Penyelesaian Persamaan Schrodinger Takgayut Waktu untuk Osilator Harmonik Menggunakan Metode Beda Hingga
Abstract
This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (?1), regression coefficient (?0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.
Abstrak
Penelitian ini menyelidiki penyelesaian numerik dari persamaan Schrodinger takgayut waktu untuk osilator harmonik kuantum dengan metode beda hingga. Osilator harmonik dinyatakan dengan potensial berbentuk fungsi kuadrat, adalah model dasar dalam mekanika kuantum karena aplikasinya yang luas mulai dari vibrasi molekul sampai teori medan kuantum. Persamaan Schrodinger bergantung waktu, adalah persamaan diferensial orde dua, yang biasanya menghadapi masalah jika diselesaikan secara analitik untuk potensial yang bentuknya komplek. Metode beda hingga menjadi pilihan yang menarik, mengubah persamaan differensial kontinyu menjadi sebuah sistem persamaan linear yang dapat diselesaikan secara komputasi dengan kode program komputer. Dalam penelitian ini domain spasial dideskritkan dan turunan kedua dihitung dengan beda sentral, mentransformasi TISE ke dalam matrik tridiagonal yang merepresentasikan Hamiltonian dari sistem. Dengan mencari penyelesaian dari masalah matrik eigenvalue ini, diperoleh fungsi gelombang dan energi. Hasil penyelidikan menunjukkan metode beda hingga mampu menyelesaikan TISE untuk osilator harmonik dengan efektif. Hasil yang diperoleh metode beda hingga sangat mendekati hasil analitik. Dari hasil regresi linear memperlihatkan secara berurutan nilai gradien (m), koefisien regresi (c) dan koefisien determinasi (R2) mendekati nilai ideal yaitu: 1, 0 dan 1. Hasil uji z juga memperlihatkan nilai z hitung < z kritis, mengindikasikan hasil perhitungan fungsi gelombang dan rapat kebolehjadian baik dengan metode beda hingga maupun analitik adalah sama dengan tingkat kepercayaan 95 persen.
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References
[2] Boas, Mary L., 2006, Mathematical Methods in the Physical Sciences, John Willey and Sons, Inc., USA.
[3] Izaac, J., Wang, J, 2018, Computational Quantum Mechanics, Springer Nature, Switzerland.
[4] Bramer, M.,2020, Principles of Data Mining, Springer-Verlag Ltd, London
[5] Ott, R.L., Longnecker, M., 2016, An Introduction to Statistical Methods and Data Analysis, Sixth edition, Cengage Learning, Boston, USA.
[6] Weinberg, S., 2021, Foundations of Modern Physics, Cambridge University Press, New York, USA.
[7] Beiser, Arthur, 2003, Concept of Modern Physics, sixth edition, McGraw-Hill Company, New York, USA.
[8] Griffiths, David J., Schroeter, Darrell F., 2018, Introduction to Quantum Mechanics, Cambridge University Press, New York, USA.
[9] Krane, Kenneth S., 2020, Modern Physics, fourth edition, John Willey and Sons, Inc., New Jersey, USA.
[10] Burden, Richard L., Faires J.D, Burden A.M, Numerical Analysis, Cengage Learning, Boston, USA.
[11] Pine, David J., 2019, Introduction to Python for Science and Engineering, CRC Press, New York, USA.
[12] Widagda, IGA, Trisnawati, NLP, Suharta IWG, Perancangan Aplikasi Metode Newton-Rapshon Termodifikasi dalam Pembuktikan Panjang Gelombang Maksimum pada Hukum Pergeseran Wien, Buletin Fisika, vol.24, no.2, 2023
[13] Montgomery, Douglas C., Peck, Elizabeth A., Vining, G. Geoffrey, 2021, Introduction to Linear Regression Analysis, sixth edition, John Wiley & Sons, Inc., New Jersey, USA.
