6110, Physics Fall 2008 Problem Set #4
Due: This problem set is due on Thursday, October 30, except for Problem 2, which is due on the 28th as part of the exam (see below). Exam: The midterm exam will be held on Tuesday, October 28. It will include (1) everything through problem set 3, and (2) Problem 2 on this problem set, which is therefore in effect a takehome exam problem. The exam will start at the regular class time and room, but you can stay until 1 PM. You are permitted to bring your lecture notes and at most one book for reference. Problem 2 from this problem set must be handed in with the exam. No collaboration is permitted on this particular problem.
1.
Fine structure of Mg and Mg-like Fe in the LS coupling scheme (based on Foot problem (5.5)) In the LS or Russell-Saunders coupling scheme, the hydrogenic fine-structure expression in Eq. (29) on p. DE-16 can still be used if the H-atom Hamiltonian is replaced by one with a semiempirical scaling parameter, H fs L S The expectation value for states LSJm can still be found as on p. DE-16 of the lecture notes. The energy levels of the 3s3p configuration of neutral magnesium (Z=12) and the magnesiumlike ion Fe14+ are given in the table below in cm-1, measured relative to the ground state. The fine structure is determined by the angular momentum and spin of the two valence electrons, since the Ne-like core formed by the other ten electrons has Score = Lcore = 0. Suggest, giving your reasons, further quantum numbers to identify these levels. Calculate the ratio of the spin-orbit interaction energies in the 3s3p configuration of Mg and Fe14+, and explain your result. Discuss the occurrence in the solar spectrum of a strong line at 41.726 nm originating from Fe14+. Would a corresponding transition be expected in neutral Mg, and if so, at what wavelength? 3s3p, Mg (cm-1) 21850 21870 21911 35051 3s3p, Fe14+ 233860 239660 253780 351930
2.
Structure of tau-deuterium
Consider a hypothetical hydrogen-like D+ - atom comprised of a deuteron and a tau lepton. The tau has a mass slightly heavier than a proton, m = 3478 me. Like any lepton, it has a spin of and a g factor close to 2 (as a multiple of the magneton for the appropriate mass). The deuteron has nuclear spin I=1 and a similar mass to the tau, 3670 me, with a g factor of 0.857 (i.e., D = 0.857 N). In practice this atom would be nearly impossible to study because the lifetime of the tau lepton is too short, about 300 fs. Ignore this limitation by pretending that the particle is stable. Being sure to take the reduced mass into account, find lowest-order expressions for each of the following, and evaluate them numerically in either SI or atomic units: (1) The 2s-3p Balmer alpha transition energy. (2) The radiative lifetime of the 2p state. (3) The fine-structure spacings of the 2p levels. (4) The hyperfine structure spacings of the ground state and the 2p3/2 state. Suggestion: if you know the scaling of a given parameter relative an to H atom, you can take advantage of this to quickly find its numerical values.
3.
Coupling of two angular momenta I=3/2 and L=1
Consider the electronic states of a hypothetical atom with nuclear spin 3/2, similar to 39K except that its electron spin interactions can be ignored. For a p state with L=1, L and I couple to form 3 eigenstates L I F mF = 1 2 F mF . Use the 3-j symbols to find explicit expressions for all of the 3 possible levels having m=1/2, written as sums over the uncoupled basis set 1 mL 2 mI . For example, if you were instead writing expressions for m=5/2 levels, there would be only one 3 5 5 3 3 possibility, 2 1 2 2 , which can be re-written in the uncoupled basis as 1 1 2 2 . You can check your results by making sure that your wave functions are properly normalized. If you evaluated all of the 3-j symbols correctly, this happens automatically. In any cases where the expressions involve more than one non-zero term, do they occur symmetrically or antisymmetrically, or are there some of each? What is the significance of this, if any? Finally, use your results to directly evaluate the following matrix element representing the hyperfine coupling, without resorting to the standard trick of writing I iL = ( F 2 L2 I 2 ) / 2 :
3 3 3 3 1 1 2 2 1 IiL 1 2 2 2 . 2
4.
Zeeman effect for hydrogen, ignoring hyperfine structure
(a) The Hamiltonian for a hydrogen atom in a static magnetic field B along the z direction can be derived using the same procedure we followed for the case of an electron, on pp. DE-6 ff. of the lecture notes. To first order in the vector potential A, there are two new terms in the Hamiltonian,
H Zeeman =
B
B ( Lz + g s S z )
Here gs is the electron g factor, which we can set equal to 2 for present purposes. The two te...