Exact energy eigenstates of the Coulomb-Stark Hamiltonian

We propose an approximation-free solution to the problem of an electron subject to a central Coulomb potential and a constant uniform electric field of high intensity i.e. the Stark-Coulomb problem. We introduce an algorithm to calculate the eigenstates of the Coulomb-Stark problem using a modest computational setup together with their proper normalization. Using this algorithm, calculating the Coulomb-Stark eigenstates is not harder than other special functions.

As a demonstrative example, we use our solution to calculate the time evolution of an electron initially in the ground state of hydrogen atom and is suddenly exposed to an external constant and uniform electric field. By calculating the transition amplitudes we find that there is a non-adiabatic part in the response of the electron wave function to the external field that cannot be captured in the commonly used adiabatic approximation schemes such as Keldysh formula. We show that this non-adiabatic response corresponds to a broad peak in the energy spectrum and is in the form of a sudden tunneling of a portion of the wave function to infinity shortly after the external field is turned on.