De Broglie waves meet Schrodinger's equation

摘要

For a quantum-mechanical free particle, de Broglie's hypothesis relates group velocity v_g to phase velocity v_p via v_gv_p = c~2. However, the relation has not yet been grounded in the usual definition v_g = dω)/dk. This article provides the missing dispersion relation to evaluate dω/dk. Direct solution of Schrodinger's equation in one dimension yields the dispersion relation co = (h/ 2 m)k~2, from which follows the relation vg = 2v_p Further implications of the dispersion relation are explored. Because the free-particle Schrodinger equation is a heat equation with an imaginary heat constant, its Green's function is a linear chirp that fills all space starting the instant after a deltafunction initial condition. Particle localization is possible only when the Green's function is spatially convolved (at a time t) with a substantially band-limited initial wavefunction. The faster expansion of a wavefunction when it is initially narrow could be a realization of the uncertainty principle. All of this Schrodinger-equation analysis, however, makes a disconcerting break with relativistic quantum mechanics. Free-particle solution of the Klein-Gordon equation vindicates V_gV_p = c~2. The nonrelativistic noncorrespondence of Schrodinger and Klein-Gordon, or more simply of v_g = 2v_p with v_gv_p = c~2, leads us to ask which can be trusted in the nonrelativistic velocity limit.