I am studying on the Ashcroft- Mermin Solid State Physics and at chap8 pag 141 it is said that for given n, the eigenstates and eigenvalues are periodic functions of k in the reciprocal lattice that is $$E_{n, k+K} = E_{n,k}$$ and same for $\psi$. I get the argument about the wave function, but not the one about energy. For example the energy of a free electron (the potential is 0 so it's periodic and we can apply bloch and develop wave functions with it) is different, $\hbar²k²/2m ≠ \hbar²(k-K)²/2m$. Can someone tell me where I am wrong?
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1$\begingroup$ What is the period of the "periodic" potential $V=0$? $\endgroup$– probably_someoneCommented May 1, 2018 at 20:32
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1$\begingroup$ The potential U=0 satisfies $U(r+R)=U(r)$ for every R , so the Bloch Theorem may be applied (it is also used as an example of OK potential in the text, but it goes not further) $\endgroup$– LenzCommented May 1, 2018 at 20:36
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$\begingroup$ $K$ is defined by one of the reciprocal lattice vectors. What are these lattice vectors for the potential $V=0$? $\endgroup$– probably_someoneCommented May 1, 2018 at 20:44
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$\begingroup$ Along the same lines, what is the first Brioullin zone for the potential $V=0$? Canonically, you would define Bloch waves by restricting to this region. $\endgroup$– probably_someoneCommented May 1, 2018 at 20:47
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