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The wave function for an electron in a crystal lattice is modeled by a Bloch wave. A Bloch wave is a function with the periodicity of the lattice multiplied times a complex exponential function. This exponential function has a wave vector k, called the crystal momentum, which can have any value. To my knowledge this k independent of the configuration of the lattice. Assume the potential of the atoms in the lattice are weak. If we take the potential to zero, the piece of the Bloch wave that holds the lattice periodicity disappears and we are left with the exponential (i.e. a plane wave).

Question:

Increasing the value of the crystal momentum by a reciprocal lattice vector does not change the wave function. But obviously, a function modulated by e^(i⋅16⋅π⋅r) would be much different than a function modulated by e^(i⋅2⋅π⋅r). The spacial frequency of the former is much higher. How could this not make a difference? I know it has something to do with breaking of symmetry, and a Bloch wave not being an eigenstate of the momentum operator. Perhaps the wavefunction is only sampled periodically so it doesn't make a difference? I'd like to see the mathematical justification, but couldn't find a clear explanation in Ashcroft or on the web. I'm mostly interested in how two functions with a different crystal momentum could mathematically be the same. Thank you.