Wave packet construction in two-dimensional quantum billiards: Blueprints for the square, equilateral triangle, and circular cases

American Journal of Physics

Volumn 71, Issue 6, 2003, Pages 541-557

  Doncheski, M A ^a   Heppelmann, S ^b   Robinett, R W ^b   Tussey, D C ^b  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038575078     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1538574     Document Type: Article

References (63)

1. For an excellent review of wave packet dynamics, see The Physics and Chemistry of Wave Packets, edited by J. A. Yeazell and T. Uzer (Wiley, New York, 2000).
2. J. R. Hiller, I. D. Johnston, and D. F. Styer, Quantum Mechanics Simulations: The Consortium for Upper-Level Physics Software (Wiley, New York, 1995); B. Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena (Springer-Verlag, New York, 2000); S. Brandt and S. Dahmen, The Picture Book of Quantum Mechanics, 3rd ed. (Springer-Verlag, New York, 2001); J. Bayfield, Quantum Evolution: An Introduction to Time-Dependent Quantum Mechanics (Wiley, New York, 1999); S. M. McMurry, Quantum Mechanics (Addison-Wesley, Reading, MA, 1993); J. Jarecki, Graphical Schrödingers Equation (Physics Academic Software, Raleigh, NC, 1998); W. Christian and M. Belloni, Physlets: Teaching Physics with Interactive Curricular Material (Prentice-Hall, Englewood Cliffs, NJ, 2000).
3. J. R. Hiller, I. D. Johnston, and D. F. Styer, Quantum Mechanics Simulations: The Consortium for Upper-Level Physics Software (Wiley, New York, 1995); B. Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena (Springer-Verlag, New York, 2000); S. Brandt and S. Dahmen, The Picture Book of Quantum Mechanics, 3rd ed. (Springer-Verlag, New York, 2001); J. Bayfield, Quantum Evolution: An Introduction to Time-Dependent Quantum Mechanics (Wiley, New York, 1999); S. M. McMurry, Quantum Mechanics (Addison-Wesley, Reading, MA, 1993); J. Jarecki, Graphical Schrödingers Equation (Physics Academic Software, Raleigh, NC, 1998); W. Christian and M. Belloni, Physlets: Teaching Physics with Interactive Curricular Material (Prentice-Hall, Englewood Cliffs, NJ, 2000).
4. J. R. Hiller, I. D. Johnston, and D. F. Styer, Quantum Mechanics Simulations: The Consortium for Upper-Level Physics Software (Wiley, New York, 1995); B. Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena (Springer-Verlag, New York, 2000); S. Brandt and S. Dahmen, The Picture Book of Quantum Mechanics, 3rd ed. (Springer-Verlag, New York, 2001); J. Bayfield, Quantum Evolution: An Introduction to Time-Dependent Quantum Mechanics (Wiley, New York, 1999); S. M. McMurry, Quantum Mechanics (Addison-Wesley, Reading, MA, 1993); J. Jarecki, Graphical Schrödingers Equation (Physics Academic Software, Raleigh, NC, 1998); W. Christian and M. Belloni, Physlets: Teaching Physics with Interactive Curricular Material (Prentice-Hall, Englewood Cliffs, NJ, 2000).
5. J. R. Hiller, I. D. Johnston, and D. F. Styer, Quantum Mechanics Simulations: The Consortium for Upper-Level Physics Software (Wiley, New York, 1995); B. Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena (Springer-Verlag, New York, 2000); S. Brandt and S. Dahmen, The Picture Book of Quantum Mechanics, 3rd ed. (Springer-Verlag, New York, 2001); J. Bayfield, Quantum Evolution: An Introduction to Time-Dependent Quantum Mechanics (Wiley, New York, 1999); S. M. McMurry, Quantum Mechanics (Addison-Wesley, Reading, MA, 1993); J. Jarecki, Graphical Schrödingers Equation (Physics Academic Software, Raleigh, NC, 1998); W. Christian and M. Belloni, Physlets: Teaching Physics with Interactive Curricular Material (Prentice-Hall, Englewood Cliffs, NJ, 2000).
