Quantum Theory of Electrical Transport in a Magnetic Field. I

Abstract

The relationship between the quantum theory of electrical transport in the presence of a magnetic field and the corresponding Boltzmann transport equation is established for a simple system. The model consists of noninteracting free electrons being elastically scattered by an arbitrary potential in the presence of uniform electric and magnetic fields. Without the aid of a representation, the exact gauge-dependent Liouville equation for the density operator of this system is transformed into a completely gauge-independent equation satisfied by a new density operator. The new density operator is shown to give the correct current density, using the ordinary gauge-independent free-particle velocity operator. No approximations are made in performing the transformations, and the physical content of the new gauge-independent transport formalism is identical in all respects with that contained in the initial gauge-dependent equations. A new density matrix is defined which is essentially the Fourier sum of the matrix of the gauge-independent density operator. By considering the scattering potential resulting from a set of fixed impurity centers, it is shown that the diagonal elements of this new density matrix satisfy the ordinary time-dependent Boltzmann transport equation in which the spatial gradient term appears explicitly. The possible existence of a spatial variation in the average density of scatterers is also taken into account. The final equation for the quantum-mechanical distribution function represents the result obtained by treating the electric field and the effective scattering potential to the lowest possible order in which they contribute.