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Bloch wave

  1. Nov 30, 2014 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations
    Bloch's theorem says given a periodic potential V(x)=V(x+a), the wave function is given by ψk(x) such that

    ψk( x + a ) = ei k ⋅ ( x + a ) uk( x + a ) = ei k ⋅ ( x + a ) uk( x ) = ei k ⋅ ( x + a ) [ ψk( x ) ei k ⋅ a ] = ei k ⋅ a [ ψk( x ) ]

    3. The attempt at a solution
    Find Φk( x+a ) and express in terms of Φk(x).

    Φk( 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.

    Φk( 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 ei k ⋅ a [ ψk( x ) ] ?
     
  2. jcsd
  3. Dec 1, 2014 #2
    I wanted to correct the last thing I said.

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