Quantum Numbers for Bloch Electrons in a Magnetic Field

Landau functions characterize the motion of an electron in a plane perpendicular to a magnetic field. They have the degeneracy parameter ky. This parameter is resolved into four quantum numbers by a periodic crystalline potential. Two are phase angles; two are restricted to a finite set of values. Only one of the latter is a degeneracy parameter. A constructive proof is given for this proposition; it is valid under wide conditions, the only restriction being the famous ‚rationality condition’︁. The two discrete quantum numbers have as their range the numerator and denominator of the representative fraction. Group theory yields three of the numbers; the fourth labels the subbands, that is the equivalent representations within the spectrum. In the Peierls‐Onsager low field approach, the same four quantum numbers arise from one (two‐dimensional) Bloch band.