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 zdirection, 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 firstorder in timedependent 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 negativeenergy bound electron in its ground state to a positiveenergy "free" electron. The wave function of the latter is actually quite complicated, since one cannot really neglect the effects of the longrange Coulomb potential. Nevertheless, you should simplify the computation by assuming the wave function of the ejected electron is a freeparticle 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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 Spring '09
 Polarization, Fundamental physics concepts, FNI, total ionization rate, 2me ni n i · 2 . x

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