Journal of Computational Electronics

, Volume 14, Issue 4, pp 907–915 | Cite as

Wigner functions, signed particles, and the harmonic oscillator

Article

Abstract

In this paper, we introduce the simple harmonic oscillator and we address it in the Wigner formulation of quantum mechanics, therefore describing the whole problem in terms of quasi-distribution functions defined over the phase-space. The harmonic oscillator represents a very important problem as it provides exact solutions in both stationary and transient regimes. Subsequently, we outline the time-dependent signed particle Wigner Monte Carlo method and simulate the oscillator problem starting from stationary initial conditions, i.e. rotationally invariant functions in the phase-space, showing no evolution in time of the distribution function as expected. This work is, thus, twofold. On the one hand, one may see it as a short review effort to demonstrate the convenience of utilizing a phase-space approach in this particular context, suggesting that it could be the case again for different interesting problems. On the other hand, it represents a further opportunity to validate the signed particle Monte Carlo method, showing that a new reliable and powerful tool is available for the time-dependent simulation of quantum systems.

Keywords

Quantum mechanics Wigner formulation Monte Carlo methods Harmonic oscillator 

References

  1. 1.
    Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–760 (1932)CrossRefGoogle Scholar
  2. 2.
    Leonhardt, U.: Measuring the Quantum State of Light. Cambridge University Press, Cambridge (1997)Google Scholar
  3. 3.
    Leibfried, D., Pfau, T., Monroe, C.: Shadows and mirrors: reconstructing quantum states of atom motion. Phys. Today 51, 22–28 (1998)CrossRefGoogle Scholar
  4. 4.
    Nedjalkov, M.: Wigner Transport in Presence of Phonons: Particle Models of the Electron Kinetics, From Nanostructures to Nanosensing Applications, vol. 160, pp. 55–103. Società Italiana Di Fisica, IOS Press, Amsterdam (2005)Google Scholar
  5. 5.
    Sellier, J.M., Nedjalkov, M., Dimov, I., Selberherr, S.: A benchmark study of the Wigner Monte-Carlo method. Monte Carlo Methods Appl. 20, 43–51 (2014)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Sellier, J.M., Dimov, I.: The many-body Wigner Monte Carlo method for time-dependent ab-initio quantum simulations. J. Comput. Phys. 273, 589–597 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sellier, J.M., Dimov, I.: On the simulation of indistinguishable fermions in the many-body Wigner formalism. J. Comput. Phys. 280, 287–294 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Rosati, R., Dolcini, F., Iotti, R.C., Rossi, F.: Wigner-function formalism applied to semiconductor quantum devices: failure of the conventional boundary condition scheme. Phys. Rev. B 88, 035401 (2013)CrossRefGoogle Scholar
  9. 9.
    Moyal, J.E.: Quantum mechanics as a statistical theory, vol. 45. In: Proceedings of the Cambridge Philosophical Society (1949)Google Scholar
  10. 10.
    Groenewold, H.: On the principles of elementary quantum mechanics. Physica 12, 405–460 (1946)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dias, N.C., Prata, J.N.: Admissible states in quantum phase space. Ann. Phys. 313, 110–146 (2004)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Tatarskii, V.I.: The Wigner representation of quantum mechanics. Sov. Phys. Usp. 26, 311–327 (1983)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sellier, J.M.: Nano-Archimedes. http://www.nano-archimedes.com. Accessed 23 June 2015

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.IICT, Bulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations