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Initial phase and free-particle wave packet evolution
2.M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 370–458.
3.V. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge U. P., Cambridge, 1984).
5.T. M. Pritchett and A. D. Trubatch, “A differential formulation of diffraction theory for the undergraduate optics course,” Am. J. Phys. 72, 1026–1034 (2004).
6.D. Bohm, Quantum Theory (Dover, New York, 1989).
10.There are restrictions and limitations to representing the transmission of a transparent object of finite thickness as a single phase value that reflects the optical path length. See, for example, E. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970)
10.and M. Totzeck and B. Kuhlow, “Validity of the Kirchhoff approximation for diffraction by weak phase objects,” Opt. Commun. 78, 13–19 (1990). For our purposes, for which we are concerned primarily with the qualitative behavior of free-particle wave packets by analogy with optical systems, the approximation is sufficiently accurate.
11.F. D. Feiock, “Wave propagation in optical systems with large apertures,” J. Opt. Soc. Am. 68, 485–489 (1978).
12.J. P. Corones and R. J. Krueger, “Higher-order parabolic approximations to time-independent wave equations,” J. Math. Phys. 24, 2301–2304 (1983);
13.H. G. Booker, J. A. Ratcliffe, and D. J. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London, Ser. A 242, 579–607 (1950);
13.R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962);
14.A. Hewish, “The diffraction of radio waves in passing through a phase-changing ionosphere,” Proc. R. Soc. London, Ser. A 209, 81–96 (1951).
17.T. L. Beach and R. V. E. Lovelace, “Diffraction by a sinusoidal phase screen,” Radio Sci. 32, 913–921 (1997).
20.Note that a quadratic phase in the spatial coordinate has been observed in the wave function of a Bose-Einsten condensate, J. E. Simsarian, J. Denschlag, M. Edwards, C. W. Clark, L. Deng, E. W. Hagley, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Imaging the phase of an evolving Bose-Einstein condensate wave function,” Phys. Rev. Lett. 85, 2040–2043 (2000).
21.I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, New York, 1980), pp. 337–496.
claims generality for asymptotic Gaussian behavior in the evolution of free-particle wave packets. M. A. Andrews
, “The evolution of free wave packets
,” arXiv:0801.0188 points
out that the claim is not true using the counterexample of an odd wave packet. (Andrews also summarizes several useful general properties of spreading wave packets and is a potential source of additional classroom exercise material.)
22.An additional counterexample appears in M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
24.W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. P., Cambridge, 1994), pp. 255–257.
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