  The influence of a magnetic field on superconductivity is usually described
either phenomenologically, using Ginzburg-Landau theory, or semiclassically
using Gor'kov theory. In this article we discuss the influence of magnetic
fields on the mean-field theory of the superconducting instability from a
completely quantum mechanical point of view. The suppression of
superconductivity by an external magnetic field is seen in this more physically
direct picture to be due to the impossibility, in quantum mechanics, of
precisely specifying both the center-of-mass state of a pair and the individual
electron kinetic energies. We also discuss the possibility of novel aspects of
superconductivity at extremely strong magnetic fields where recent work has
shown that the transition temperature may be enhanced rather than suppressed by
a magnetic field and where a quantum treatment is essential.
