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Foundations of Physics Letters

, Volume 18, Issue 5, pp 455–475 | Cite as

Simple Examples of Position-Momentum Correlated Gaussian Free-Particle Wave Packets in One Dimension With the General Form of the Time-Dependent Spread in Position

  • R. W. RobinettEmail author
  • M. A. DoncheskiEmail author
  • L. C. BassettEmail author
Original Article

Abstract

We provide simple examples of closed-form Gaussian wave packet solutions of the free-particle Schrödinger equation in one dimension which exhibit the most general form of the time-dependent spread in position, namely (Δx t )2 = (Δx0)2 + At + (Δp0)2t2/m2, where A ≡ ‹(?? − ‹??›0)(?? − ‹??›0) + (?? − ‹??›0)(?? − ‹??›0)›0 contains information on the position-momentum correlation structure of the initial wave packet. We exhibit straightforward examples corresponding to squeezed states, as well as quasi-classical cases, for which A < 0 so that the position spread can (at least initially) decrease in time because of such correlations. We discuss how the initial correlations in these examples can be dynamically generated (at least conceptually) in various bound state systems. Finally, we focus on providing different ways of visualizing the xp correlations present in these cases, including the time-dependent distribution of kinetic energy and the use of the Wigner quasi-probability distribution. We discuss similar results, both for the time-dependent Δx t and special correlated solutions, for the case of a particle subject to a uniform force.

Key words:

