Abstract
The Hamiltonian of a Bloch electron in a static magnetic field is , where is the periodic potential, , and A is the vector potential giving rise to the magnetic field . We consider the case of a nondegenerate band . It is then shown that, with an error vanishing with like ( arbitrary), the eigenstates of can be calculated from an equivalent Hamiltonian with the following properties: (1) It is a one-band Hamiltonian, obtained by transforming away all relevant interband matrix elements. (2) It depends only on the gauge-covariant operators . (3) It has the periodicity property , where K is an arbitrary reciprocal lattice vector. (4) It can be written as a series where and the functions are completely symmetrized in the noncommuting operators . Properties (3) and (4) can also be summarized in the equations , where the are lattice vectors and the can be expanded as . An algorithm is given for the construction of the and carried through for . The formalism is not restricted to the neighborhood of the bottom and top of the band. We believe that the equivalent Hamiltonian provides a sound basis for a discussion of wave functions and energy levels of Bloch electrons in a magnetic field.
- Received 26 February 1959
DOI:https://doi.org/10.1103/PhysRev.115.1460
©1959 American Physical Society