Skip to main content
SpringerLink
Log in

Deterministic Solution of the Discrete Wigner Equation

  • Conference paper
  • First Online:
Numerical Methods and Applications (NMA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8962))

Included in the following conference series:

Abstract

The Wigner formalism provides a convenient formulation of quantum mechanics in the phase space. Deterministic solutions of the Wigner equation are especially needed for problems where phase space quantities vary over several orders of magnitude and thus can not be resolved by the existing stochastic approaches. However, finite difference schemes have been problematic due to the discretization of the diffusion term in this differential equation. A new approach, which uses an integral formulation of the Wigner equation that avoids the problematic differentiation, is shown here. The results of the deterministic method are compared and validated with solutions of the Schrödinger equation. Furthermore, certain numerical aspects pertaining to the demanded parallel implementation are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Dimov, I.T.: Monte Carlo Methods for Applied Scientists. World Scientific, London (2008)

    MATH  Google Scholar 

  2. Fu, Y., Willander, M.: Electron wave-packet transport through nanoscale semiconductor device in time domain. J. Appl. Phys. 97(9), 094311 (2005)

    Article  Google Scholar 

  3. Griffiths, D.: Introduction to Quantum Mechanics. Pearson Prentice Hall, Upper Saddle River (2005)

    Google Scholar 

  4. Kosik, R.: Numerical challenges on the road to NanoTCAD. Ph.D. thesis, Institut für Mikroelektronik (2004)

    Google Scholar 

  5. Nedjalkov, M., Kosina, H., Selberherr, S., Ringhofer, C., Ferry, D.K.: Unified Particle approach to Wigner-Boltzmann transport in small semiconductor devices. Phys. Rev. B 70, 115319 (2004)

    Article  Google Scholar 

  6. Nedjalkov, M., Querlioz, D., Dollfus, P., Kosina, H.: Wigner function approach. In: Vasileska, D., Goodnick, S.M. (eds.) Nano-electronic Devices. Semiclassical and Quantum Transport Modeling, pp. 289–358. Springer, New York (2011)

    Chapter  Google Scholar 

  7. Sellier, J.M.D., Nedjalkov, M., Dimov, I., Selberherr, S.: A benchmark study of the Wigner Monte Carlo method. Monte Carlo Method Appl. 20(1), 43–51 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Sudiarta, I.W., Geldart, D.J.W.: Solving the Schrödinger equation using the finite difference time domain method. J. Phys. A: Math. Theor. 40(8), 1885 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Microelectronics, TU Wien, Vienna, Austria

    Johann Cervenka, Paul Ellinghaus & Mihail Nedjalkov

Authors
  1. Johann Cervenka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Ellinghaus
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mihail Nedjalkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johann Cervenka .

Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Stefka Fidanova

  3. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Cervenka, J., Ellinghaus, P., Nedjalkov, M. (2015). Deterministic Solution of the Discrete Wigner Equation. In: Dimov, I., Fidanova, S., Lirkov, I. (eds) Numerical Methods and Applications. NMA 2014. Lecture Notes in Computer Science(), vol 8962. Springer, Cham. https://doi.org/10.1007/978-3-319-15585-2_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-15585-2_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15584-5

  • Online ISBN: 978-3-319-15585-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation