The spreading of a one-dimensional wavepacket solution of Schrödinger’s equation is related to the diffraction of light, as can be verified by considering the three-dimensional spreading of a wavepacket for an arbitrary dispersion relation. This investigation uncovers a special property of Schrödinger’s equation for a free particle: A wavepacket with initial spherical symmetry will preserve this symmetry in all Galilean reference frames. This property leads to a derivation of de Broglie’s postulate that wave number is proportional to momentum (or velocity). The application to non-Gaussian wavepackets and to Fraunhoffer diffraction also is discussed.
REFERENCES
1.
Richard L. Liboff, Introductory Quantum Mechanics (Addison-Wesley, Reading, MA, 1992), pp. 56–57, 157–161, 208–209.
2.
Albert Messiah, Quantum Mechanics (North-Holland, Amsterdam, The Netherlands, 1965), Vol. I, pp. 218–222.
3.
E.
MacKinnon
, “De Broglie’s thesis: A critical retrospective
,” Am. J. Phys.
44
, 1047
–1055
(1976
).4.
F.
Crawford
, “Derivation of the de Broglie relation from the Doppler effect
,” Am. J. Phys.
50
, 269
–270
(1982
).5.
Max Born and Emil Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), pp. 392–395, 738–746.
6.
Charles Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976), pp. 212–219.
7.
Steven
Weinberg
, “Eikonal method in magnetohydrodynamics
,” Phys. Rev.
126
, 1899
–1909
(1962
).8.
Ronald Adler, Maurice Bazin, and Menahem Schiffer, Introduction to General Relativity (McGraw-Hill, New York, 1965), pp. 220–224.
9.
George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, CA, 1995), pp. 439–453.
This content is only available via PDF.
© 2004 American Association of Physics Teachers.
2004
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
