Scaling properties of composite information measures and shape complexity for hydrogenic atoms in parallel magnetic and electric fields

The scaling properties of various composite information-theoretic measures (Shannon and R\'enyi entropy sums, Fisher and Onicescu information products, Tsallis entropy ratio, Fisher-Shannon product and shape complexity) are studied in position and momentum spaces for the non-relativistic hydrogenic atoms in the presence of parallel magnetic and electric fields. Such measures are found to be invariant at the fixed values of the scaling parameters given by $s_1 = B \hbar^3(4\pi\epsilon_0)^2 / (Z^2m^2e^3)$ and $s_2 = F \hbar^4(4\pi\epsilon_0)^3 / (Z^3e^5m^2)$. Numerical results which support the validity of the scaling properties are shown by choosing the representative example of the position space shape complexity. Physical significance of the resulting scaling behaviour is discussed.