Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T01:09:48.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Path Integral Approach to Quantum Dynamics and Stationary Quantum States

Part of: Quantum theory Integral equations, integral transforms

Published online by Cambridge University Press:  03 July 2015

Ilkka Ruokosenmäki  and
Tapio T. Rantala
Ilkka Ruokosenmäki
Affiliation:
Department of Physics, Tampere University of Technology, Finland
Tapio T. Rantala*
Affiliation:
Department of Physics, Tampere University of Technology, Finland
*
*Corresponding author. Email addresses: [email protected] (I. Ruokosenmäki), [email protected] (T. T. Rantala)
Get access

Abstract

Applicability of Feynman path integral approach to numerical simulations of quantum dynamics of an electron in real time domain is examined. Coherent quantum dynamics is demonstrated with one dimensional test cases (quantum dot models) and performance of the Trotter kernel as compared with the exact kernels is tested. Also, a novel approach for finding the ground state and other stationary sates is presented. This is based on the incoherent propagation in real time. For both approaches the Monte Carlo grid and sampling are tested and compared with regular grids and sampling. We asses the numerical prerequisites for all of the above.

Keywords

71.15.-m31.15.X-73.21.-bPath integralreal time domainquantum dynamicsincoherent propagationstationary states

MSC classification

Secondary: 81-08: Computational methods 65R20: Integral equations
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 1 , July 2015 , pp. 91 - 103
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Feynman, R.P. and Hibbs, A.R., Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).Google Scholar
[2]Feynman, R.P., Rev. Mod. Phys. 20, 367 (1948).Google Scholar
[3]Duru, I.H. and Kleinert, H., Phys. Lett. 84B, 185 (1979) and Kleinert, H., Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets. World Scientific Publishing Co. Pte. Ltd. Singapore (2004). The 5th edition.Google Scholar
[4]Schulman, L.S., Techniques and Applications of Path Integration (Wiley, New York, 1981).Google Scholar
[5]Wong, K.-Y., Commun. Comput. Phys. 15, 853 (2014).Google Scholar
[6]Ceperley, D.M., Rev. Mod. Phys. 67, 279 (1995).Google Scholar
[7]Kylänpää, I., PhD Thesis (Tampere University of Technology 2011).Google Scholar
[8]Kylänpää, I. and Rantala, T.T., J. Chem. Phys. 133, 044312 (2010), Kylänpää, I. and Rantala, T.T., J. Chem. Phys. 135, 104310(2011) and Kylänpää, I. and Rantala, T.T., Phys. Rev. A 80, 024504 (2009).Google Scholar
[9]Militzer, and Ceperley, D.M., Phys. Rev. B 63, 066404 (2001).Google Scholar
[10]Weiss, S. and Egger, R., Phys. Rev. B 72, 245301 (2005).Google Scholar
[11]Gull, E.et al., Rev. Mod. Phys. 83, 349 (2011).Google Scholar
[12]Makri, N., Comp. Phys. Comm. 63, 389–414.Google Scholar
[13]Makri, N., Chem. Phys. Lett. 193, 435 (1992).Google Scholar
[14]Filinov, V.S., Nucl. Phys. B 271, 717–725 (1986).Google Scholar
[15]Wang, H.et al., J. Chem. Phys. 115, 6317(2001).Google Scholar
[16]Makri, N., Ann. Rev. Phys. Chem. 50, 167–191 (1999) and Jadhao, V. and Makri, N., J. Chem. Phys. 132, 104110 (2010).Google Scholar
[17]Marchioro, T.L. and Beck, T.L., J. Chem. Phys. 96, 2966 (1992).Google Scholar
[18]Makri, N., Comp. Phys. Comm. 63, 389–414 (1991) and Makri, N., J. Math. Phys. 36, 2430–56 (1995).Google Scholar
[19]Lambert, R. and Makri, N., J. Chem. Phys. 137 22A552 and 22A553 (2012).Google Scholar
[20]Makarov, D.E. and Makri, N., Chem. Phys. Lett. 221, 482 (1994).Google Scholar
[21]Kolmogorov, A., G.Ist.Ital.Attuari 4, 83 (1933).Google Scholar
[22]Suzuki, M., Phys. Lett. A 201, 425–428 (1995).Google Scholar
[23]Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, H. and Teller, E., J. Chem. Phys. 21, 1087 (1953).Google Scholar
[24]Atkins, P. and Friedman, R., Molecular Quantum Mechanics (Oxford University Press Inc., New York, 2005). The 4th edition.Google Scholar
[25]Schulten, K., “Notes on Quantum Mechanics” (University of Illinois at Urbana Champaign, 2000).Google Scholar