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≫ℏ2/ma2. *Hint:* This is just the ground state energy of the *infinite* square well (of width 2a).
(b) Now introduce a perturbation H′=−αx (for an electron in an electric field E=−Eexti^ we would have α=eEext). Assume it is relatively weak (αa≪ℏ2/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). *Answer:* γ=8mV03/3αℏ, τ=(8ma2/πℏ)e2γ.
(d) Put in some reasonable numbers: V0=20 eV (typical binding energy for an outer electron), a=10−10 m (typical atomic radius), Eext=7×106 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.