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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  $$i\hbar\partial_t\psi_n(r,t)=E_n^0(k)\psi_n(r,t)+U(x)\psi_n(r,t)+\sum_{n\neq $$i\hbar\partial_t\psi_n(r,t)=E_n^0(k)\psi_n(r,t)+\phi(x)\psi_n(r,t)+\sum_{n\neq  n'} W_{n\,n'}\psi_{n'}(r,t)$$ where the effects 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 $W_{n\neq n'}$.