Excitations from a filled Landau level in the two-dimensional electron gas

We consider an interacting two-dimensional electron system, with a uniform positive background, in a strong perpendicular magnetic field at zero temperature, under conditions where an integral number of Landau levels are filled and the Coulomb energy $\frac{e^2}{\epsilon l_0}$ is smaller than the cyclotron energy $\hslash {\omega }_c$. The elementary neutral excitations may be described alternatively as magnetoplasma modes, or as magnetic excitons—i.e., a bound state of a hole in a filled Landau level and one electron in an otherwise empty level—and they are characterized by a conserved wave vector $\overrightarrow{\mathrm{k}}$. The dispersion relations may be calculated exactly, to first order in $\frac{\frac{(e^2}{\epsilon l_0)}}{\hslash {\omega }_c}$, for the lowest magnetoplasmon band, which comes in to the cyclotron frequency at $k=0\mathrm{}$. We also calculate the spin-wave dispersion relations for the case where one spin state of a Landau level is completely occupied, and we discuss qualitatively the exciton spectrum for a partially filled Landau level, under the conditions of the fractional quantized Hall effect.