# Free electron energy levels

nista
In many textbooks and sites of solid state physics energy levels for the free electron approximation in band calculations are displayed. The result is a collection of many parabolas, each of these parabolas being centered on a site of the reciprocal lattice.
This is not anyway the picture that should emerge from the free electron solution where we should find a single parabola with vertex at k=0.
Can anyone help me explain this discrepancy?
Thanks a lot

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I'm not quite sure what you're asking, but it sounds like you're comparing the extended zone scheme to the free electron theory? The extended zone scheme (or any zone scheme) assumes that there is a fixed periodicity in your real space lattice, ie. the potential V(x+a) = V(x) with a > 0. This gives a periodicity in reciprocal space, in 1D the periodicity is $$g = 2\pi/a$$. This gives us a periodicity in the energy bands, $$\varepsilon(k) = \varepsilon(k+g)$$. But the truly free electron model doesn't have just a finite periodicity, it has an infinitesimal periodicity, so V(x+a) = V(x) for any arbitrarily small a. If you put this into reciprocal space by taking the limit as a -> 0, you would find that $$g \rightarrow \infty$$, destroying the periodicity in reciprocal space.