p137bfin - Physics 137B Fall 2004 Quantum Mechanics II...

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Physics 137B, Fall 2004 Quantum Mechanics II Final exam 12/20/2004, 8:10-11:00 a.m. Please attempt to solve all 6 problems. No books, notes, or calculators are allowed, but some helpful equations are on later pages. Please start each of the 6 problems on a fresh side. Problems III and V are worth 20 points; all others are worth 15. I. Consider the 3d levels ( n = 3, l = 2) of the electron in a hydrogen atom. Each part of this question is worth 3 points. (a) What is the eigenvalue of the orbital squared angular momentum operator L 2 in all these states? (b) Now include the electron spin. Give the possible eigenvalues of the total squared angular mo- mentum operator J 2 , where J = L + S , and the number of states corresponding to each eigenvalue. Remember to state the connection between the total angular momentum quantum number j and the eigenvalue of J 2 . (c) For each of the states you found above, give the expectation value of L · S . Hint: can you express this operator using J 2 , L 2 , and S 2 ? (d) Use the atomic selection rules to list all the atomic orbitals to which a 3d electron can decay by spontaneous emission . (e) The element manganese has 5 electrons in 3d. Suppose that all these electrons have spin-up, so that the total S z quantum number for these 5 electrons is m s = +5 / 2, i.e., the eigenvalue of total S z is 5¯ h/ 2. What must be the eigenvalue of total S 2 for these 5 electrons, and why? II. (15 points) At t = 0, unpolarized light shines on 10 5 hydrogen atoms that are initially distributed evenly over all 2 p excited states ( n = 2 , l = 1). The light has intensity per unit frequency interval I ( ω ). Give the total number of atoms found in n = 1 states after a time t that is short enough that most of the hydrogen atoms have not made any transition. Estimate also the time beyond which your answer will cease to apply. You may leave integrals in your answer and use the symbol ψ nlm rather than write the whole wavefunctions, but you should state clearly if an integral will turn out to be 0.
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