As students on solid state physics, we are all taught to use the periodic boundary condition, taking 1D as an example: $\psi(x)=\psi(x+L)$ where $L$ is the length of the 1D crystal.
My question is:
Why this boundary condition is acceptable? Only because we think we are dealing with the bulk, and the surface are not relevant too much?
How about other boundary condations, such as zero boundary condtion, grain boundary condition etc? If ignoring their complexity, can they get the same answer as the periodic boundary condition?
How about other artificial boundary conditions, for example, if I let $\psi(x)=2 \psi(x+L)$, could I get the same energy band?