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INDONESIAN JOURNAL OF APPLIED PHYSICS

Analisis Energi Osilator Harmonik Menggunakan Metode Path Integral Hypergeometry dan Operator

Fuzi Marati Sholihah, Suparmi Suparmi, Viska Inda Variani

Published May 22, 2016• 2 citations
Publisher

Abstract

Solution of the harmonic oscillator equation has a goal to get the energy levels of particles moving harmonic. The energy spectrums of one dimensional harmonic oscillator are analyzed by 3 methods: path integral, hypergeometry and operator. Analysis of the energy spectrum by path integral method is examined with Schrodinger equation. Analysis of the energy spectrum by operator method is examined by Hamiltonian in operator. Analysis of harmonic oscillator energy by 3 methods: path integral, hypergeometry and operator are getting same results 𝐸 = ℏ𝜔 (𝑛 + 1 2)

DOI
10.13057/ijap.v2i02.1280

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The harmonic oscillator has been described numerically using Delphi 7.0 programming language. It’s wave function and probabilistic density expressedby hermitte polynomial are visualized graphically for n = 0 until n = 10. The graph can be used to describethe behaviorof particles that moves under the influence of periodic force (oscillation). In addition tothesolution using polynomial hermitte, the wave function also can be derived using operator method (algebraic) which is expressed asfirst order differential expression and solved using Maple 9.5 programming. Keywords: harmonic oscillator, operator method, hermitte polynomial

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Solution of the harmonic oscillator equation has a goal to get the energy levels of particles moving harmonic. The energy spectrums of one dimensional harmonic oscillator are analyzed by 3 methods: path integral, hypergeometry and operator. Analysis of the energy spectrum by path integral method is examined with Schrodinger equation. Analysis of the energy spectrum by operator method is examined by Hamiltonian in operator. Analysis of harmonic oscillator energy by 3 methods: path integral, hypergeometry and operator are getting same results 𝐸 = ℏ𝜔 (𝑛 + 1 2)
The path integral is a method that often used in the quantum problems calculation. For example; the calculation of quantum system energy that has complex potential form. The method gives more easily than perturbation method. The method is also used to derive Green function, which usually used Fourier transformation. The Green function has widely application in quantum physics, since it used to compute solution of inhomogen differential equation as Schrodinger equation. In the particle physics, the Green function used as propagator in Feynman’s diagram. Considering the importance of Green function, and the powerfull of path integral method, in the paper, the method used to derive the formula of Green function for quantum harmonic oscillator system. The system has widely application to give more information of physical phenomena, for example, the atomic vibration in solid state. The result was also compared with Fourier transformation method and both give the same result as hoped.
Harmonic oscillator is a part of quantum mechanics which is commonly used as an approximation model to solve cases in physics that are considered complicated. The harmonic oscillator has a function as an energy source, namely as a description of anatomic particles. This research focuses on solving the Schrödinger equation for particles under the influence of a harmonic oscillator potential. This study aims to visualize the wave function and particle probability under the influence of a harmonic oscillator potential. The wave function is obtained by solving the Schrödinger equation. The Schrödinger equation can be solved using a second order differential equation that satisfies the solution of the hermite polynomial. The nature of the first hermite polynomial is the relationship between polynomials that have adjacent orders, the second is the derivative of hermite polynomials and the third is the orthogonality of hermite polynomials. The solution to the Schrödinger equation is in the form of a wave function which is then developed to determine the probability of particles in a harmonic oscillator which is then visualized using Matlab programming. The wave function and particle probability visualization is obtained by plotting the wave function and probability using Matlab programming. Osilator harmonik menjadi salah satu bagian dari mekanika kuantum yang biasa digunakan sebagai model pendekatan untuk menyelesaikan kasus-kasus dalam fisika yang dianggap rumit. Osilator harmonik memiliki fungsi dalam sumber energi yaitu sebagai pendeskripsi partikel anatomik. Penelitian ini memfokuskan bahasan mengenai penyelesaian persamaan Schrödinger untuk partikel dibawah pengaruh potensial osilator harmonik. Penelitian ini bertujuan memvisualisasikan fungsi gelombang dan probabilitas partikel dalam pengaruh potensial osilator harmonik. Fungsi gelombang diperoleh dengan penyelesaian persamaan Schrödinger. Persamaan Schrödinger tersebut dapat diselesaikan menggunakan penyelesaian persamaan diferensial orde 2 yang memenuhi solusi polinomial hermitte. Sifat dari polinom hermitte yang pertama adalah hubungan antar polinom yang memiliki orde berdekatan, kedua adalah derivatif polinom hermitte dan ketiga adalah keortogonalan polinom hermitte. Penyelesaian persamaan Schrödinger tersebut berupa fungsi gelombang yang kemudian dikembangkan untuk mengetahui probabilitas pada partikel dalam osilator harmonik yang kemudian dilakukan visualisasi menggunakan pemrograman Matlab. Visualisasi fungsi gelombang dan probabilitas partikel diperoleh dengan melakukan ploting terhadap fungsi gelombang dan probabilitas menggunakan pemrograman Matlab.
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
A simple harmonic oscillator, or oscillator in short, is one of the most important systems in physics as well as in quantum mechanics. The path integral and Hamiltonian of a simple harmonic oscillator are leading exemplars for the description of a wide class of physical systems. An oscillator is a theoretical model of great utility, primarily due to its simplicity that allows for the exact derivation of many results. The simple harmonic oscillator can be generalized to the case of infinitely many oscillators, and is also the starting point for analysis of many nonlinear quantum systems. Path integrals for the simple harmonic oscillator are all based on Gaussian integration, which has been discussed in Section 7.2. The harmonic oscillator is described by Gaussian path integrals , and is a bedrock of the general theory of path integration. Moreover, all perturbation expansions of nonlinear path integrals about the oscillator path integral, which includes the semi-classical expansion, are based on results of Gaussian integration. The properties of the harmonic oscillator are analyzed in the coordinate space representation, and its path integral as well as its correlation functions are analyzed. All formulas and derivations are for Euclidean time. In Sections 11.1 and 11.2 the Hamiltonian and state space of the oscillator are studied and the correlator is derived using state space methods. In Section 11.2 the infinite time oscillator is introduced and in Sections 11.3–11.5 the path integral for the oscillator is evaluated.
The harmonic oscillator has been described numerically using Delphi 7.0 programming language. It’s wave function and probabilistic density expressedby hermitte polynomial are visualized graphically for n = 0 until n = 10. The graph can be used to describethe behaviorof particles that moves under the influence of periodic force (oscillation). In addition tothesolution using polynomial hermitte, the wave function also can be derived using operator method (algebraic) which is expressed asfirst order differential expression and solved using Maple 9.5 programming. Keywords: harmonic oscillator, operator method, hermitte polynomial