Neural Structures Using the Eigenstates of a Quantum Harmonic Oscillator

2006 ◽  
Vol 13 (01) ◽  
pp. 27-41 ◽  
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.

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

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.

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

2007 ◽  
Vol 21 (08) ◽  
pp. 475-480 ◽  
Author(s):  
HONG-YI FAN ◽  
TENG-FEI JIANG
Keyword(s):  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Transition Amplitude ◽  
Initial Number ◽  
Quantum Harmonic Oscillator ◽  
Physical Explanation ◽  
Final State ◽  
Number State

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.

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

1997 ◽  
Vol 12 (19) ◽  
pp. 3335-3346 ◽  
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].

Attractors and Energy Spectrum of Neural Structures Based on the Model of the Quantum Harmonic Oscillator

2009 ◽  
pp. 376-410
Author(s):  
G.G. Rigatos ◽  
S.G. Tzafestas
Keyword(s):  
Neural Networks ◽  
Quantum Mechanics ◽  
Brownian Motion ◽  
Harmonic Oscillator ◽  
Quantum Physics ◽  
Neural Model ◽  
Neural Computation ◽  
Quantum Harmonic Oscillator ◽  
Brain Functioning ◽  
Computing Machines

Neural computation based on principles of quantum mechanics can provide improved models of memory processes and brain functioning and is of primary importance for the realization of quantum computing machines. To this end, this chapter studies neural structures with weights that follow the model of the quantum harmonic oscillator. The proposed neural networks have stochastic weights which are calculated from the solution of Schrödingers equation under the assumption of a parabolic (harmonic) potential. These weights correspond to diffusing particles, which interact with each other as the theory of Brownian motion (Wiener process) predicts. The learning of the stochastic weights (convergence of the diffusing particles to an equilibrium) is analyzed. In the case of associative memories the proposed neural model results in an exponential increase of patterns storage capacity (number of attractors). It is also shown that conventional neural networks and learning algorithms based on error gradient can be conceived as a subset of the proposed quantum neural structures. Thus, the complementarity between classical and quantum physics is also validated in the field of neural computation.

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