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Using translational properties of the Hamiltonian, a standard form is derived for the wave functions describing conduction electrons moving in a uniform magnetic field. This is used to obtain a solution of the Schrödinger equation in order to examine the magnetic energy levels. For non-overlapping bands the notion of effective mass is shown adequate (neglecting level broadening) but in an overlap region neighbouring bands interact strongly and distort the single band levels. A brief application to the diamagnetism of metals is considered.
P.G. Harper (Sat,) studied this question.