Energy and polarizability of atoms in a weak magnetic field: Hydrogenic atoms

The ground-state wave function of a hydrogenic atom is approximated by ${\psi }_0\left(\mathrm{H}\right)=\exp \{-{\left[k{(x^2+y^2+s{z}^2)}^{\frac{1}{2}}\right]}^{\frac{1}{p}}\}$, and the expectation value of the Hamiltonian, including diamagnetic and spin-orbit terms in a magnetic field $H_z$, is minimized with respect to $k$, $s$, and $p$ so as to determine these $H_z$-dependent parameters, and the energy is evaluated. In the presence of an infinitesimal electric field ${\mathcal{E}}_t$, in the $t$ direction, the wave function is taken to be ${\psi }_0\left(\mathrm{H}\right)(1+bt+c\rho t)$, where $\rho ={(x^2+y^2+s{z}^2)}^{\frac{1}{2}}$. The energy is minimized with respect to $b$ and $c$, and the coefficient of ${{\mathcal{E}}_t}^2$ is identified as $-\left(\frac{1}{2}\right){\alpha }_t$, the polarizability component, which contains diamagnetic and weakly spin-dependent terms. Comparison with experiments of Castner and Lee on donors in Si is briefly discussed.