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I am trying to figure out the width of bands in a 1-dimensional lattice. Here is a short derivation from the book I am reading: if we approximate the free electrons as being in a square well then the energy levels are ## \frac{\pi^2 \hbar^2 n^2}{2mL^2}##. If there are ##N## ions separated by a distance ##a## in the metal, then ##N = L/a##. The number of energy levels in the band is ##N##, so $$E_\text{max} \approx \frac{\pi^2 \hbar^2 N^2}{2mL^2} = \frac{ \pi^2 \hbar^2}{2 ma^2},$$

since the lowest energy is nearly 0. Thus, the width of the band depends not on ##N##, but on ##a##.

I'm a bit confused about what the calculation actually says. My understanding was that there are many bands, and that the energy width of each successive band increases. Is this the calculation for the width of the lowest energy band? Or is this just a nonsense derivation?

Thank you for the help. In case you are curious, the source isEisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, which I have found to be a good source for nearly everything except results requiring math. (In fact, nearly every calculation requiring more than a page is relegated to a ~100 page long appendix.)

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# I Basic band theory question, free electron model

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