Quantum Mechanics and Superconductivity in a Magnetic Field

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.