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An efficient and accurate quantum lattice-gas model for the many-body Schrödinger wave equation. (English) Zbl 0998.81124
Summary: We presented a quantum lattice-gas model for simulating the time-dependent evolution of a many-body quantum mechanical system of particles governed by the non-relativistic Schrödinger wave equation with an external scalar potential. A variety of computational demonstrations are given where the numerical predictions are compared with exact analytical solutions. In all cases, the model results accurately agree with the analytical predictions and we show that the model’s error is second order in the temporal discretization and fourth order in the spatial discretization. The difficult problem of simulating a system of fermionic particles is also treated and a general computational formulation of this problem is given. For pedagogical purposes, the two-particle case is presented and the numerical dispersion of the simulated wave packets is compared with the analytical solutions.

MSC:
81V70 Many-body theory; quantum Hall effect
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81-04 Software, source code, etc. for problems pertaining to quantum theory
81P68 Quantum computation
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References:
[1] Feynman, R.P., Simulating physics with computers, Internat. J. theoret. phys., 21, 6/7, 467-488, (1982)
[2] Feynman, R.P., Quantum mechanical computers, Optics news, 11, 2, 11-20, (1985)
[3] ()
[4] Feynman, R.P., Space-time approach to non-relativistic quantum mechanics, Rev. modern phys., 20, 2, 367-387, (1948) · Zbl 1371.81126
[5] Feynman, R.P.; Hibbs, A.R., Quantum mechanics and path integrals, (1965), McGraw-Hill · Zbl 0176.54902
[6] Bialynicki-Birula, I., Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata, Phys. rev. D, 49, 12, 6920-6927, (1994)
[7] Meyer, D.A., From quantum cellular automata to quantum lattice gas, J. statist. phys., 85, 5,6, 551-574, (1996) · Zbl 0952.37501
[8] Meyer, D.A., Quantum mechanics of lattice gas automata: one-particle plane waves and potentials, Phys. rev. E, 55, 5, 5261-5269, (1997)
[9] Meyer, D.A., Quantum mechanics of lattice gas automata. II. boundary conditions and other inhomogeneities, 1997
[10] Succi, S.; Benzi, R., Lattice Boltzmann equation for quantum mechanics, Physica D, 69, 327-332, (1993) · Zbl 0798.76080
[11] Succi, S., Numerical solution of the Schrödinger equation using discrete kinetic theory, Phys. rev. E, 53, 2, 1969-1975, (1996)
[12] Boghosian, B.M.; Taylor, W., Quantum lattice gas models for the many-body Schrödinger equation, Internat. J. modern phys. C, 8, 705-716, (1997)
[13] Boghosian, B.M.; Taylor, W., Simulating quantum mechanics on a quantum computer, Physica D, 120, 30-42, (1998) · Zbl 1040.81505
[14] Boghosian, B.M.; Taylor, W., A quantum lattice-gas model for the many-particle Schrödinger equation in d-dimensions, Phys. rev. E, 57, 54-66, (1998)
[15] Succi, S.; Benzi, R.; Higuera, F., The lattice Boltzmann equation: A new tool for computational fluid dynamics, Physica D, 47, 219-230, (1991)
[16] Qian, Y.H.; d’Humiéres, D.; Lallemand, P., Lattice BGK models for navier – stokes equation, Europhys. lett., 17, 6BIS, 479-484, (1992) · Zbl 1116.76419
[17] Chen, H.; Chen, S.; Mattaeus, W.H., Recovery of the navier – stokes equations using a lattice-gas Boltzmann method, Phys. rev. A, 45, 8, R5339-R5342, (1992)
[18] Polley, L., Schrödinger equation as the universal continuum limit of nonrelativistic coherent hopping on a cubic spatial lattice, Los Alamos National Laboratory Electronic Archive
[19] Yepez, J., Quantum lattice-gas model for the diffusion equation, Internat. J. modern phys. C, 12, 9, 1-19, (2001), Presented at the 9th International Conference on Discrete Simulation of Fluid Dynamics, Santa Fe, NM, August 22, 2000
[20] Boghosian, B.M.; Taylor, W., A quantum lattice gas models for the many-body Schrödinger equation, 1996
[21] Schrödinger, E., Naturwiss, 14, 664, (1926)
[22] Schiff, L.I., Quantum mechanics, International series in pure and applied physics, (1968), McGraw-Hill New York
[23] Yepez, J., A quantum lattice-gas model for computational fluid dynamics, Phys. rev. E, (2001), 046702-1-046702-18, APS E-Print: aps1999Oct22_002
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