In 3D period lattice, can we separate variable and write potential as V=V(x)+V(y)+V(z)?Then we can reduce the 3D problems into 1D problems. I ask this question because in Solid State Physics books they often consider the 1D problems.
Yes, it is possible to separate variables and write potential as V=V(x)+V(y)+V(z) in a 3D period lattice. This is done frequently in solid state physics when dealing with periodic systems that have symmetry in certain directions. By separating the variables and writing the potential as the sum of one-dimensional potentials, the 3D problem can be reduced to several 1D problems and solved more easily.