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Wigner functions, signed particles, and the harmonic oscillator


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.

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  1. Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–760 (1932)

    Article  Google Scholar 

  2. Leonhardt, U.: Measuring the Quantum State of Light. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  3. Leibfried, D., Pfau, T., Monroe, C.: Shadows and mirrors: reconstructing quantum states of atom motion. Phys. Today 51, 22–28 (1998)

    Article  Google Scholar 

  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. 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)

    MATH  MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  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)

    Article  Google Scholar 

  9. Moyal, J.E.: Quantum mechanics as a statistical theory, vol. 45. In: Proceedings of the Cambridge Philosophical Society (1949)

  10. Groenewold, H.: On the principles of elementary quantum mechanics. Physica 12, 405–460 (1946)

    MATH  MathSciNet  Article  Google Scholar 

  11. Dias, N.C., Prata, J.N.: Admissible states in quantum phase space. Ann. Phys. 313, 110–146 (2004)

    MATH  MathSciNet  Article  Google Scholar 

  12. Tatarskii, V.I.: The Wigner representation of quantum mechanics. Sov. Phys. Usp. 26, 311–327 (1983)

    MathSciNet  Article  Google Scholar 

  13. Sellier, J.M.: Nano-Archimedes. Accessed 23 June 2015

Download references


The authors acknowledge financial support from the European Project EC AComIn (FP7-REGPOT-2012-2013-1).

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Correspondence to J. M. Sellier.

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Sellier, J.M., Dimov, I. Wigner functions, signed particles, and the harmonic oscillator. J Comput Electron 14, 907–915 (2015).

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  • Quantum mechanics
  • Wigner formulation
  • Monte Carlo methods
  • Harmonic oscillator