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We diagonalize the 8-band Kane Hamiltonian with a proper inclusion of theinterband matrix elements of the scalar and vector potentials. This leads,among other results, to a modification of the conventional expression for thespin-orbit coupling (SOC) strength in narrow-gap semiconductors with the zincblende symmetry. We find that in GaAs, at low temperatures, the correctexpression for the SOC strength is actually twice as large as usuallyconsidered. In InSb it is $1.76$ times larger. We also provide a propertreatment of the interband matrix elements of the position operator. We showthat the velocity operator in a crystal should be defined as a time-derivativeof a fictitious position operator rather than the physical one. We compute theexpressions for both these position operators projected to the conduction bandof the 8-band Kane model. We also derive an expression for the velocityoperator and demonstrate that the SOC strength in it differs from the SOCstrength in the Hamiltonian. The ratio between them is not equal to $1$, as itis often assumed for the Rashba model. It does not equal $2$ either. Thecorrect result for this ratio is given by a rational function of the parametersof the model. This function takes values between $4(23+3\sqrt{2})/73\approx1.49$ and $2$. Our findings modify a vast number of research results obtainedusing the Rashba model and provide a path for a consistent treatment of thelatter in future applications.

Position operators and interband matrix elements of scalar and vector potentials in the 8-band Kane model