Abstract
The implementation of Lévy path integral generated by Lévy stochastic process on fractional Schrödinger equation has been investigated in the framework of fractional quantum mechanics. As the comparison, the implementation of Feynmann path integral generated by Wiener stochastic process on Schrödinger equation also has been investigated in the framework of standard quantum mechanics. There are two stochastic processes. There are Lévy stochastic and Wiener stochastic process. Both of them are able to produce fractal. In fractal’s concept, there is a value known as fractal dimension. The implementation of fractal dimension is the diffusion equation obtained by using Fokker Planck equation. In this paper, Lévy and Wiener fractal dimension have been obtained. There are for Lévy and 2 for Wiener/Brown fractal dimension. Fractional quantum mechanics is generalization of standard quantum mechanics. A fractional quantum mechanics state is represented by wave function from fractional Schrödinger equation. Fractional Schrödinger equation is obtained by using kernel of Lévy path integral generated by Lévy stochastic process. Otherwise, standard quantum mechanics state is represented by wave function from standard Schrödinger equation. Standard Schrödinger equation is obtained by using kernel of Feynmann path integral generated by Wiener/Brown stochastic process. Both Lévy and Feynmann Kernel have been investigated and the outputs are the Fourier Integral momentum phase of those kernels. We find that the forms of those kernels have similiraty. Therefore, we obtain Schrödinger equation from Lévy and Feynmann Kernel and also the comparison of Lévy energy in fractional quantum mechanics and particle energy in standard quantum mechanics.
Highlights
The implementation of Lévy path integral generated by Lévy stochastic process on fractional Schrödinger equation has been investigated in the framework of fractional quantum mechanics
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The Quantum Mechanics Based on a General Kinetic Energy, Int. J
Summary
Metode penelitian yang digunakan adalah studi pustaka yakni dengan meninjau secara teoritis terkait mekanika kuantum fraksional yang didekati dengan penerapan integral lintasan Lévy dan mengaitkannya dengan mekanika kuantum standar yang selama ini telah diketahui (Laskin 2000). 1. Kajian tentang mekanika kuantum secara umum serta urgensinya dalam menghasilkan persamaan. 2. Mengkaji lebih dalam tentang proses stokastik Lévy yang menjadi pondasi dasar dalam memperoleh integral lintasan Lévy. 3. Mengkaji keterkaitan antara integral lintasan Lévy dan integral lintasan Feynman dalam keberlakuannya pada mekanika kuantum. 4. Merumuskan bentuk umum persamaan Schrödinger yaitu persamaan Schrödinger fraksional dalam mekanika kuantum fraksional. 5. Membuktikan bahwa mekanika kuantum fraksional merupakan bentuk umum dari mekanika kuantum standar dengan membandingkan persamaan Schrödinger standar dan persamaan Schrödinger fraksional
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