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A Model of a Quantum Particle in a Quantum Environment: A Numerical Study

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems General quantum mechanics and problems of quantization Partial differential equations, boundary value problems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  03 July 2015

Raffaele Carlone ,
Rodolfo Figari  and
Claudia Negulescu
Raffaele Carlone
Affiliation:
Università Federico II di Napoli, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, MSA I-80126 Napoli, Italy
Rodolfo Figari
Affiliation:
Università Federico II di Napoli, Dipartimento di Fisica e INFN Sezione di Napoli, MSA I-80126 Napoli, Italy
Claudia Negulescu*
Affiliation:
Université de Toulouse & CNRS, UPS, Institut de Mathématiques de Toulouse UMR 5219, F-31062 Toulouse, France
*
Email addresses: [email protected] (R. Carlone), [email protected] (R. Figari), [email protected] (C. Negulescu)
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Abstract

We define and investigate, via numerical analysis, a one dimensional toy-model of a cloud chamber. An energetic quantum particle, whose initial state is a superposition of two identical wave packets with opposite average momentum, interacts during its evolution and exchanges (small amounts of) energy with an array of localized spins. Triggered by the interaction with the environment, the initial superposition state turns into an incoherent sum of two states describing the following situation: or the particle is going to the left and a large number of spins on the left side changed their states, or the same is happening on the right side. This evolution is reminiscent of what happens in a cloud chamber where a quantum particle, emitted as a spherical wave by a radioactive source, marks its passage inside a supersaturated vapour-chamber in the form of a sequence of small liquid bubbles arranging themselves around a possible classical trajectory of the particle.

Keywords

Schrödinger equationquantum particle+environment modelmulti-channel point interactionsWilson cloud chambertrace formationdecoherencenumerical discretization

MSC classification

Secondary: 35Q41: Time-dependent Schrödinger equations, Dirac equations 65N06: Finite difference methods 65M06: Finite difference methods 81S22: Open systems, reduced dynamics, master equations, decoherence 68U20: Simulation
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 1 , July 2015 , pp. 247 - 262
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Adami, R., Hauray, M., Negulescu, C., Decoherence for a heavy particle interacting with a light one: new analysis and numerics, submitted.Google Scholar
[2]Bauer, M., Bernard, D., Convergence of repeated quantum non demolition measurements and wave function collapse Phys. Rev. A 84 044103, (2011).Google Scholar
[3]Cacciapuoti, C., Carlone, R., Figari, R. Perturbations of eigenvalues embedded at threshold: Two-dimensional solvable models, J. Math. Phys., 52, 8, 083515, 2011.Google Scholar
[4]Cacciapuoti, C., Carlone, R., Figari, R.A solvable model of a tracking chamber Rep. Math. Phys., 59, 3, 2007.Google Scholar
[5]Cacciapuoti, C., Carlone, R., Figari, R., Spin-dependent point potentials in one and three dimensions, J. Phys. A: Math. Theor., 40, 249–261, 2007.Google Scholar
[6]Darwin, C.G., A collision problem in the wave mechanics. Proc. R. Soc. Lond. A, 124, 375–394, 1929.Google Scholar
[7]Dell’Antonio, G., Figari, R., Teta, A., A brief review on point interactions. Inverse problems and imaging (Martinafranca, 2002), Bonilla, L. ed. 171–189, Springer LNM n. 1843, 2008.Google Scholar
[8]Dell’Antonio, G., Figari, R., Teta, A., Joint excitation probability for two harmonic oscillators in dimension one and the Mott problem. J. Math. Phys., 49, n. 4, 042105, 2008.Google Scholar
[9]Dell’Antonio, G., Figari, R., Teta, A., A time dependent perturbative analysis for a quantum particle in a cloud chamber. Ann. H. Poincaré, 11, n. 3, 539–564, 2010.Google Scholar
[10]Demkov, Y.N., Ostrovskii, V.N., Zero-Range Potentials and Their Applications in Atomic Physics, Plenum Pub Corp, 1988.Google Scholar
[11]Figari, R., Teta, A., Emergence of classical trajectories in quantum systems: the cloud chamber problem in the analysis of Mott (1929). Arch. Hist. Ex. Sci., 67, no. 2, 215–234, 2013.Google Scholar
[12]Figari, R. and Teta, A., Quantum Dynamics of a Particle in a Tracking Chamber, Springer-Verlag (2013).Google Scholar
[13]Heisenberg, W., über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Z. Phys., 33, 879–893, 1925. Eng. trans. reprinted in: van der Waerden, B.L., Source of Quantum Mechanics, Dover Publications, Inc., 1967.Google Scholar
[14]Heisenberg, W., über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys., 43, 172–198, 1927. Eng. trans. reprinted in: Wheeler, J.A., Zurek, W., Quantum Theory and Measurement, Princeton University Press, 1983.Google Scholar
[15]Heisenberg, W., The Physical Principles of the Quantum Theory, The University of Chicago Press, 1930.Google Scholar
[16]Hepp, K., Quantum Theory of Measurement and Macroscopic Observable. Helv. Phys. Acta, 45, 237–248, 1972.Google Scholar
[17]Hornberger, K, Sipe, J. E., Collisional decoherence reexamined. Phys. Rev. A, 68, 2003.Google Scholar
[18]Joos, E., Zeh, H.D., The emergence of classical properties through interaction with the environment. Z. Phys. B, 59, 223243, 1985.Google Scholar
[19]Joos, E., Zeh, H. D., Kiefer, C., Giulini, D., Kupsch, J., Stamatescu, I. O., Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed., Springer, 2003.Google Scholar
[20]Lovesey, S.W., Theory of Neutron Scattering from Condensed Matter, vol. I, Claredon Press, 1984.Google Scholar
[21]Mott, N.F., The wave mechanics of α-ray tracks. Proc. R. Soc. Lond. A, 126, 79–84, 1929.Google Scholar
[22]Recchia, C, Teta, A., Semiclassical wave-packets emerging from interaction with an environment, J. Math. Phys., 55,1012104, (2014).Google Scholar
[23]Schlosshauer, M., Decoherence and the Quantum-To-Classical Transition Springer-Verlag (2007).Google Scholar
[24]Seba, P., Exner, P., Pichugin, K.N., Vyhnal, A., Streda, P., Two-component interference effect: model of a spin-polarized transport. Phys. Rev. Lett., 86, 1598–1601, 2001.Google Scholar
[25]Sewell, G., On the Mathematical Structure of Quantum Measurement Theory. Rep. Math. Phys., 56, n. 2, 271–290, 2005.Google Scholar
[26]Teta, A., Classical behavior in quantum systems: the case of straight tracks in a cloud chamber. Eur. J. Phys., 31, n. 1, 215–227, 2010.Google Scholar