Putting this into the Schrödinger equation, we get: [ ℏ ℏ 2 ( k + K ) 2 2 m − − E k ] u ~ ~ k ( K ) + ∑ ∑ K ′ V ~ ~ ( K − − K ′ ) u ~ ~ k ( K ′ ) = 0 or rather: [ ℏ ℏ 2 ( k + K ) 2 2 m − − E k ] u ~ ~ k ( K ) + A a ∑ ∑ K ′ u ~ ~ k ( K ′ ) = 0