Simulation of Time-Dependent Schrödinger Equation in the Position and Momentum Domains
Author(s) Michael Jennings
Affiliation(s)
Chicago, IL, USA.
ABSTRACT
The paper outlines the development of a new, spectral method of simulating the Schr&ouml;dinger equation in the momentum domain. The kinetic energy operator is approximated in the momentum domain by exploiting the derivative property of the Fourier transform. These results are compared to a position-domain simulation generated by a fourth-order, centered-space, finite-difference formula. The time derivative is approximated by a four-step predictor-corrector in both domains. Free-particle and square-well simulations of the time-dependent Schr&ouml;dinger equation are run in both domains to demonstrate agreement between the new, spectral methods and established, finite-difference methods. The spectral methods are shown to be accurate and precise.

Cite this paper
Jennings, M. (2015) Simulation of Time-Dependent Schrödinger Equation in the Position and Momentum Domains. American Journal of Computational Mathematics, 5, 291-303. doi: 10.4236/ajcm.2015.53027.
References
[1]   Griffiths, D.J. (2005) Introduction to Quantum Mechanics. 2nd Edition, Prentice Hall, New Jersey.

[2]   Hamming, R.W. (1973) Numerical Methods for Scientists and Engineers. 2nd Edition, Dover, New York.

[3]   Kahaner, D., Moler, C. and Nash, S. (1989) Numerical Methods and Software. Prentice Hall, Upper Saddle River.

[4]   Moxley III, F.I., Zhu, F. and Dai, W. (2012) A Generalized FDTD Method with Absorbing Boundary Condition for Solving a Time-Dependent Linear Schrödinger Equation. American Journal of Computational Mathematics, 2, 163-172.
http://dx.doi.org/10.4236/ajcm.2012.23022

[5]   Bracewell, R.N. (2000) The Fourier Transform and Its Applications. 3rd Edition, McGraw-Hill, Boston.

[6]   Frigo, M. and Johnson, S.G. (2005) The Design and Implementation of FFTW3. Proceedings of the IEEE, 93, 216-231.
http://dx.doi.org/10.1109/JPROC.2004.840301

[7]   Askar, A. and Cakmak, A. (1978) Explicit Integration Method for the Time-Dependent Schrödinger Equation for Collision Problems. Journal of Chemical Physics, 68, 2794-2798.
http://dx.doi.org/10.1063/1.436072

[8]   Maestri, J.J., Landau, R.H. and Páez, M.J. (2000) Two-Particle Schrödinger Equation Animations of Wavepacket Wave-Packet Scattering. American Journal of Physics, 68, 1113-1119.
http://dx.doi.org/10.1119/1.1286310

[9]   Soriano, A., Navarro, E.A., Porti, J.A. and Such, V. (2004) Analysis of the Finite Difference Time Domain Technique to Solve the Schrödinger Equation for Quantum Devices. Journal of Applied Physics, 95, 8011-8018.
http://dx.doi.org/10.1063/1.1753661

[10]   Sullivan, D.M. (2012) Quantum Mechanics for Electrical Engineers. John Wiley and Sons, Hoboken.
http://dx.doi.org/10.1002/9781118169780

[11]   Visscher, P. (1991) A Fast Explicit Algorithm for the Time-Dependent Schrödinger Equation. Computers in Physics, 5, 596-598.
http://dx.doi.org/10.1063/1.168415

[12]   Chen, Z.D., Zhang, J.Y. and Yu, Z.P. (2009) Solution of the Time-Dependent Schrödinger Equation with Absorbing Boundary Conditions. Journal of Semiconductors, 30, 012001-1-012001-6.

[13]   Hunter, J.D. (2007) Matplotlib: A 2D Graphics Environment. Computing in Science & Engineering, 9, 90-95.
http://dx.doi.org/10.1109/MCSE.2007.55

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