Ground state energy of the hydrogen atom inside penetrable spherical cavities; variational approach

In this work we calculate the ground state energy of the hydrogen atom confined in a sphere of penetrable walls of radius Rc. Inside the sphere the system is subject to a Coulomb potential, whereas outside of it the potential is a finite constant V0. The energy is obtained as a function of  Rc and V0  by means of the Rayleigh-Ritz variational method, in which, the trial function is proposed as a free particle wave function within a finite square well potential but including an exponential factor that takes into account the electron-nucleus Coulomb attraction. For an impenetrable sphere, , the energy grows fast as Rc approaches zero. On the other hand, when the height of the barrier V0 is finite, the energy increases slowly as Rc goes to zero. We also compute the Fermi contact term, nuclear magnetic screening, polarizability, pressure and tunneling as a function of  Rc and V0. As expected, these physical quantities approach the corresponding values of the free hydrogen atom as Rc grows. We also discuss the pressure-induced ionization of the hydrogen atom. The present results are found in good agreement with those previously published in the literature.