The Galilean covariance of quantum mechanics in the case of external fields

Textbook treatments of the Galilean covariance of the time-dependent Schrod inger equation for a spinless particle seem invariably to cover the case of a free particle or one in the presence of a scalar potential. The principa l objective of this paper is to examine the situation in the case of arbitr ary forces, including the velocity-dependent variety resulting from a vecto r potential. To this end, we revisit the 1964 theorem of Jauch which purpor ts to determine the most general form of the Hamiltonian consistent with '' Galilean-invariance," and argue that the proof is less than compelling. We then show systematically that the Schrodinger equation in the case of a Jau ch-type Hamiltonian is Galilean covariant, so long as the vector and scalar potentials transform in a certain way. These transformations, which to our knowledge have appeared very rarely in the literature on quantum mechanics , correspond in the case of electrodynamical forces to the ''magnetic" nonr elativistic limit of Maxwell's equations in the sense of Le Bellac and Levy -Leblond (1973). Finally, this Galilean covariant theory sheds light on Fey nman's ''proof" of Maxwell's equations, as reported by Dyson in 1990. (C) 1 999 American Association of Physics Teachers.