Bloch theorem?

[phyCIAA]

Hi guys I have these question, please someone help me to answer?
What is Bloch theorem? Why we use it? Explain?
What is the consequence of this theorem?

[kanato]

The Bloch theorem allows us to partially diagonalize a periodic Hamiltonian. A periodic Hamiltonian has a discrete translational symmetry, you can translate by any lattice vector and get the same Hamiltonian back. Because of this symmetry, there will be a good quantum number for the wavefunction associated with it, usually called the pseudomomentum and given the letter k. This allows us to take a problem of N atoms in M cells and solve M independent problems with N atoms, rather than to solve the whole crystal of N*M atoms. The consequences of Bloch's theorem are pervasive in solid-state physics, so it is difficult to give a concise summary.

[phyCIAA]

Thank you for your response.