Simulation of Time-Dependent Schrödinger Equation in the Position and Momentum Domains

Author(s)
Michael Jennings

ABSTRACT

The paper outlines the development of a new, spectral method of simulating the Schrödinger 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 Schrödinger 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.

The paper outlines the development of a new, spectral method of simulating the Schrödinger 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 Schrödinger 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.

Jennings, M. (2015) Simulation of Time-Dependent Schrödinger Equation in the Position and Momentum Domains.

References

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[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

[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