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In crystals, a periodic potential appears because the ions are arranged with a periodicity of their Bravais lattice. The periodic potential can be given by,
U(r + R) = U(r)  [2204a]
where,
R  The lattice vector.
This potential can be applied into the Schrödinger equation,
 [2204b]
The eigenstates ψ of the Hamitonian Ĥ in Equation 2204b can be chosen to have the form of a plane wave times a function with the periodicity of the Bravais lattice so that each eigenstate ψ satisfies,
ψ_{nk}(r) = e^{ik•r}u_{nk}(r)  [2204c]
where,
ħk is the crystal momentum (but not true momentum);
n  The quantum number (also called the band index, are equal to 1, 2, 3, . . .);
u_{nk}(r + R) = u_{nk}(r)  [2204d]
The quantum number (n) corresponds to the appearance of independent eigenstates of different energies but with the same k. Equation 2204d means that u_{nk}(r) has the same periodicity as the Hamitonian Ĥ for any lattice vector R of the Hamitonian.
An alternative formulation of Bloch’s theorem is that the eigenstates of a periodic Hamitonian Ĥ can be chosen so that for any lattice vector R of the Hamitonian, associated with each ψ is a wave vector k and thus each eigenstate satisfies,
ψ_{nk}(r+R) = e^{ik•R}ψ_{nk}(r)  [2204e]
