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- W2089563660 abstract "The Hamiltonian of a Bloch electron in a static magnetic field is $H=frac{1}{2}{mathrm{P}}^{2}+V(mathrm{r})$, where $V(mathrm{r})$ is the periodic potential, $mathrm{P}=mathrm{p}+frac{mathrm{A}}{c}$, and A is the vector potential giving rise to the magnetic field $mathcal{H}$. We consider the case of a nondegenerate band $m$. It is then shown that, with an error vanishing with $mathcal{H}$ like ${mathcal{H}}^{N+1}$ ($N$ arbitrary), the eigenstates of $H$ can be calculated from an equivalent Hamiltonian ${overline{H}}_{m}(mathrm{P})$ 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 ${P}^{ensuremath{alpha}}$. (3) It has the periodicity property ${overline{H}}_{m}(mathrm{P}+mathrm{K})={overline{H}}_{m}(mathrm{P})$, where K is an arbitrary reciprocal lattice vector. (4) It can be written as a series ${overline{H}}_{m}(mathrm{P})={{ensuremath{Sigma}}_{i=0}}^{N}{s}^{i}{overline{H}}_{m;i}(mathrm{P})$ where $sensuremath{equiv}frac{mathcal{H}}{c}$ and the functions ${overline{H}}_{m;i}(mathrm{P})$ are completely symmetrized in the noncommuting operators ${P}^{ensuremath{alpha}}$. Properties (3) and (4) can also be summarized in the equations ${overline{H}}_{m}(mathrm{P})={ensuremath{Sigma}}_{l}{a}^{(l)}ifmmodetimeselsetexttimesfi{}mathrm{exp}[i{mathrm{R}}^{(l)}ifmmodecdotelsetextperiodcenteredfi{}mathrm{P}]$, where the ${mathrm{R}}^{(l)}$ are lattice vectors and the ${a}^{(l)}$ can be expanded as ${a}^{(l)}={{ensuremath{Sigma}}_{i=0}}^{N}{s}^{i}{{a}_{i}}^{(l)}$. An algorithm is given for the construction of the ${overline{H}}_{m;i}$ and carried through for $i=0, 1, 2$. The formalism is not restricted to the neighborhood of the bottom and top of the band. We believe that the equivalent Hamiltonian ${overline{H}}_{m}(mathrm{P})$ provides a sound basis for a discussion of wave functions and energy levels of Bloch electrons in a magnetic field." @default.
- W2089563660 created "2016-06-24" @default.
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- W2089563660 date "1959-09-15" @default.
- W2089563660 modified "2023-10-01" @default.
- W2089563660 title "Theory of Bloch Electrons in a Magnetic Field: The Effective Hamiltonian" @default.
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- W2089563660 doi "https://doi.org/10.1103/physrev.115.1460" @default.
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