Tunneling in the Stark Effect. When you turn on an external electric field, the electron in an atom can, in principle, tunnel out, ionizing the atom. Question: Is this likely to happen in a typical Stark effect experiment? We can estimate the probability using a crude one-dimensional model, as follows. Imagine a particle in a very deep finite square well (Section 2.6).
(a) What is the energy of the ground state, measured up from the bottom of the well? Assume V₀ >> ћ²/ma².
(b) Now introduce a perturbation H' = ax (for an electron in an electric field E = -E_extî we would have a = e E_ext). Assume it is relatively weak (aa << ћ²/ma²). Sketch the total potential, and note that the particle can now tunnel out, in the direction of positive x.
(c) Calculate the tunneling factor γ (Equation 9.23), and estimate the time it would take for the particle to escape (Equation 9.29).
(d) Put in some reasonable numbers: V₀ = 20 eV (typical binding energy for an outer electron), a = 10^-10 m (typical atomic radius), E_ext = 7 x 10⁶ V/m (strong laboratory field), e and m the charge and mass of the electron. Calculate τ, and compare it to the age of the universe.