Abstract
We discuss the role of boundary conditions in determining the physical content of the solutions of the Schrödinger equation. We study the standing-wave, the “in,” the “out,” and the purely outgoing boundary conditions. As well, we rephrase Feynman’s +iε prescription as a time-asymmetric, causal boundary condition, and discuss the connection of Feynman’s +iε prescription with the arrow of time of Quantum Electrodynamics. A parallel of this arrow of time with that of Classical Electrodynamics is made. We conclude that, in general, the time evolution of a closed quantum system has indeed an arrow of time built into the propagators.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. de la Madrid: Quantum Mechanics in Rigged Hilbert Space Language, PhD Thesis, Universidad de Valladolid, Valladolid (2001). Available at http://www.isi.it/~rafa/.
E. Brändas, M. Rittby, N. Elander: J. Math. Phys. 26, 2648 (1985).
E. Engdahl, E. Brändas, M. Rittby, N. Elander: Phys. Rev. A 37, 3777 (1988).
G. Gamow: Z. Phys. 51, 204 (1928).
A. F. J. Siegert: Phys. Rev. 56, 750 (1939).
E. Hernńdez, A. Mondragón: Phys. Rev. C 29, 722 (1984); A. Mondragón, E. Hernández, J. M. Velázquez Arcos: Ann. Phys. 48, 503 (1991).
M. Baldo, L. S. Ferreira, L. Streit: Phys. Rev. C 36, 1743 (1987); L. S. Ferreira, E. Maglione, R. J. Liotta: Phys. Rev. Lett. 78, 1640 (1997).
R. de la Madrid, M. Gadella: Am. J. Phys. 70, 626 (2002); quant-ph/0201091.
A. Bohm, M. Gadella (eds.): Dirac Kets, Gamow Vectors and Gelfand Triplets. In Lecture Notes in Physics, LNP 348 (Springer, Berlin, Heidelberg, 1989)
L. I. Schiff: Quantum Mechanics (McGraw-Hill, New York 1968).
R. P. Feynman: Quantum Electrodynamics (Benjamin, New York 1961).
N. Dunford, J. Schwartz: Linear operators, vol. II (Interscience Publishers, New York 1963).
R. de la Madrid: J. Phys. A: Math. Gen. 35, 319 (2002); quant-ph/0110165.
R. de la Madrid: Chaos, Solitons & Fractals 12, 2689 (2001); quant-ph/0107096.
R. de la Madrid, A. Bohm, M. Gadella: Fortsch. Phys. 50, 185 (2002); quant-ph/0109154.
R. Ritz: Physikalische Zeitschrift 9, 903 (1908); ibid. 10, 224 (1909); R. Ritz, A. Einstein: Physikalische Zeitschrift 10, 323 (1909).
R. Peierls: Surprises in Theoretical Physics (Princeton University Press, Princeton 1979).
R. Penrose: ‘Singularities and time-asymmetry’. In General Relativity: An Einstein Centenary Survey, ed. by S. W. Hawking, W. Israel (Cambridge University Press, Cambridge 1979), pp. 581–638.
M. Gell-Mann, J. B. Hartle: ‘Time Symmetry and Asymmetry in Quantum Mechanics and Quantum Cosmology’. In [11], pp. 311–345.
J. Preskill: Physics 229: Advanced Mathematical Methods of Physics— Quantum Computation and Information (California Institute of Technology, Pasadena, CA, 1998). Available at http://www.theory.caltech.edu/people/preskill/ph229/.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
de la Madrid, R. (2003). The Importance of Boundary Conditions in Quantum Mechanics. In: Benatti, F., Floreanini, R. (eds) Irreversible Quantum Dynamics. Lecture Notes in Physics, vol 622. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44874-8_17
Download citation
DOI: https://doi.org/10.1007/3-540-44874-8_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40223-7
Online ISBN: 978-3-540-44874-7
eBook Packages: Springer Book Archive