Polarization in a semiconductor

A system of many identical particles is most conveniently described by the if) field operators (5). The generic form of the electronic charge density operator in a semiconductor is 

Taking into account only two nondegenerate bands, one can split the electronic i/j-operator into two parts, corresponding to the conduction and valence bands  

Consider the first term (the second one being just its hermitian conjugate). The matrix element of the operator J(r ) c(r) between the ground state 0) and some eigenstate s) of a single electron-hole pair is 

Here vn is the unit cell volume, u,(r) and uv(r) are the conduction and valence band extremes Bloch functions,68 and (re, r/,j is the envelope wavefunction of the electron-hole pair state s), normalized to the unit integral 

Now consider the interband matrix element of the charge density, corresponding to the first term in the expression (B.8). One is interested in the long-wavelength Fourier components of the density, given by 

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