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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.:

Schenk's paper

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.

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$\hat{X} = i\partial_{k_x}$ in momentum space

is wrong. The left hand side is an operator, the right hand side is the representation of an operator when acting on functions, taken with respect to the scalar product with the basis $|\ p\rangle$. In particular we have $$ \langle x\,|\,\hat{x}\,|\,\phi\rangle = x\phi(x) $$ and $$ \langle p\,|\,\hat{x}\,|\,\phi\rangle = i\hbar\frac{\partial}{\partial k_x}\phi(p). $$ As you can see the two above are different things, as they arise after taking scalar products with different basis elements.

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  • $\begingroup$ Thank you for this answer. I completely agree. I'm still trying to understand how they go from $e\phi(x)$ [which equals $eF\hat x$] in Eq. (25) to it's representation as $i \frac{\partial}{\partial k_x}$ in Eq. (26) given that everything is in position space. Any ideas? $\endgroup$ – JTT Jul 22 '16 at 2:30

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