We present a method for solving both the time-independent and time-dependent Schroedinger equations based on the von Neumann (vN) lattice of phase space Gaussians. By incorporating periodic boundary conditions into the vN lattice 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. Formally the method defeats exponential scaling with dimensionality. In the classical limit the method reaches the remarkable efficiency of 1 converged eigenstate per 1 basis functions. We illustrate the method by calculating the vibrational eigenstates of a polyatomic with 104 bound states and by simulating attosecond electron dynamics in the presence of combined strong XUV and NIR laser fields. The method also has applications to signal and image processing. This will be illustrated with several audio and image examples where large compression factors are achieved.1. A. Shimshovitz and D. J. Tannor, Phase Space Approach to Solving the Time-independent Schr�dinger Equation, Phys. Rev. Lett. 109 , 070402 (2012).2. N. Takemoto, A. Shimshovitz and D. J. Tannor, Phase Space Approach to Solving the Time-dependent Schr�dinger Equation: Application to the Simulation and Control of Attosecond Electron Dynamics in the Presence of a Strong Laser Field, J. Chem. Phys. 137 , 011102 (2012) (Communication).3. A. Shimshovitz and D. J. Tannor, Phase Space Wavelets for Solving Coulomb Problems, J. Chem. Phys. 137 , 101103 (2012) (Communication).

Phase space approach to quantum mechanical calculations for large systems: Thinking inside the box

246th National Meeting (2013)

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