By David Tannor
Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100 Israel

We present 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 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 for several polyatomic molecules 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.

References

1. Shimshovitz and D. J. Tannor, Phase Space Approach to Solving the Time-independent Schrödinger Equation, Phys. Rev. Lett. 109, 070402 (2012).
2. Shimshovitz and D. J. Tannor, Phase Space Wavelets for Solving Coulomb Problems, J. Chem. Phys. 137, 101103 (2012) (Communication).
3. N. Takemoto, A. Shimshovitz and D. J. Tannor, Phase Space Approach to Laser-driven Electronic Wavepacket Propagation, J. Chem. Phys. 137, 011102 (2012) (Communication).
4. J. Tannor, N. Takemoto and A. Shimshovitz, Phase Space Approach to Solving the Schrödinger Equation: Thinking Inside the Box, Adv. Chem. Phys. 156, 1 (2014). 
5. Assémat, S. Machnes and D. J. Tannor, Double Ionization of Helium from a Phase Space (preprint).
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