, (ii) the basis functions are the eigenstates of the quantum harmonic oscillator, and stem from the solution of Schrödinger’s diffusion equation. The proposed feed-forward neural networks belong to the general category of nonparametric estimators and can be used for function approximation, system modelling and image processing.

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Neural Structures Using the Eigenstates of a Quantum Harmonic Oscillator

Open Systems & Information Dynamics ◽

10.1007/s11080-006-7265-6 ◽

2006 ◽

Vol 13 (01) ◽

pp. 27-41 ◽

Cited By ~ 11

Author(s):

Gerasimos Rigatos ◽

Spyros Tzafestas

Keyword(s):

Neural Networks ◽

Harmonic Oscillator ◽

Hermite Polynomials ◽

Feedforward Neural Networks ◽

Basis Functions ◽

System Modelling ◽

Quantum Harmonic Oscillator ◽

Wave Nature ◽

Change Of Scale ◽

The Fourier Transform

The main result of the paper is the use of orthogonal Hermite polynomials as the basis functions of feedforward neural networks. The proposed neural networks have some interesting properties: (i) the basis functions are invariant under the Fourier transform, subject only to a change of scale, (ii) the basis functions are the eigenstates of the quantum harmonic oscillator, and stem from the solution of Schrödinger's diffusion equation. The proposed feed-forward neural networks demonstrate the particle-wave nature of information and can be used in nonparametric estimation. Possible applications of the proposed neural networks include function approximation, image processing and system modelling.

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