[Table of Contents]

Plasma and Fusion Research

Volume 7, 2401106 (2012)

Regular Articles

Numerical Analysis of Schrödinger Equation for a Magnetized Particle in the Presence of a Field Particle
Shun-ichi OIKAWA, Emi OKUBO1) and Poh Kam CHAN1)
Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan
Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan
(Received 9 December 2011 / Accepted 22 May 2012 / Published 26 July 2012)


We have solved the two-dimensional time-dependent Schödinger equation for a magnetized proton in the presence of a fixed field particle with an electric charge of 2×10−5 e, where e is the elementary electric charge, and of a uniform megnetic field of B = 10 T. In the relatively high-speed case of v0 = 100 m/s, behaviors are similar to those of classical ones. However, in the low-speed case of v0 = 30 m/s, the magnitudes both in momentum mv = |mv|, where m is the mass and v is the velocity of the particle, and position r = |r| are appreciably decreasing with time. However, the kinetic energy K = m⟨v2⟩/2 and the potential energy U = ⟨qV⟩ , where q is the electric charge of the particle and V is the scalar potential, do not show appreciable changes. This is because of the increasing variances, i.e. uncertainty, both in momentum and position. The increment in variance of momentum corresponds to the decrement in the magnitude of momentum: Part of energy is transfered from the directional (the kinetic) energy to the uncertainty (the zero-point) energy.


uncertainty, field particle, uniform magnetic field, magnetic length, quantum mechanical effect

DOI: 10.1585/pfr.7.2401106


  • [1] S. Oikawa, T. Shimazaki and E. Okubo, Plasma Fusion Res. 6, 2401058 (2011).
  • [2] S. Oikawa, T. Oiwa and T. Shimazaki, Plasma Fusion Res. 5, S1050 (2010).
  • [3] S. Oikawa, T. Oiwa and T. Shimazaki, Plasma Fusion Res. 5, S2024 (2010).
  • [4] S. Oikawa, T. Shimazaki and T. Oiwa, Plasma Fusion Res. 5, S2025 (2010).
  • [5] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, 3rd ed., translated from the Russian by J.B. Sykes and J.S. Bell (Pergamon Press, Oxford, 1977).
  • [6] R.E. Marshak, Ann. N.Y. Acad. Sci. 410, 49 (1941).
  • [7] R.S. Cohen, L. Spitzer, Jr. and P. McR. Routly, Phys. Rev. 80, 230 (1950).
  • [8] S.I. Braginskii, Reviews of Plasma Physics, M.A. Leontovich (ed.), (Consultants Bureau, New York, 1965).
  • [9] R.J. Hawryluk, Rev. Mod. Phys. 70, 537 (1998).
  • [10] J.J. Sakurai, Modern Quantum Mechanics, Rev. ed., (Addison-Wesley, Reading, 1994).
  • [11] H. Natori and T. Munehisa, J. Phys. Soc. Jpn. 66, 351 (1997).
  • [12] P.K. Chan, S. Oikawa and E. Okubo, Plasma Fusion Res. 7, 2401034 (2012).
  • [13] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

This paper may be cited as follows:

Shun-ichi OIKAWA, Emi OKUBO and Poh Kam CHAN, Plasma Fusion Res. 7, 2401106 (2012).