I Calculating Momentum Operator Matrix Elements from <φ|dH/dkx|ψ>

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TL;DR Summary

quantum physics in solids - momentum operator

Is there a relationship between the momentum operator matrix elements and the following:

<φ|dH/dk_x|ψ>

where k_x is the Bloch wave number

such that if I have the latter calculated for the x direction as a matrix, I can get the momentum operator matrix elements from it?

 

[azntoon]

Yes, it is possible to obtain the momentum operator matrix elements from the <φ|dH/dkx|ψ> matrix if you take the Fourier transform of the matrix. The Fourier transform of the matrix will give you the momentum operator matrix elements in the x direction, which you can then use to construct the momentum operator matrix in full.