6. J. R. Hiller, I. D. Johnston, and D. F. Styer, Quantum Mechanics Simulations: The Consortium for Upper-Level Physics Software (Wiley, New York, 1995); B. Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena (Springer-Verlag, New York, 2000); S. Brandt and S. Dahmen, The Picture Book of Quantum Mechanics, 3rd ed. (Springer-Verlag, New York, 2001); J. Bayfield, Quantum Evolution: An Introduction to Time-Dependent Quantum Mechanics (Wiley, New York, 1999); S. M. McMurry, Quantum Mechanics (Addison-Wesley, Reading, MA, 1993); J. Jarecki, Graphical Schrödingers Equation (Physics Academic Software, Raleigh, NC, 1998); W. Christian and M. Belloni, Physlets: Teaching Physics with Interactive Curricular Material (Prentice-Hall, Englewood Cliffs, NJ, 2000).
7. J. R. Hiller, I. D. Johnston, and D. F. Styer, Quantum Mechanics Simulations: The Consortium for Upper-Level Physics Software (Wiley, New York, 1995); B. Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena (Springer-Verlag, New York, 2000); S. Brandt and S. Dahmen, The Picture Book of Quantum Mechanics, 3rd ed. (Springer-Verlag, New York, 2001); J. Bayfield, Quantum Evolution: An Introduction to Time-Dependent Quantum Mechanics (Wiley, New York, 1999); S. M. McMurry, Quantum Mechanics (Addison-Wesley, Reading, MA, 1993); J. Jarecki, Graphical Schrödingers Equation (Physics Academic Software, Raleigh, NC, 1998); W. Christian and M. Belloni, Physlets: Teaching Physics with Interactive Curricular Material (Prentice-Hall, Englewood Cliffs, NJ, 2000).
8. J. R. Hiller, I. D. Johnston, and D. F. Styer, Quantum Mechanics Simulations: The Consortium for Upper-Level Physics Software (Wiley, New York, 1995); B. Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena (Springer-Verlag, New York, 2000); S. Brandt and S. Dahmen, The Picture Book of Quantum Mechanics, 3rd ed. (Springer-Verlag, New York, 2001); J. Bayfield, Quantum Evolution: An Introduction to Time-Dependent Quantum Mechanics (Wiley, New York, 1999); S. M. McMurry, Quantum Mechanics (Addison-Wesley, Reading, MA, 1993); J. Jarecki, Graphical Schrödingers Equation (Physics Academic Software, Raleigh, NC, 1998); W. Christian and M. Belloni, Physlets: Teaching Physics with Interactive Curricular Material (Prentice-Hall, Englewood Cliffs, NJ, 2000).
9. C. Singh, "Student understanding of quantum mechanics," Am. J. Phys. 69, 885-895 (2001).
10. E. Cataloglu and R. W. Robinett, "Testing the development of student conceptual and visualization understanding in quantum mechanics through the undergraduate career," Am. J. Phys. 70, 238-251 (2002).
11. A. Goldberg, H. M. Schey, and J. L. Schwartz, "Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena," Am. J. Phys. 35, 177-186 (1967).
12. M. H. Bramhall and B. M. Casper, "Reflections on a wave packet approach to quantum mechanical barrier penetration," Am. J. Phys. 38, 1136-1145 (1970).
13. B. Diu, "Plane waves and wave packets in elementary quantum mechanics problems," Eur. J. Phys. 1, 231-240 (1980).
14. A. Edgar, "Reflection of wave packets from a quantum well with a tunneling transmission resonance," Am. J. Phys. 63, 136-141 (1995).
15. J. S. Boleman and S. B. Haley, "More time-dependent calculations for the Schrödinger equation," Am. J. Phys. 43, 270-271 (1975).
16. M. Andrews, "Wave packets bouncing off walls," Am. J. Phys. 66, 252-254 (1998).
17. M. A. Doncheski and R. W. Robinett, "Anatomy of a quantum bounce," Eur. J. Phys. 20, 29-37 (1999).
18. J. V. Greenman, "Non-dispersive mirror wave packets," Am. J. Phys. 40, 1193-1201 (1972).
19. C. U. Segre and J. D. Sullivan, "Bound-state wave packets," Am. J. Phys. 44, 729-732 (1976). This paper also contains one of the first references to what have become known as wave packet revivals.
