Quantum revivals versus classical periodicity in the infinite square well

American Journal of Physics

Volumn 69, Issue 1, 2001, Pages 56-62

  Styer, Daniel F ^a  

^a OBERLIN COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035585177     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1287355     Document Type: Article

References (32)

1. P. Ehrenfest, "Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik," Z. Phys 45, 455-457 (1927). Reprinted in Martin J. Klein, Ed., Paul Ehrenfest: Collected Scientific Papers (North-Holland, Amsterdam, 1959), pp. 556-558.
2. P. Ehrenfest, "Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik," Z. Phys 45, 455-457 (1927). Reprinted in Martin J. Klein, Ed., Paul Ehrenfest: Collected Scientific Papers (North-Holland, Amsterdam, 1959), pp. 556-558.
3. Michael Berry, "Some quantum-to-classical asymptotics." in Chaos and Quantum Physics (Les Houches, 1989), edited by M.-J. Giannoni, A. Voros, and J. Zinn-Justin (North-Holland, Amsterdam, 1991), pp. 250-303.
4. W. H. Zurek, "Decoherence and the transition from quantum to classical," Phys. Today 44, 36-44 (October 1991); Serge Haroche, "Entanglement, decoherence and the quantum/classical boundary," ibid. 51, 36-42 (July 1998).
5. W. H. Zurek, "Decoherence and the transition from quantum to classical," Phys. Today 44, 36-44 (October 1991); Serge Haroche, "Entanglement, decoherence and the quantum/classical boundary," ibid. 51, 36-42 (July 1998).
6. Martin C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990).
7. Eric J. Heller and Steven Tomsovic, "Postmodern quantum mechanics," Phys. Today 46, 38-46 (July 1993).
8. R. Bocchieri and A. Loinger, "Quantum recurrence theorem," Phys. Rev. 107, 337-338 (1957).
9. James Bayfield, Quantum Evolution: An Introduction to Time-Dependent Quantum Mechanics (Wiley, New York, 1999).
10. Tamar Seideman, "Revival structure of aligned rotational wave packets," Phys. Rev. Lett. 83, 4971-4974 (1999).
11. Robert Bluhm, V. Alan Kostelecký, and James A. Porter, "The evolution and revival structure of localized quantum wave packets," Am. J. Phys. 64, 944-953 (1996).
12. David L. Aronstein and C. R. Stroud, Jr., "Fractional wave-function revivals in the infinite square well," Phys. Rev. A 55, 4526-4537 (1997).
13. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory (Pergamon, London, 1958), particularly §48 and Eq. (48.5) on p. 165.
14. R. W. Robinett, "Visualizing the collapse and revival of wave packets in the infinite square well using expectation values," Am. J. Phys. 68, 410-420 (2000).
15. Julio Gea-Banacloche, "A quantum bouncing ball," Am. J. Phys. 67, 776-782 (1999).
16. Daniel F. Styer, "The motion of wave packets through their expectation values and uncertainties," Am. J. Phys. 58, 742-744 (1990).
17. 2N and the correspondence principle," Int. J. Theor. Phys. 18, 185-191 (1979); R. W. Robinett, "Wave packet revivals and quasirevivals in one-dimensional power law potentials," J. Math. Phys. 41, 1801-1813 (2000).
18. 2N and the correspondence principle," Int. J. Theor. Phys. 18, 185-191 (1979); R. W. Robinett, "Wave packet revivals and quasirevivals in one-dimensional power law potentials," J. Math. Phys. 41, 1801-1813 (2000).
19. 2N and the correspondence principle," Int. J. Theor. Phys. 18, 185-191 (1979); R. W. Robinett, "Wave packet revivals and quasirevivals in one-dimensional power law potentials," J. Math. Phys. 41, 1801-1813 (2000).
20. David K. Ferry and Stephen M. Goodnick, Transport in Nanostructures (Cambridge University Press, Cambridge, UK, 1997).
21. Elmar Schreiber, Femtosecond Real-Time Spectroscopy of Small Molecules and Clusters (Springer, Berlin, 1998).
22. J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, "Coherent dynamics of excitonic wave packets," Phys. Rev. Lett. 70, 3027-3030 (1993).
23. A. Bonvalet, J. Nagle, V. Berger, A. Migus, J.-L. Martin, and M. Joffre, "Femtosecond infrared emission resulting from coherent charge oscillations in quantum wells," Phys. Rev. Lett. 76, 4392-4395 (1996).
24. R. L. Liboff, "The correspondence principle revisited," Phys. Today 37, 50-55 (February 1984).
25. I represent an arbitrary wave function by ψ(x) and an energy eigenfunction by η(x), following the admirable convention established by Daniel T. Gillespie in A Quantum Mechanics Primer (International Textbook Company, Scranton, PA, 1970).
26. This result has been known for some time and rediscovered frequently. The earliest printed mention that I can find is Carlo U. Segre and J. D. Sullivan, "Bound-state wave packets," Am. J. Phys. 44, 729-732 (1976). See also Refs. 9 and 10, and Siegmund Brandt and Hans Dieter Dahmen, The Picture Book of Quantum Mechanics, 2nd ed. (Springer-Verlag, Berlin, 1995), pp. 107-108.
27. This result has been known for some time and rediscovered frequently. The earliest printed mention that I can find is Carlo U. Segre and J. D. Sullivan, "Bound-state wave packets," Am. J. Phys. 44, 729-732 (1976). See also Refs. 9 and 10, and Siegmund Brandt and Hans Dieter Dahmen, The Picture Book of Quantum Mechanics, 2nd ed. (Springer-Verlag, Berlin, 1995), pp. 107-108.
28. The applicability of Ehrenfest's theorem to the pathological infinite square well is treated in the delightful paper D. S. Rokhsar, "Ehrenfest's theorem and the particle-in-the-box," Am. J. Phys. 64, 1416-1418 (1996).
29. This example also shows that for a general wave function the periodicity of 〈x(t)〉 is not equal to the classical period. The period of such expectation values can possess the expected classical limit but does not need to. To find a correspondence and resolve the paradox, we must restrict our attention to a small subset of wave functions, namely those that are in some sense "nearly classical."
30. rev/3. Remarkably, it turns out that in all such cases the full wave function recurs with precisely the same period: see Theorem 6 in Appendix B.
31. Notice that such a quasiclassical state does not need to be a conventional wave packet, although all conventional wave packets are quasiclassical states. The term "wave packet" usually refers to a wave function that is well-localized in space, with a probability density that decreases from the center. The term is also used for a wave function that is well-defined in energy, with an energy probability that decreases from the central energy. A quasiclassical state as defined here need not be well-localized in space, and, while it is a superposition of energy states in a particular range, within that range the energy probability may vary in any imaginable way.
32. D. F. Styer, The Strange World of Quantum Mechanics (Cambridge University Press, Cambridge, UK, 2000), Appendix B 2.