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2. Homework Equations

ψ_k (x)=u_k (x)e^ikx , solution to the SE for the periodic potential.

My understanding of it is that the e^ikx is the 'envelope' for the solution and takes the shape of the solution of the SE for an equivalent single well and the u_k(x) is the periodic function that modulates the wavefunction with the same periodicity of the lattice.

So for the lower energy band, is the envelope function the familiar 1/2 wave for all states in the lower band and the 1 wavelength wavfunction the envelope for all the states in the higher band?