TEKNIK PEMISAHAN OPERATOR DAN PENDEKATAN SPEKTRAL SEBAGAI SOLUSI PERSAMAAN SCHRÖDINGER BERGANTUNG WAKTU PADA ATOM HIDROGEN
DOI:
https://doi.org/10.25077/jfu.4.3.%25p.2015Abstract
ABSTRAK
Telah  dikembangkan sebuah metode numerik dengan teknik pemisahan operator dan pendekatan oleh fungsi basis spektral (Sorevik dkk., 2009) untuk menyelesaikan persamaan Schrödinger bergantung waktu pada koordinat bola. Penelitian Sorevik dkk. mendefinisikan fungsi gelombang tereduksi, teknik splitting Strang, metode kolokasi Chebyshev, variabel transformasi di arah radial, rumus umum metode swadekomposisi pada swanilai di arah radial, swanilai fungsi basis harmonik bola, dan teknik memulihkan fungsi basis Chebyshev. Sedangkan, penelitian ini memberikan analisis fisis dan matematis untuk menjabarkan metode ini disertai penerapannya pada sistem atom hidrogen. Hasil penelitian ini melengkapi penelitian Sorevik dkk. dalam memberikan seluruh tafsiran dan penyusunan persamaan numerik meliputi nilai inisialisasi swafungsi, diskritisasi fungsi Chebyshev dan polinomial Legendre terasosiasi yang dinormalisasi, ortogonalitas koefisien fungsi basis Chebyshev dan harmonik bola, transformasi Householder dan Metode QR untuk menghitung swanilai operator di arah radial, dan propagasi waktu imajiner.
Kata kunci : persamaan Schrödinger bergantung waktu, model atom kuantum, teknik pemisahan operator, pendekatan spektral, atom hidrogen
Abstract
It has been developed a numerical method with operator-splitting technique and spectral approximation by spectral basis functions (Sorevik et al., 2009) to solve the time-dependent Schrödinger equation on spherical coordinates. Sorevik et al.’s research defines the reduced wave function, Strang splitting technique, Chebyshev collocation method, variable transformation in the radial direction, the general formula of eigen decomposition method on eigen values in the radial direction, eigen values of spherical harmonic basis function, and a technique to restore the Chebyshev basis function. While this study provides physical and mathematical analysis to describe this method with its application in hydrogen atom system. This study complements the results of research Sorevik et al in giving the whole interpretation and compilation of numerical equations include initialization value of eigen function, discretization Chebyshev function and normalized associated Legendre polynomial, orthogonality coefficient of Chebyshev and spherical harmonic basis functions, Householder transformation and QR method for calculating eigen values of operator in the radial direction, and the propagation of imaginary time.
Keywords : the time dependent Schrödinger equation, quantum atomic model, splitting-operator technique, spectral approximation, hydrogen atom
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