Cross-post: Matrix elements <n,k|x|n',k'> for Bloch states

Question

Cross posted at Physics.SE

I believe this is just elementary QM, but I'm getting awfully confused. The question is drawn from this paper on Wannier-Stark localization (but is self-contained).

Let: \begin{equation} |n,k\rangle = e^{ikx}u_{n,k}(x),\tag{1} \end{equation} be a Bloch state, where $u_{n,k}(x+a)=u_{n,k}(x)$ is the periodic part. We wish to calculate the matrix element $\langle n,k|x|n',k'\rangle$, which the authors claim can be written as: \begin{equation} \langle n,k|x|n',k'\rangle = i\delta_{n,n'}\delta_{k,k'}\tag{2}\frac{\partial}{\partial k} + i\delta_{k,k'}X_{n,n'}, \end{equation} in which: \begin{equation} X_{n,n'} = iN\int_0^ae^{i(k-k')x}u_{n,k}^*(x)\frac{\partial}{\partial k}u_{n',k'}(x) \;\mathrm{d}x.\tag{3} \end{equation} This latter term is physically the inter-band coupling. Note that I am not sure what $N$ is, as I can't see it defined in the paper. Also, the equation for $X_{n,n'}$ is as written in the paper, with $\partial_k$ acting on $u_{n',k'}$.

Question: show the above.

I've tried writing $x=-i\partial_k e^{ikx}$ and integrating by parts, but do not get their expression. I am also generally confused as to how a matrix element $\langle n,k|x|n',k'\rangle$ (which should be a number...?) can be equal to a derivative $\partial_k$.

Answer 1

If you look at the paper, they cite the work of M. Saitoh in J. Phys. C: Solid State Phys. 5, 914 (1972) for the matrix element.

Saitoh writes in Eqs 6 and 7 $$ X_{j{\bf p},c{\bf k}} = \int {\rm d}^3 r \psi_{j{\bf p}}^* ({\bf r}) x \psi_{c{\bf k}}({\bf r}) = i\delta_{jc} \delta_{\bf pk} \frac {\partial} {\partial k_x} $$ which for sure is abuse of notation since the elements of ${\bf X}$ clearly should be scalars. However, Saitoh then proceeds to use the action of ${\mathbf X}$ onto the eigenvalue equation in his Eq 5, and proceeds by integrating the resulting differential equation.

You should thus not think of the elements of ${\bf X}$ as scalars, but as operators: ${\bf X}$ only makes sense when applied to a function, such as $\sum_{j p_x} X_{j{\bf p},c{\bf k}} A_j({\bf p})$ in Saitoh's Eq 5 or $\sum_{n'} X_{nn'} C(n',k)$ in Leo and McKinnon's article.