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when I'm going through the online Density of States(DOS) deriving courses, I find that there seem to be 2 set of boundary condition which will lead to the same result.

Note: K is the one in p=(h/2π)K L is the length of the material

Conditon 1: Thinking of the material as a 1D/2D/3D potential well. And electron moving inside is considered as stand wave. So K=π/L*N N= 1,2,3,4,5.... Because it's stand wave, so the k=1 and k=-1 is the same wave.

see here at 18:08 https://www.youtube.com/watch?v=smKQtZUhtBQ&list=PLbMVogVj5nJSvhvgcBfT3e6HFFuhq2xqz&index=4

Condition 2: Considering the material as a 1D/2D/3D potential well. And electron moving inside is under Bon-Von Karman condition. So K=2π/L*N, N=±1,±2,±3,±4...... Because when the electron reach one end of the material it is considered to instantly moved the begin place. So K=1 is not the same as k=-1

See here http://electrons.wikidot.com/density-of-states

which one is more reasonable?

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In general, the fixed boundary conditions (condition 1) are physically more reasonable, as the material will end at some point. For calculations, however, it is often easier to deal with periodic boundary conditions (condition 2). For large $L$, the descriptions become equivalent, which is why pbc are mostly used.

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  • $\begingroup$ The periodic boundary conditions can also be understood as an infinite material with an infra-red cut-off for wavenumbers at around $k_0=2\pi/L$. I.e., you can trust the results to hold in an infinite material at wavenumbers $k \gg k_0$. $\endgroup$ – Void Oct 9 '17 at 11:14

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