Phase Space Approach to Solving the Time-independent and Time-dependent Schrödinger Equations: Toward Vibrational Calculations for Large Polyatomics and Multi-electron Dynamics
Asaf Shimshovitz, Norio Takemoto and David J. Tannor
Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100 Israel
We propose a method for solving both the time-independent and time-dependent Schrödinger equations based on the von Neumann (vN) lattice of phase space Gaussians. By incorporating periodic boundary conditions into the vN lattice [F. Dimler et al., New. J. Phys. 11, 105052 (2009)] we solve a longstanding problem of convergence of the vN method. This opens the door to tailoring quantum calculations to the underlying classical phase space structure while retaining the accuracy of the Fourier grid basis. The method has the potential to provide enormous numerical savings as the dimensionality increases. In the classical limit the method reaches the remarkable efficiency of 1 basis function per 1 eigenstate. We illustrate the time-independent method for a model of polyatomic vibrations with 104 bound states, and the time-dependent method by simulating the attosecond (10-18s) electron dynamics that follows excitation with combined strong XUV and NIR laser fields.
1. F. Dimler, S. Fechner, A. Rodenberg, T. Brixner and D. J. Tannor, Accurate and Efficient Implementation of the von Neumann Representation for Laser Pulses with Discrete and Finite Spectra, New J. Phys. 11, 105052 (2009).
2. A. Shimshovitz and D. J. Tannor, Phase Space Approach to Solving the Time-independent Schrödinger Equation (Phys. Rev. Lett. (2012); arXiv:1201.2299v1 [quant-ph]).
3. N. Takemoto, A. Shimshovitz and D. J. Tannor, Phase Space Approach to Solving the Time-dependent Schrödinger Equation: Application to the Simulation and Control of Attosecond Electron Dynamics in the Presence of a Strong Laser Field (manuscript in preparation).
