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\section{Kane's model}  Whereas Sze's approach could be described as a time-independent WKB method within the effective mass picture of transport, Kane instead applies time-dependent perturbation theory to the the Wannier equation  $$i\hbar\partial_t\psi_n(r,t)=E_n^0(k)\psi_n(r,t)-e\phi(x)\psi_n(r,t)+\sum_{n\neq n'} W_{n\,n'}\psi_{n'}(r,t)$$  where the effect of the electric field $-\partial_x\phi(x)$ is two-fold. First, by removing translation symmetry, it mixes the $k$-states within a band, and, secondly, it provides matrix elements $W_{n\neq n'}$ that mix bands together. Kane solves for the wavefunctions with $W$ set to zero, and then applies Fermi's golden rule to determine the transition rate between bands. In Kane's approach, the Wannier equation is expressed not as above, but rather in the crystal momentum basis.