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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 [tex]ψ_{k’}(r)=exp(i{k’}r) u_{k’}(r)[/tex], 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

[tex]ψ_{k’}(r)=exp(ik’r) u_{k’}(r)=exp(ikr) [exp(-iGr) u_{k’}(r)][/tex]

[tex]=exp(ikr) u_k(r)=ψ_k(r)[/tex]"

I wonder why [tex]exp(-iGr) u_{k’}(r)=u_k(r)[/tex], how to derive this relation?

"If we encounter a Bloch function written as [tex]ψ_{k’}(r)=exp(i{k’}r) u_{k’}(r)[/tex], 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

[tex]ψ_{k’}(r)=exp(ik’r) u_{k’}(r)=exp(ikr) [exp(-iGr) u_{k’}(r)][/tex]

[tex]=exp(ikr) u_k(r)=ψ_k(r)[/tex]"

I wonder why [tex]exp(-iGr) u_{k’}(r)=u_k(r)[/tex], how to derive this relation?

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