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ck00
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The classical textbook, Introduction to solid state physics by Charles Kittle said:
"If we encounter a Bloch function written as ψ_{k’}(r)=exp(i{k’}r) u_{k’}(r), with k’ outside the first zone, we may find a suitable reciprocal lattice vector G such that k=k’+G lies within the first Brillouin zone. Then
ψ_{k’}(r)=exp(ik’r) u_{k’}(r)=exp(ikr) [exp(-iGr) u_{k’}(r)]
=exp(ikr) u_k(r)=ψ_k(r)"
I wonder why exp(-iGr) u_{k’}(r)=u_k(r), how to derive this relation?
"If we encounter a Bloch function written as ψ_{k’}(r)=exp(i{k’}r) u_{k’}(r), with k’ outside the first zone, we may find a suitable reciprocal lattice vector G such that k=k’+G lies within the first Brillouin zone. Then
ψ_{k’}(r)=exp(ik’r) u_{k’}(r)=exp(ikr) [exp(-iGr) u_{k’}(r)]
=exp(ikr) u_k(r)=ψ_k(r)"
I wonder why exp(-iGr) u_{k’}(r)=u_k(r), how to derive this relation?
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