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H = p^2/2m*

, where m* is the effective mass in the conduction band and p^2 is the usual differential operator. I am not sure how this is derived rigourously. I suppose you solve the band structure and show that as a function of k the band structure is parabolic in k about the minimum of the conduction band:

E ≈ E0 + ħ^2k^2/2m*

But how do you rigorously go from this expression, which contains the wave numbers k = (kx,ky,kz) back to differential operators? I hope you understand my question.