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International Conference on Engineering and Emerging Technologies (ICEET)

Calculating Energies of 1D Quantum Harmonic Oscillator with Artificial Neural Networks

Huber Nieto-Chaupis

Published Oct 22, 2025• 0 citations
Publisher

Abstract

We employ the simplest representation of perceptron and orthogonal polynomials, to derive the quantized energies of an one-dimension harmonic oscillator by using mathematical methodologies that are interpreted in terms of quantum language. In contrast to standard mechanisms based in the usage of annihilation and creation operators, the oscillator turns out to be quantized. Indeed, a generalization is presented, fact that might be interpreted as a path to quantize 2D and 3D oscillators. From the Schrödinger equation, the respective solution namely the wave function for present case is also estimated in a closed-form. In the present case, the well-known integer-order Bessel functions are employed. The resultant energy spectra are in agree with quantum mechanics calculations.

DOI
10.1109/iceet67911.2025.11424047

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We employ the simplest representation of perceptron and orthogonal polynomials, to derive the quantized energies of an one-dimension harmonic oscillator by using mathematical methodologies that are interpreted in terms of quantum language. In contrast to standard mechanisms based in the usage of annihilation and creation operators, the oscillator turns out to be quantized. Indeed, a generalization is presented, fact that might be interpreted as a path to quantize 2D and 3D oscillators. From the Schrödinger equation, the respective solution namely the wave function for present case is also estimated in a closed-form. In the present case, the well-known integer-order Bessel functions are employed. The resultant energy spectra are in agree with quantum mechanics calculations.
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Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
Abstract Liénard-type nonlinear one dimensional oscillator is quantized using van Roos symmetric ordering recipe for the kinetic-like part of the new derived Hamiltonian. The corresponding Schrödinger equation is exactly solved in momentum space via the approach of supersymmetric quantum mechanics (SUSYQM). The bound-states energy spectra and corresponding wave functions are given explicitly in terms of the ambiguity parameters. The limiting case of no deformation agrees exactly with the eigenenergies and eigenfunctions of the ordinary quantum harmonic oscillator.
One dimensional quantum harmonic oscillator is well studied in elementary textbooks of quantum mechanics. The wave function of one-dimensional oscillator harmonic can be written in term of Hermite polynomial. Due to the symmetry of the spring energy, the wave functions of two-dimensional and three-dimensional harmonic oscillators can be written as products of the one-dimensional case. Because of that, the wave functions of two- and three-dimensional cases are focused on cartesian coordinates. In this article, we utilize polar and spherical coordinates to describe the wave function of two- and three-dimensional harmonic oscillators, respectively. The radial part of the wave functions can be written in term of associated Laguerre polynomials.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.
The study of quantum properties of the harmonic oscillator is important in physics as many real-world systems oscillate harmonically. Motion of systems in a confined space is often modeled as being a quantized harmonic motion or, in first instance, is approximated by a harmonic motion. This chapter helps the reader to solve the equation employing an algebraic operator technique, which is based on the Dirac notation. This approach has several definite advantages and exploits the commutation relations among the operators involved and their properties. The chapter transforms the stationary Schrodinger equation into a second-order differential equation and finds the solution to the equation with the aid of a more advanced mathematical technique that involves special functions. The energy of a harmonic oscillator is quantized, with the sequence of values. The difference in energy between adjacent energy levels is equal to the energy of a single photon, ħω.