Verifying Bloch's Theorem with Φk(x)

[loginorsinup]

Homework Statement

Show that Φ k (x)= ∑_n f( x-na ) exp ( ikna ) is a Bloch wave, where f(x) is an arbitrary function with period a.

Homework Equations

Bloch's theorem says given a periodic potential V(x)=V(x+a), the wave function is given by ψ_k(x) such that

ψ_k( x + a ) = e^{i k ⋅ ( x + a )} u_k( x + a ) = e^{i k ⋅ ( x + a )} u_k( x ) = e^{i k ⋅ ( x + a )} [ ψ_k( x ) e^{i k ⋅ a} ] = e^{i k ⋅ a} [ ψ_k( x ) ]

The Attempt at a Solution

Find Φ_k( x+a ) and express in terms of Φ_k(x).

Φ_k( x+a ) = ∑_n f( [ x+a ]-na ) exp ( ikna ) = ∑_n f( x+a-na ) exp ( ikna ) = ∑_n f( x-[ n-1 ]a ) exp ( ik[ n-1 ]a ) exp ( ika ) = exp ( ika ) ∑_n f( x-[ n-1 ]a ) exp ( ik[ n-1 ]a

Since f is a periodic function.

Φ_k( x+a ) = exp ( ika ) ∑_n f( x-[ n-1 ]a ) exp ( ik[ n-1 ]a ) = exp ( ika ) ∑_n f( x-na ) exp ( ik[ n-1 ]a )

Now what can I do to say that this equals e^{i k ⋅ a} [ ψ_k( x ) ] ?

 

[loginorsinup]

I wanted to correct the last thing I said.

"Now what can I do to say that this equals ei k ⋅ a [Φk( x ) ] ?"