In chapter 7 of Jackson’s Classical Electrodynamics (page 323) he speaks of ‘‘rms deviations from mean, $Delta x$ and $Delta k$, defined with respect to $|u(x,0)|^2$ and $|A(k)|^2$’’. I do not understand how do I calculate $Delta x$ and $Delta k$?
What Jackson calls the root-mean-square (rms) is really the standard deviation$$Delta xequivsigma_x=sqrt{langle (x-langle xrangle)^2rangle}=sqrt{langle x^2rangle-langle xrangle^2},$$
and similarly for $Delta k$. The rms is really $sqrt{langle x^2rangle}$ but physicists sometimes use both terms as the same, as can be seen here, here, and mentioned here at the end.
To find $Delta x$ and $Delta k$ in terms of $|u(x,0)|^2$ and $|A(k)|^2$ respectively, notice the expressions for $u(x,0)$ and $A(k)$$$u(x,0)=frac{1}{sqrt{2pi}}int_{-infty}^infty A(k)e^{ikx}dk$$$$A(k)=frac{1}{sqrt{2pi}}int_{-infty}^infty u(x,0)e^{-ikx}dx.$$$u(x,0)$ and $A(k)$ are Fourier transforms of each other, and basically $|A(k)|^2$ tells you how much is the plain wave with wavenumber $k$ contributing to the wavepacket $u(x,0)$ and similarly for $|u(x,0)|^2$ and $A(k)$. So when calculating the means $langle (x-langle xrangle)^2rangle$ and $langle (k-langle krangle)^2rangle$, $|u(x,0)|^2$ and $|A(k)|^2$ act as probability density functions:
$$Delta x^2=langle (x-x_0)^2rangle=int (x-x_0)^2p(x)dx$$$$Delta k^2=langle (k-k_0)^2rangle=int (k-k_0)^2p(k)dk$$
with $p(x)=|u(x,0)|^2$ and $p(k)=|A(k)|^2$.