It is a known fact that electrons in the conduction band of a semiconductor can (in certain scenario's) be described as having an approximate parabolic dispersion relation of the form $E_c(k) = E_c + \frac{\hbar^2k^2}{2m^*}$ where $m^*$ is the so called effective mass, which increases with the size of the band gap. This effective mass is often measured as a fraction of the standard electron mass $m_e$ and it can be much smaller: for example, in GaAs we have that $m^* = 0.067m_e$.
Now, the way I was taught, this was just a result of standard $\vec{k}\cdot{\vec{p}}$ perturbation theory, which somehow involves the crystal lattice structure and related periodicity to look at band structure near band extrema. This formulation turns out to be effective, and thus it is used.
But for me, the origin of this effective mass was never explained. Because isn't this amazing? Why does an electron suddenly behave as if it is much, much lighter, when placed into a lattice? I suppose it is a quantum effect, maybe having to do with interference? That is just a guess though. I would be very grateful if someone could help me gain some insight into how this effect comes about.
