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Symplectic integrators for the numerical solution of the Schrödinger equation. (English) Zbl 1027.65171

Summary: The solution of the one-dimensional time-independent Schrödinger equation is considered by symplectic integrators. The Schrödinger equation is first transformed into a Hamiltonian canonical equation. The concept of asymptotic symplecticness is introduced and asymptotically symplectic methods of order up to 3 are developed. Numerical results are obtained for the one-dimensional harmonic oscillator, the hydrogen atom and the one dimensional double-well anharmonic oscillator.

MSC:

65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
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References:

[1] Arnold, V., Mathematical methods of classical mechanics, (1978), Springer-Verlag New York · Zbl 0386.70001
[2] Blatt, J.M., Practical points concerning the solution of the Schrödinger equation, J. comput. phys., 1, 382-396, (1967) · Zbl 0182.49702
[3] Killingbeck, J., Shooting methods for the Schrödinger equation, J. phys. A: math. gen., 20, 1411-1417, (1987) · Zbl 0627.65096
[4] Liu, X.S.; Liu, X.Y.; Zhou, Z.Y.; Ding, P.Z.; Pan, S.F., Numerical solution of the one-dimensional time-independent Schrödinger equation by using symplectic schemes, Int. J. quant. chem., 79, 343-349, (2000)
[5] Raptis, A.D., On the numerical solution of the Schrödinger equation, Comput. phys. commun., 24, 1-4, (1981)
[6] Raptis, A.D., Two step methods for the numerical solution of the Schrödinger equation, Computing, 28, 373-378, (1982) · Zbl 0473.65060
[7] Sanz-Serna, J.M.; Calvo, M.P., Numerical Hamiltonian problem, (1994), Chapman & Hall London · Zbl 0816.65042
[8] T.E. Simos, Numerical methods for 1D, 2D and 3D differential equations arising in chemical problems, Chemical Modelling: Application and Theory, The Royal Society of Chemistry, in press.
[9] Yoshida, H., Construction of higher order symplectic integrators, Phys. lett. A, 150, 262-268, (1990)
[10] Zhu, W.; Zhao, X.; Tang, Y., Numerical methods with a high order of accuracy in the quantum system, J. chem. phys., 104, 2275-2286, (1996)
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