For the position operator $\hat X$:
$\hat X \rightarrow x$ in position space
$\hat X \rightarrow i \frac{\partial}{\partial k_x}$ in momentum space
Question:
Can $\hat X \rightarrow i \frac{\partial}{\partial k_x}$ be used in position space?
Example: If $\Psi(x)=e^{-ikx}$ then both representations of $\hat X$ yield $\hat X\ | \Psi(x)>=x\ |\Psi(x)> $.
Edit: As I've gotten a better understanding, it seems that the $i \frac{\partial}{\partial k_x}$ representation of $\hat X$ can be used in position space but only for specific functions?
Background on why I want to do this:
The matrix element for the band-to-band tunneling of electrons from the valence band (VB) to the conduction band (CB) of a semiconductor is given by
(1) $M =<\Psi_{CB}\ | \ q\phi(x) \ |\ \Psi_{VB}>$,
where $q\phi(x)$ is the potential along the tunneling path and $\Psi$ is the wavefunction in the CB and VB. ($\Psi=u_k(r)f_k(r)$, where $u$ is the Bloch function and $f$ is the envelope wavefunction.) Assuming a constant electric field ($F$) across the tunneling region,
(2) $M =<\Psi_{CB}\ | \ qF\hat X \ |\ \Psi_{VB}>$.
Eq. (1) above matches Eq. (25) from Schenk et al.:
Expressing $\hat X$ as $ i \frac{\partial}{\partial k_x}$ (shown in Eq. (26) in Schenk), allows the matrix element to be written in the form of Eq. (27), which is convenient for comparison to matrix elements of other potentials.
Reference: A. Schenk, M. Stahl, and H.-J. Wünsche, “Calculation of Interband Tunneling in Inhomogeneous Fields,” phys. stat. sol. (b), vol. 154, no. 2, pp. 815–826, Aug. 1989.