Abstract
In quantum physics the free particle and the harmonically trapped particle are arguably the most important systems a physicist needs to know about. It is little known that, mathematically, they are one and the same. This knowledge helps us to understand either from the viewpoint of the other. Here we show that all general time-dependent solutions of the free-particle Schrödinger equation can be mapped to solutions of the Schrödinger equation for harmonic potentials, both the trapping oscillator and the inverted “oscillator”. This map is fully invertible and therefore induces an isomorphism between both types of system, they are equivalent. A composition of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant and different center position. The map is independent of the state of the system, consisting only of a coordinate transformation and multiplication by a form factor, and can be chosen such that the state is identical in both systems at one point in time. This transition point in time can be chosen freely, the wave function of the particle evolving in time in one system before the transition point can therefore be linked up smoothly with the wave function for the other system and its future evolution after the transition point. Such a cut-and-paste procedure allows us to describe the instantaneous changes of the environment a particle finds itself in. Transitions from free to trapped systems, between harmonic traps of different spring constants or center positions, or, from harmonic binding to repulsive harmonic potentials are straightforwardly modelled. This includes some time-dependent harmonic potentials. The mappings introduced here are computationally more efficient than either state-projection or harmonic oscillator propagator techniques conventionally employed when describing instantaneous (non-adiabatic) changes of a quantum particle’s environment.
Article PDF
Similar content being viewed by others
References
O. Steuernagel, Am. J. Phys. 73, 625 (2005) arXiv:physics/0312116v2.
O. Steuernagel, D. Kakofengitis, G. Ritter, Phys. Rev. Lett. 110, 030401 (2013) arXiv:1208.2970.
O. Steuernagel, arXiv:1109.1818 (2011).
A. Yariv, Quantum Electronics (Wiley, New York, 1967).
U. Niederer, Helv. Phys. Acta 45, 802 (1972).
S. Takagi, Prog. Theor. Phys. 84, 1019 (1990).
H.R. Lewis, Phys. Rev. Lett. 18, 510 (1967).
G. Nienhuis, L. Allen, Phys. Rev. A 48, 656 (1993).
G. Barton, Ann. Phys. 166, 322 (1986).
C. Yuce, A. Kilic, A. Coruh, Phys. Scr. 74, 114 (2006) arXiv:quant-ph/0703234.
K. Andrzejewski, J. Gonera, P. Kosinski, arXiv:1310.2799 (2013).
T. Kiss, J. Janszky, P. Adam, Phys. Rev. A 49, 4935 (1994).
H. Moya-Cessa, M.F. Guasti, Phys. Lett. A 311, 1 (2003).
H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, 1981).
W.P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).
L.E. Gendenshteïn, Sov. J. Exp. Theor. Phys. Lett. 38, 356 (1983).
F. Cooper, J.N. Ginocchio, A. Wipf, J. Phys. A: Math. Gen. 22, 3707 (1989).
R.N.C. Filho, G. Alencar, B.S. Skagerstam, J.S. Andrade, EPL 101, 10009 (2013).
H.R. Lewis, Jr., J. Math. Phys. 9, 1976 (1968).
J.R. Ray, Phys. Rev. A 26, 729 (1982).
G.S. Agarwal, S.A. Kumar, Phys. Rev. Lett. 67, 3665 (1991).
M.A. Lohe, J. Phys. A: Math. Theor. 42, 035307 (2009).
G. Bluman, V. Shtelen, J. Phys. A: Math. Gen. 29, 4473 (1996).
A. Mostafazadeh, J. Math. Phys. 40, 3311 (1999).
S. Chu, Rev. Mod. Phys. 70, 685 (1998).
R. Grimm, M. Weidemüller, Y.B. Ovchinnikov, arXiv:physics/9902072 (1999).
A. Walther, F. Ziesel, T. Ruster, S.T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, U. Poschinger, Phys. Rev. Lett. 109, 080501 (2012).
R. Bowler, J. Gaebler, Y. Lin, T.R. Tan, D. Hanneke, J.D. Jost, J.P. Home, D. Leibfried, D.J. Wineland, Phys. Rev. Lett. 109, 080502 (2012).
E.S. Shuman, J.F. Barry, D. Demille, Nature 467, 820 (2010) arXiv:1103.6004.
I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, M. Allegrini, Phys. Rev. Lett. 109, 183001 (2012) arXiv:1211.2652.
T.J. Kippenberg, K.J. Vahala, Science 321, 1172 (2008).
F. Marquardt, S.M. Girvin, Physics 2, 40 (2009).
M. Fernández Guasti, H. Moya-Cessa, Phys. Rev. A 67, 063803 (2003) quant-ph/0212073.
J.F. Cariñena, J. de Lucas, M.F. Rañada, SIGMA 4, 031 (2008) math-ph/0803.1824.
J. Guerrero, F.F. López-Ruiz, V. Aldaya, F. Cossío, J. Phys. A Math. Gen. 44, 44537 (2011) quant-ph/1010.5525.
Author information
Authors and Affiliations
Corresponding author
Additional information
This article is published with open access at Springerlink.com
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Steuernagel, O. Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes. Eur. Phys. J. Plus 129, 114 (2014). https://doi.org/10.1140/epjp/i2014-14114-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2014-14114-3