20. P. A. Deutchmann, "Tunneling between two square wells-Computer movie," Am. J. Phys. 39, 952-954 (1971).
21. E. A. Johnson and H. Thomas Williams, "Quantum solutions for a symmetric double square well," Am. J. Phys. 50, 239-243 (1981).
22. J. C. Hamilton, J. L. Schwartz, and W. A. Bowers, "Computer generated films for solid state physics," Am. J. Phys. 40, 1657-1661 (1972).
23. G. Friedmann and W. A. Little, "A study of a wave function of a particle striking a crystal interface," Am. J. Phys. 61, 835-843 (1993).
24. J. R. Merrill, "The propagation of quantum mechanical wave packets," Am. J. Phys. 41, 1101-1103 (1973).
25. R. Bluhm, V. A. Kostelecký, and J. Porter, "The evolution and revival structure of localized wave packets," Am. J. Phys. 64, 944-953 (1996).
26. 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).
27. D. F. Styer, "Quantum revivals versus classical periodicity in the infinite square well," Am. J. Phys. 69, 56-62 (2001).
28. F. Großmann, J.-M. Rost, and W. P. Schleich, "Spacetime structures in simple quantum systems," J. Phys. A A30, L277-L283 (1997).
29. P. Stifter, W. E. Lamb Jr. and W. P. Schleich, "The particle in a box revisited," in Frontiers of Quantum Optics and Laser Physics, Proceedings of the International Conference on Quantum Optics and Laser Physics, edited by S. Y. Zhu, M. S. Zubairy, and M. O. Scully (Springer, Singapore, 1997), pp. 236-246.
30. J. Gea-Banacloche, "A quantum bouncing ball," Am. J. Phys. 67, 776-782 (1999).
31. M. A. Doncheski and R. W. Robinett, "Expectation value analysis of wave packet solutions for the quantum bouncer: Short-term classical and long-term revival behaviors," Am. J. Phys. 69, 1084-1090 (2001).
32. D. L. Aronstein and C. R. Stroud, Jr., "Fractional wave-function revivals in the infinite square well," Phys. Rev. A 55, 4526-4537 (1997).
33. K. Razi Naqvi, S. Waldenstrom, and T. Jaji Hassan, "Fractional revival of wave packets in an infinite square well: A Fourier perspective," Eur. J. Phys. 22, 395-402 (2001).
34. W. Loinaz and T. J. Newman, "Quantum revivals and carpets in some exactly solvable systems," J. Phys. A 32, 8889-8895 (1999).
35. C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, "Conductance fluctuations and chaotic scattering in ballistic microstructures," Phys. Rev. Lett. 69, 506-509 (1992).
36. N. Friedman, A. Kaplan, D. Carasso, and N. Davison, "Observation of chaotic and regular dynamics in atom-optics billiards," Phys. Rev. Lett. 86, 1518-1521 (2001).
37. R. W. Robinett and S. Heppelmann, "Quantum wave packet revivals in circular billiards," Phys. Rev. A 65, 062103-062113 (2002).
38. M. A. Doncheski and R. W. Robinett, "Quantum mechanical analysis of the equilateral triangle billiard: periodic orbit theory and wave packet revivals," Ann. Phys. (N.Y.) 299, 208-227 (2002).
39. M. Nauenberg, "Autocorrelation function and quantum recurrence of wave packets," J. Phys. B 23, L385-L390 (1990).
40. R. W. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples (Oxford University Press, New York, 1997), pp. 142-143.
41. R. Bluhm, V. Alan Kostelecký, and B. Tudose, "Wave-packet revivals for quantum systems with nondegenerate energies," Phys. Lett. A 222, 220-226 (1996).
42. G. S. Agarwal and J. Banerji, "Fractional revivals in systems with two time scales," Phys. Rev. A 57, 3880-3884 (1998).
43. K. Richter, D. Ullmo, and R. A. Jalabert, "Orbital magnetism in the ballistic regime: geometrical effects," Phys. Rep. 276, 1-83 (1996).
44. R. W. Robinett, "Visualizing classical periodic orbits from the quantum energy spectrum via the Fourier transform: Simple infinite well examples," Am. J. Phys. 65, 1167-1175 (1997).
