Matrix elements of non-normalizable states

[taishizhiqiu]

Although strictly quantum mechanics is defined in $L_2$ (square integrable function space)， non normalizable states exists in literature.

In this case, textbooks adopt an alternative normalization condition. for example, for $\psi_p(x)=\frac{1}{2\pi\hbar}e^{ipx/\hbar}$
$\langle\psi_p|\psi_{p'}\rangle=\delta(p-p')$

However, it is not easy calculating matrix elements this way. For example, how to calculate
$A(k)=i\langle u(k)|\partial_k|u(k)\rangle$
$A(k)$ is actually berry connection in solid state band theory and $u(k)$ is periodic part of bloch wave function.

Can anyone tell me how to define this matrix elements?

 

[vanhees71]

I do not understand your notation, defining $A(k)$. It simply doesn't make any sense to me. Where does this come from?

 

[taishizhiqiu]

According to bloch theorem, wave function in crystals should be like $\psi_k(x)=e^{ikx}u_k(x)$, where $u_k(x+a)=u_k(x)$ and $a$ is lattice constant.

So $\langle u(k)|\partial_k|u(k)\rangle$ should be something like $\int u^*_k(x)\partial_k u_k(x)dx$, although it doesn't make sense because this integral is infinite.

$A$ is berry connection where the adiabatic parameter is $k$.(https://en.wikipedia.org/wiki/Berry_connection_and_curvature)This quantity is heavily used in topological insulators