scholarly journals Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates

1997 ◽  
Vol 38 (10) ◽  
pp. 5031-5043 ◽  
Author(s):  
Jorge Sánchez-Ruiz
Keyword(s):  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Logarithmic Potential

The quantum relativistic harmonic oscillator: generalized Hermite polynomials

1991 ◽  
Vol 156 (7-8) ◽  
pp. 381-385 ◽  
Author(s):  
V. Aldaya ◽  
J. Bisquert ◽  
J. Navarro-Salas
Keyword(s):  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Relativistic Harmonic Oscillator

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.

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].

scholarly journals On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 88
Author(s):  
Elchin I. Jafarov ◽  
Aygun M. Mammadova ◽  
Joris Van der Jeugt
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Short Communication ◽  
Jacobi Polynomials ◽  
Direct Limit ◽  
Hermite Polynomials ◽  
Limit Relation ◽  
Exactly Solvable ◽  
Transformation Formulas ◽  
Oscillator Models

In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of this limit relation is based upon 2F1-type hypergeometric transformation formulas, which are applicable to even and odd polynomials separately. This limit opens the way to studying new exactly solvable harmonic oscillator models in quantum mechanics in terms of pseudo-Jacobi polynomials.

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.

scholarly journals Quantization of the one-dimensional free harmonic oscillator as an example of the application of the node theorem and the Macdonald-Hylleraas-Undheim theorem

2020 ◽  
Vol 22 (1) ◽  
pp. 87-90
Author(s):  
Kunle Adegoke ◽  
A. Olatinwo
Keyword(s):  
Differential Equation ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Hermite Polynomials ◽  
Traditional Methods ◽  
One Dimensional ◽  
Energy Eigenvalues ◽  
The One

Using heuristic arguments alone, based on the properties of the  wavefunctions, the energy eigenvalues and the corresponding eigenfunctions of the one-dimensional harmonic oscillator are obtained. This approach is considerably simpler and is perhaps more intuitive than the traditional methods of solving a differential equation and manipulating operators. Keywords: Time-independent Schrödinger equation, MacDonald-Hylleraas-Undheim theorem, Node theorem, Hermite polynomials,  energy eigenvalues

THE q-SCHRÖDINGER EQUATION

1990 ◽  
Vol 05 (31) ◽  
pp. 2625-2632 ◽  
Author(s):  
JOSEPH A. MINAHAN
Keyword(s):  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Hermite Polynomials ◽  
Normal Derivative ◽  
One Dimensional ◽  
The One

We consider the Schrödinger equation for the one-dimensional harmonic oscillator, but with the normal derivative replaced by a q-derivative. The normalizable solutions are found and the q-generalization of the Hermite polynomials is given. The free equation is also considered, but no normalizable eigenstates exist even if the system is in a box.

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.

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