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Jurnal Matematika Sains Dan Teknologi

Solusi Persamaan SchröDinger Osilator Harmonik Dalam Ruang Momentum

Paken Pandiangan

Published Aug 15, 2005• 0 citations
OpenAlex

Abstract

The solution of Schrödinger equation for the simple harmonic oscillator has been investigated and explored. The solutions are coordinate representation or momentum representation. The eigen energy is . The expectation value is respectively. Both Method will produce similar result.

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Visualisasi osilator harmonik kuantum dengan polinomial hermitte menggunakan simulasi pemrograman matlab

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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.
Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick,the rigorous solutions of stationary Schro¨dinger equation for one-dimensional harmonic oscillator are solved with the methods of matrix mechanics and wave mechanics respectively.It is showed that the matrix mechanics and the wave mechanics are equivalent in phase space.The eigenfunctions in position and momentum spaces can be obtained through the Fourier-like projection transformation from the phase-space eigenfunction.
By the Fourier transform ation,we change the Schr dinger equation of one-dimensional linear harmonic oscillator to the first-order differential equation which can be easily solved.The progressional solution of the Schr dinger equation can be obtained by the inverse Fourier transform ation after we solve the first-order differential equation.Finally,we can fix the eigen-value and eigen-function of energy based on the boundary condition that wave function is zero when the variable becomes infinite.
The Schr\"odinger equation for the two-dimensional simple harmonic oscillator is solved using elliptic coordinates where it is separable. The separability of the problem in such coordinates is independent of the selection of the focal distance. The solutions are labeled by the total number of quanta N and by a set of characteristic values b corresponding to the eigenvalues of an observable B^, which does not commute with L^, the total angular momentum or H${^}_{x}$, the energy associated with the x degree of freedom. The well-known quantum energies as well as the characteristic values are obtained by imposing physical polynomial solutions.
The quantization of orbital angular momentum can be introduced simply through the model of the two- and three-dimensional simple harmonic oscillator. Most of the analysis is carried out in the familiar Cartesian coordinate system in which the simple harmonic oscillator Schrödinger equation is separable. Angular momentum eigenfunctions are constructed by superposition, and the resulting functions easily transformed into spherical coordinates. The angle-dependent parts of the solutions are the spherical harmonics, which are the orbital angular momentum eigenfunctions for all central potentials including the Coulomb potential of the hydrogen atom. The method also introduces the separation of variables technique and provides practice in applying the concepts of completeness, orthogonality, superposition of states, and alternative representations.