For the initial and final states in valence and conduction bands, we have | i ⟩ ⟩ = Ψ Ψ v , k i , s i ( r ) and | f ⟩ ⟩ = Ψ Ψ c , k f , s f ( r ) , respectively and if the H ′ operator does not act on the spin, the electron stays in the same spin state and hence we can write the Bloch wavefunction of the initial and final states as Ψ Ψ v , k i ( r ) = 1 N Ω Ω 0 u n v , k i ( r ) e i k i ⋅ ⋅ r , Ψ Ψ c , k f ( r ) = 1 N Ω Ω 0 u n c , k f ( r ) e i k f ⋅ ⋅ r , where N is the number of unit cells with volume Ω Ω 0 . Calculating using these wavefunctions, and focusing on emission (photoluminescence) rather than absorption, we are led to the transition rate Γ Γ c v = 2 π π ℏ ℏ ( e A 0 m 0 ) 2 | ε ε ⋅ ⋅ μ μ c v ( k ) | 2 δ δ ( E c − − E v − − ℏ ℏ ω ω ) , where μ μ c v defined as the optical transition dipole moment is qualitatively the expectation value ⟨ ⟨ c | ( charge ) × × ( distance ) | v ⟩ ⟩ and in this situation takes the form μ μ c v = − − i ℏ ℏ Ω Ω 0 ∫ ∫ Ω Ω 0 d r ′ u n c , k ∗ ∗ ( r ′ ) ∇ ∇ u n v , k ( r ′ ) .