Exploring the propagator of a particle in a box

American Journal of Physics

Volumn 71, Issue 1, 2003, Pages 55-63

  Fulling, S A ^a   Gunturk K S ^a  

^a TEXAS A AND M UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037237106     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1509415     Document Type: Article

References (17)

1. R. W. Robinett, "Visualizing the collapse and revival of wave packets in the infinite square well using expectation values," Am. J. Phys. 68 (5), 410-420 (2000).
2. D. F. Styer, "Quantum revivals versus classical periodicity in the infinite square well," Am. J. Phys. 69 (1), 56-62 (2001).
3. I. Sh. Averbukh and N. F. Perelman, "Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics," Phys. Lett. A 139, 449-453 (1989); "Fractional revivals of wave packets," Acta Phys. Pol. A 78, 33-40 (1990).
4. I. Sh. Averbukh and N. F. Perelman, "Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics," Phys. Lett. A 139, 449-453 (1989); "Fractional revivals of wave packets," Acta Phys. Pol. A 78, 33-40 (1990).
5. R. Bluhm, V. A. Kostelecký, and J. A. Porter, "The evolution and revival structure of localized quantum wave packets," Am. J. Phys. 64 (7), 944-953 (1996).
6. D. L. Aronstein and C. R. Stroud, "Fractional wave-function revivals in the infinite square well," Phys. Rev. A 55, 4526-4537 (1997).
7. in(x-y)/2π may diverge (in the literal, pointwise sense), but if we multiply it by ψ(y) and integrate term by term, we obtain the Fourier series of ψ(x); thus the sum converges in the distributional sense to the periodic extension of the Dirac delta function.
8. To avoid technical issues, assume that ψ(0,x) is continuous inside the interval and that its limit as x approaches -L is equal to ψ(0,L).
9. Whenever we write t=p/q, we shall assume that the fraction is in lowest terms. Thus the case of p and q both even does not need to be considered.
10. http://www.math.tamu.edu/̃fulling/box/. The figures in this paper were made with MATHEMATICA. Some computations also were done with MAPLE. MATHEMATICA produces compact graphics that are more suitable for publication. MAPLE produces larger graphics whose greater detail was sometimes easier to interpret on the screen and in printouts.
11. F. U. Chaos and L. Chaos, "Comment on 'The harmonic oscillator propagator,' by Barry R. Holstein [Am. J. Phys. 66 (7), 583-589 (1998)]," Am. J. Phys. 67 (7), 643 (1999); N. S. Thornber and E. F. Taylor, "Propagator for the simple harmonic oscillator," ibid. 66 (11), 1022-1024 (1998).
12. F. U. Chaos and L. Chaos, "Comment on 'The harmonic oscillator propagator,' by Barry R. Holstein [Am. J. Phys. 66 (7), 583-589 (1998)]," Am. J. Phys. 67 (7), 643 (1999); N. S. Thornber and E. F. Taylor, "Propagator for the simple harmonic oscillator," ibid. 66 (11), 1022-1024 (1998).
13. J. J. Duistermaat and V. W. Guillemin, "The spectrum of positive elliptic operators and periodic bicharacteristics," Invent. Math. 29, 39-79 (1975), especially pp. 45-56.
14. R. Bellman, A Brief Introduction to Theta Functions (Holt-Rinehart-Winston, New York, 1961).
15. H. Helson, Harmonic Analysis (Addison-Wesley, Reading, MA, 1983).
16. G. H. Hardy, Divergent Series (Chelsea, New York, 1991), Sec. 4.10.
17. Mathematical experts will note that Eq. (41) is not a complete triviality. It will hold in some reasonable topology, which may depend on the smoothness of ψ(0,x).