Question

(5) (40 points) The ground state (n = 1, l = 0, as in SN 2nd edition, problem 5.35) of a hydrogen atom is subjected to the time-dependent potential \begin{equation} V(x, t) = V_0 \cos(kz - \omega t). \end{equation} Using time-dependent perturbation theory, obtain an expression for the transition rate at which the electron is emitted with momentum $p$. Show, in particular, how you may compute the angular distribution of the ejected electron (in terms of $\theta$ and $\phi$ with respect to the z axis). Discuss briefly the similarities and the differences between this problem and the (more realistic) photoelectric effect. The final free electron wave function should be \begin{equation} \psi_f(x) = \frac{1}{L^{3/2}}e^{i p \cdot x/\hbar}, \end{equation} and with L very large, show that the observable effects are independent of L.

          (5) (40 points) The ground state (n = 1, l = 0, as in SN 2nd edition, problem 5.35) of a hydrogen atom is subjected to the time-dependent potential \begin{equation} V(x, t) = V_0 \cos(kz - \omega t). \end{equation} Using time-dependent perturbation theory, obtain an expression for the transition rate at which the electron is emitted with momentum $p$. Show, in particular, how you may compute the angular distribution of the ejected electron (in terms of $\theta$ and $\phi$ with respect to the z axis). Discuss briefly the similarities and the differences between this problem and the (more realistic) photoelectric effect. The final free electron wave function should be \begin{equation} \psi_f(x) = \frac{1}{L^{3/2}}e^{i p \cdot x/\hbar}, \end{equation} and with L very large, show that the observable effects are independent of L.

(5) (40 points) The ground state (n = 1, l = 0, as in SN 2nd edition, problem 5.35) of a hydrogen atom is subjected to the time-dependent potential
V(x, t) = V0 cos(kz - ω t).
Using time-dependent perturbation theory, obtain an expression for the transition rate at which the electron is emitted with momentum p. Show, in particular, how you may compute the angular distribution of the ejected electron (in terms of θ and ϕ with respect to the z axis). Discuss briefly the similarities and the differences between this problem and the (more realistic) photoelectric effect. The final free electron wave function should be
(x) = (1)/(L^3/2)e^i p · x/ħ,
and with L very large, show that the observable effects are independent of L.

Added by Shannon D.

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