pset5 - PH 216 Problem set 5 Due 5/4 MIDTERM REMINDER The...

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Unformatted text preview: PH 216, Problem set 5, Due 5/4 MIDTERM REMINDER: The course midterm starts after class wednesday 5/4, and concludes at 9 am Saturday 5/7. 1. In class we derived for a plane EM wave of frequency with linear polarization direction incident upon ^ a bound electron in state |i. (a) Define the oscillator strength fni Show that 2me ni |n| |i |2 . x ^ n abs = 4 2 ni |n| |i |2 ( - ni ), x ^ fni = 1, (b) Show that Under what conditions is this valid? dabs () = 2 c 2 e2 me c2 . 2. This problem provides a crude model for the photoelectric effect. Consider the hydrogen atom in its ground state (you may neglect the spins of the electron and proton). At time t = 0, the atom is placed in a high frequency uniform electric field that points in the z-direction, E(t) = E0 z sin t . ^ We wish to compute the transition probability per unit time that an electron is ejected into a solid angle lying between and + d. (a) Determine the minimum frequency, 0 , of the field necessary to ionize the atom. (b) Using Fermi's golden rule for the transition rate at first-order in time-dependent perturbation theory, obtain an expression for the transition rate per unit solid angle as a function of the polar angle of the ejected electron (measured with respect to the direction of the electric field). HINT: The matrix element that appears in Fermi's golden rule describes a transition of the negative-energy bound electron in its ground state to a positive-energy "free" electron. The wave function of the latter is actually quite complicated, since one cannot really neglect the effects of the long-range Coulomb potential. Nevertheless, you should simplify the computation by assuming the wave function of the ejected electron is a free-particle plane wave, with wave number vector (Note that the direction of corresponds to that of the ejected electron). k. k 6 (c) Integrate the result of part (b) over all solid angles to obtain the total ionization rate as a function of the frequency of the field. Determine the value of [in terms of 0 obtained in part (a)] for which the total ionization rate is maximal. 3. In class, we computed the (first) Born approximation to the scattering amplitude. Consider now the second Born approximation; i.e., the second term in the Born series. Compute the scattering amplitude in the forward direction, = 0, in the second Born approximation. 7 ...
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