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Advances in intelligent systems and computing

Studying the Schrödinger Equation, Implementation of Qubits and Visualization of Matrices Using Matlab

Alexey Samoylov, Sergey M. Gushanskiy, Н М Коробейникова

Published Jan 1, 2019• 0 citations
Publisher
DOI
10.1007/978-3-030-19807-7_27

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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.