Quantum Theory of Transport in a Magnetic Field

A quantum-mechanical theory of transport of charge for an electron gas in a magnetic field is presented that takes account of the quantization of the electron orbits. A transport equation for the necessary elements of the density matrix is developed for arbitrary values of the magnetic field. The scattering is taken to be elastic and is treated only in the Born approximation. The effect of both the magnetic and electric fields on the collisions is taken into account. The influence of the latter has been neglected in previous theories. It is proven, however, to be important in high magnetic fields. It is established that the effect of a transverse electric field on the scattering can be described as the tendency of the electrons to "relax" to a distribution characteristic of thermal equilibrium in the presence of the electric field.Previous theories of transport for large Hall angles are consistent with this theory. They can be obtained as a special solution, found by iteration, of this transport equation.The special case of isotropic scattering has been considered in detail. In this case it is demonstrated that for small enough magnetic fields the usual classical result obtains.