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

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: $$$$|n,k\rangle = e^{ikx}u_{n,k}(x),\tag{1}$$$$ be a Bloch state, where $$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$$, which the authors claim can be written as: $$$$\langle n,k|x|n',k'\rangle = i\delta_{n,n'}\delta_{k,k'}\tag{2}\frac{\partial}{\partial k} + i\delta_{k,k'}X_{n,n'},$$$$ in which: $$$$X_{n,n'} = iN\int_0^ae^{i(k-k')x}u_{n,k}^*(x)\frac{\partial}{\partial k}u_{n',k'}(x) \;\mathrm{d}x.\tag{3}$$$$ This latter term is physically the inter-band coupling. Note that I am not sure what $$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 $$\partial_k$$.

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.