wave packets time-development correlations Gaussian 

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References

  1. 1.
    1. L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1949), 1st edn., pp. 57–59. D. S. Saxon, Elementary Quantum Mechanics (McGraw-Hill, New York, 1968), pp. 62–66. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. 1 (Wiley, New York, 1977), pp. 61–66. R. L. Liboff, Introductory Quantum Mechanics (Addison-Wesley, Reading, 1980), pp. 151–156. R. A. Serway, C. J. Moses, and C. A. Moyer, Modern Physics (Saunders, Philadelphia, 1989), pp. 142–143. M. A. Morrison, Understanding Quantum Mechanics (Prentice-Hall, Englewood Cliffs, 1990), pp. 206–213. A. Goswami, Quantum Mechanics (Brown, Dubuque, 1992), pp. 37–43. D. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, Englewood Cliffs, 1995), p. 50. S. Gasiorowicz, Quantum Physics (Wiley, New York, 1996), 2nd edn., pp. 30–31. R. W. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples (Oxford University Press, New York, 1997), pp. 57–60, 86–87, 90, 106.Google Scholar
  2. 2.
    2. L. C. Baird, “Moments of a wave packet,” Am. J. Phys. 40, 327–329 (1972).CrossRefGoogle Scholar
  3. 3.
    3. M. Nicola, “Position uncertainty of a free particle,” Am. J. Phys. 40, 342 (1972).CrossRefGoogle Scholar
  4. 4.
    4. H. M. Bradford, “Propagation and spreading of a pulse or wave packet,” Am. J. Phys. 44, 1058–1063 (1976).CrossRefGoogle Scholar
  5. 5.
    5. H. C. Woodsum and K. R. Brownstein, “Tumbling motion of free three-dimensional wave packets,” Am. J. Phys. 45, 667–670 (1977).CrossRefGoogle Scholar
  6. 6.
    6. J. E. G. Farina, “Classical and quantum spreading of position probability,” Am. J. Phys. 45, 1200–1202 (1977).CrossRefGoogle Scholar
  7. 7.
    7. J. R. Klein, “Do free quantum mechanical wave packets always spread?,” Am. J. Phys. 48, 1035–1037 (1980).CrossRefGoogle Scholar
  8. 8.
    8. D. F. Styer, “The motion of wave packets through their expectation values and uncertainties,” Am. J. Phys. 58, 742–744 (1990).Google Scholar
  9. 9.
    9. M. Andrews, “Invariant operators for quadratic Hamiltonians,” Am. J. Phys. 67, 336–343 (1999).CrossRefGoogle Scholar
  10. 10.
    10. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), pp. 155, 165.Google Scholar
  11. 11.
    11. D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, 1963), pp 200–205.Google Scholar
  12. 12.
    12. J.-M. Lévy-Leblond, “Correlation of quantum properties and the generalized Heisenberg inequalities,” Am. J. Phys. 54, 135–136 (1986). R. A. Campos, “Correlation coefficient for incompatible observables of the quantum harmonic oscillator,” Am. J. Phys. 66, 712–718 (1998).CrossRefGoogle Scholar
  13. 13.
    13. M. V. Berry and N. L. Balazs, “Non-spreading wave packets,” Am. J. Phys. 47, 264–267 (1979). See also the very relevant comment on this paper by D. M. Greenberger, Am. J. Phys. 48, 256 (1980). We note that the special free-particle but ‘accelerating’ solutions involving Airy packets found by Berry and Balazs are not square integrable, but localized wave packets constructed from them exhibit the standard result for ‹xt consistent with Ehrenfest’s theorem in Eq. (9) and an expression for Δxt consistent with Eq. (2).CrossRefGoogle Scholar
  14. 14.
    14. G. W. Ford and R. F. O’Connell, “Wave packet spreading: temperature and squeezing effects with applications to quantum measurement and decoherence,” Am. J. Phys. 70, 319–324 (2002).CrossRefGoogle Scholar
  15. 15.
    15. R. W. Robinett and L. C. Bassett, “Analytic results for Gaussian wave packets in four model systems: I. Visualization of the kinetic energy,” Found. Phys. Lett. 17, 607–625 (2004) [arXiv:quant-ph/0408049].CrossRefGoogle Scholar
  16. 16.
    16. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).CrossRefGoogle Scholar
  17. 17.
    17. V. I. Tatarskii, “The Wigner representation of quantum mechanics,” Sov. Phys. Usp. 26, 311–327 (1983). N. L. Balaczs and B. K. Jennings, “Wigner’s function and other distribution functions in mock phase space,” Phys. Rep. 105, 347–391 (1984). P. Carruthers and F. Zachariasen, “Quantum collision theory with phase-space distributions,” Rev. Mod. Phys. 55, 245–285 (1983). M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: Fundamentals,” Phys. Rep. 106, 121–167 (1984). J. Bertrand and P. Bertrand, “A tomographic approach to Wigner’s function,” Found. Phys. 17, 397–405 (1987). Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics: Group Theoretical Approach (Lecture Notes in Physics Series, Vol. 40) (World Scientific, Singapore, 1990). H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995). A. M. Ozorio de Almeida, “The Weyl representation in classical and quantum mechanics,” Phys. Rep. 296, 265–342 (1998).Google Scholar
  18. 18.
    18. J. Snygg, “Wave functions rotated in phase space,” Am. J. Phys. 45, 58–60 (1977). N. Mukunda, “Wigner distribution for angle coordinates in quantum mechanics,” Am. J. Phys. 47, 192–187 (1979). S. Stenholm, “The Wigner function: I. The physical interpretation,” Eur. J. Phys. 1, 244–248 (1980). G. Mourgues, J. C. Andrieux, and M. R. Feix, “Solution of the Schroedinger equation for a system excited by a time Dirac pulse of potential. An example of the connection with the classical limit through a particular smoothing of the Wigner function,” Eur. J. Phys. 5, 112–118 (1984). Y. S. Kim and E. P. Wigner, “Canonical transformations in quantum mechanics,” Am. J. Phys. 58, 439–448 (1990). M. Casas, H. Krivine, and J. Martorell, “On the Wigner transforms of some simple systems and their semiclassical interpretations,” Eur. J. Phys. 12, 105–111 (1991).CrossRefGoogle Scholar
  19. 19.
    19. H.-W. Lee, “Spreading of a free wave packet,” Am. J. Phys., 50, 438–440 (1982).CrossRefGoogle Scholar
  20. 20.
    20. See Y. S. Kim and M. E. Noz, Ref. [17].Google Scholar
  21. 21.
    21. R. W. Robinett and L. C. Bassett, “Analytic results for Gaussian wave packets in four model systems: II. Autocorrelation functions,” Found. Phys. Lett. 17, 645–661 (2004) [arXiv:quant-ph/0408050].CrossRefGoogle Scholar
  22. 22.
    22. D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, “Generation of nonclassical motional states of a trapped atom,” Phys. Rev. Lett. 76 1796–1799 (1996).CrossRefPubMedGoogle Scholar
  23. 23.
    23. D. J. Heinzen and D. J. Wineland, “Quantum-limited cooling and detection of radio-frequency oscillations by laser-cooled atoms,” Phys. Rev. A42, 2977–2994 (1990).Google Scholar
  24. 24.
    24. See D. A. Saxon in Ref. [1].Google Scholar
  25. 25.
    25. For a recent review, see R. W. Robinett, “Quantum wave packet revivals,” Phys. Rep. 392, 1–119 (2004).CrossRefGoogle Scholar
  26. 26.
    26. D. L. Aronstein and C. R. Stroud, Jr., “Fractional wave-function revivals in the infinite square well,” Am. J. Phys. 55, 4526–4537 (1997)Google Scholar
  27. 27.
    27. M. Belloni, M. A. Doncheski, and R. W. Robinett, “Wigner quasi-probability distribution for the infinite well: Energy eigenstates and time-dependent wave packets,” Am. J. Phys. 72, 1183–1192 (2004) [arXiv:quant-ph/0312086].CrossRefGoogle Scholar
  28. 28.
    28. M. A. Doncheski and R. W. Robinett, “Anatomy of a quantum ‘bounce’,” Eur. J. Phys, 20, 29–37 (1999) [arXiv:quant-ph/0307010].CrossRefGoogle Scholar
  29. 29.
    29. V. V. Dodonov and M. A. Andreata, “Deflection of quantum particles by impenetrable boundary,” Phys. Lett. A275, 173–181 (2000). M. A. Andreata and V. V. Dodonov, “The reflection of narrow slow quantum packets from mirrors,” J. Phys. A: Math. Gen. 35, 8373–8392 (2002). V. V. Dodonov and M. A. Andreata, “Quantum deflection of ultracold atoms from mirrors,” Laser Physics, 12, 57–70 (2002).Google Scholar
  30. 30.
    30. M. Belloni, M. A. Doncheski, and R. W. Robinett, “Exact results for ‘bouncing’ Gaussian wave packets,” Phys. Scripta 71, 136–140 (2005) [arXiv:quant-ph/0408182].Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of PhysicsPennsylvania State UniversityUniversity Park
  2. 2.Department of PhysicsPennsylvania State UniversityMont Alto
  3. 3.Department of Applied Mathematics and Theoretical Physics (DAMTP)University of CambridgeCambridgeUK

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