TIME-DEPENDENT SCHRODINGER-EQUATIONS WITH VECTOR POTENTIALS - NUMERICAL-CALCULATIONS AND VISUALIZATIONS

We study algorithm to calculate the time dependent Schrodinger equatio n numerically and to visualize the results. We extend the product form ula by the Trotter-Suzuki decomposition to cases of Hamiltonians with vector potentials. This method is unconditionally stable and it can be used for a time dependent Hamiltonian so that one can apply it to var ious problems without any difficulty. After examining numerical errors for time evolutions of one dimensional wave packets, we present the e rrors under a constant magnetic field. Also we discuss a cyclotron mot ion in quantum mechanics, where an advantage for the unconditional sta bility is demonstrated. In order to visualize the results we make anim ations of the time evolutions, where the phase of the wave function is represented by colors and its absolute value is shown by brightness. Here we suggest a new way to represent a complex number by colors.