Abstract
The classical two-dimensional motion of a parabolically confined charged particle in presence of a perpendicular magnetic is studied. The resulting equations of motion are solved exactly by using a mathematical method which is based on the introduction of complex variables. The two-dimensional motion of a parabolically charged particle in a perpendicular magnetic field is strikingly different from either the two-dimensional cyclotron motion, or the oscillator motion. It is found that the trajectory of a parabolically confined charged particle in a perpendicular magnetic field is closed only for particular values of cyclotron and parabolic confining frequencies that satisfy a given commensurability condition. In these cases, the closed paths of the particle resemble Lissajous figures, though significant differences with them do exist. When such commensurability condition is not satisfied, path of particle is open and motion is no longer periodic. In this case, after a sufficiently long time has elapsed, the open paths of the particle fill a whole annulus, a region lying between two concentric circles of different radii.
[1] G. R. Fowles, G. L. Cassiday, Analytical mechanics, Sixth Edition (Brooks/Cole, Belmont, 1999) Search in Google Scholar
[2] L. Jacak, P. Hawrylak, A. Wojs, Quantum dots (Springer, Berlin, 1997) 10.1007/978-3-642-72002-4Search in Google Scholar
[3] R. C. Ashoori, Nature 379, 413 (1996) http://dx.doi.org/10.1038/379413a010.1038/379413a0Search in Google Scholar
[4] L. P. Kouwenhoven, C. M. Marcus, Phys. World 11, 35 (1998) 10.1088/2058-7058/11/6/26Search in Google Scholar
[5] M. A. Kastner, Phys. Today 46, 24 (1993) http://dx.doi.org/10.1063/1.88139310.1063/1.881393Search in Google Scholar
[6] V. Fock, Z. Phys. 47, 446 (1928). http://dx.doi.org/10.1007/BF0139075010.1007/BF01390750Search in Google Scholar
[7] C. G. Darwin, Math. Proc. Cambridge 27, 86 (1930) http://dx.doi.org/10.1017/S030500410000937310.1017/S0305004100009373Search in Google Scholar
[8] O. Ciftja, M. G. Faruk, Phys. Rev. B 72, 205334 (2005) http://dx.doi.org/10.1103/PhysRevB.72.20533410.1103/PhysRevB.72.205334Search in Google Scholar
[9] O. Ciftja, J. Phys.-Condens. Mat. 19, 046220 (2007) http://dx.doi.org/10.1088/0953-8984/19/4/04622010.1088/0953-8984/19/4/046220Search in Google Scholar
[10] S. T. Thornton, J. B. Marion, Classical dynamics of particles and systems, Fifth Edition (Brooks/Cole, Belmont, 2004). Search in Google Scholar
[11] H. Goldstein, C. Poole, J. Safko, Classical mechanics, Third Edition (Addison Wesley, San Francisco, 2001). Search in Google Scholar
[12] J. Lissajous, Ann. Chim. Phys. 51, 147 (1857) Search in Google Scholar
[13] A. P. French, Vibrations and Waves (Norton, New York, 1971) 29 Search in Google Scholar
[14] J. D. Lawrence, A catalog of special plane curves (Dover, New York, 1972) 178 Search in Google Scholar
[15] J. D. Lawrence, A catalog of special plane curves (Dover, New York, 1972) 181 Search in Google Scholar
[16] E. F. Fahy, F. G. Karioris, Am. J. Phys. 20, 121 (1952) http://dx.doi.org/10.1119/1.193314210.1119/1.1933142Search in Google Scholar
[17] T. B. Greenslade Jr, Phys. Teach. 32, 364 (1993) http://dx.doi.org/10.1119/1.234380210.1119/1.2343802Search in Google Scholar
[18] J. R. Taylor, Classical Mechanics (University Science Books, Mill Valley, 2005) 65 Search in Google Scholar
[19] H. R. Lewis Jr, Phys. Rev. 172, 1313 (1968) http://dx.doi.org/10.1103/PhysRev.172.131310.1103/PhysRev.172.1313Search in Google Scholar
© 2013 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