45. M. V. Berry and M. Wilkinson, "Diabolical points in the spectra of triangles," Proc. R. Soc. London, Ser. A 392, 15-43 (1984).
46. W.-K. Li, "A particle in an isosceles right triangle," J. Chem. Educ. 61, 1034 (1984).
47. R. W. Robinett, "Isolated versus non-isolated periodic orbits in variants of the two-dimensional square and circular billiards," J. Math. Phys. 40, 101-122 (1999).
48. C. Jung, "An exactly soluble three-body problem in one-dimension," Can. J. Phys. 58, 719-728 (1980). This paper discusses the group theoretical properties of the solutions of the triangular well in some detail.
49. P. J. Richens and M. V. Berry, "Pseudointegrable systems in classical and quantum mechanics," Physica D 2, 495-512 (1981).
50. A version of this problem is also discussed in J. Mathews and R. L. Walker, Mathematical Methods of Physics, 2nd ed. (W. A. Benjamin, Menlo Park, CA, 1970), pp. 237-239.
51. W.-K. Li and S. M. Blinder, "Particle in an equilateral triangle: Exact solution of a nonseparable problem," J. Chem. Educ. 64, 130-132 (1987).
52. I. Galbraith, Y. S. Ching, and E. Abraham, "Two-dimensional time-dependent quantum-mechanical scattering event," Am. J. Phys. 52, 60-68 (1984).
53. See, for example, M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (McGraw-Hill, New York, 1964), p. 364.
54. A. B. Migdal and V. Krainov, Approximation Methods in Quantum Mechanics (Benjamin, New York, 1992), pp. 106-115.
55. S. M. Reimann, M. Brack, A. G. Magner, and M. V. N. Murthy, "Applications of classical periodic orbit theory to circular billiards with small scattering centers," Surf. Rev. Lett. 3, 19-23 (1996); R. W. Robinett, "Energy eigenvalues and periodic orbits for the circular disk or annular infinite well," ibid. 5, 519-526 (1998); R. W. Robinett, "Periodic orbit theory analysis of the circular disk or annular billiard," Am. J. Phys. 67, 67-77 (1999).
56. S. M. Reimann, M. Brack, A. G. Magner, and M. V. N. Murthy, "Applications of classical periodic orbit theory to circular billiards with small scattering centers," Surf. Rev. Lett. 3, 19-23 (1996); R. W. Robinett, "Energy eigenvalues and periodic orbits for the circular disk or annular infinite well," ibid. 5, 519-526 (1998); R. W. Robinett, "Periodic orbit theory analysis of the circular disk or annular billiard," Am. J. Phys. 67, 67-77 (1999).
57. S. M. Reimann, M. Brack, A. G. Magner, and M. V. N. Murthy, "Applications of classical periodic orbit theory to circular billiards with small scattering centers," Surf. Rev. Lett. 3, 19-23 (1996); R. W. Robinett, "Energy eigenvalues and periodic orbits for the circular disk or annular infinite well," ibid. 5, 519-526 (1998); R. W. Robinett, "Periodic orbit theory analysis of the circular disk or annular billiard," Am. J. Phys. 67, 67-77 (1999).
58. R. Balian and C. Bloch, "Distribution of eigenfrequencies for the wave equation in a finite domain III. Eigenfrequency density oscillations," Ann. Phys. (Paris) 69, 76-160 (1972).
59. For a discussion and references, see M. V. Berry and K. E. Mount, "Semi-classical approximations in wave mechanics," Rep. Prog. Phys. 35, 315-397 (1972).
60. D. L. Kauffman, I. Kosztin, and K. Schulten, "Expansion method for stationary states of quantum billiards," Am. J. Phys. 67, 133-141 (1999). The authors use the exact results for the equilateral triangle as a benchmark to test their methods.
61. For a review of the connection between the Coulomb and 3D oscillator problems in classical and quantum mechanics, see A. K. Grant and J. L. Rosner, "Classical orbits in power law potentials," Am. J. Phys. 62, 310-315 (1994).
62. Y. Zarmi, "The Bertrand theorem revisited," Am. J. Phys. 70, 446-449 (2002).
63. D. S. Saxon, Elementary Quantum Mechanics (McGraw-Hill, New York, 1968), p. 182.