ps6 - Physics 212 Problem Set 6 Spring 2010 1 More Fine...

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Physics 212 – Problem Set 6 – Spring 2010 1. More Fine Structure for the Hydrogen Atom. (a) For levels n = 1, 2, and 3, evaluate the effect of the spin orbit coupling. Sketch the corrected levels. (b) Combine the three fine structure corrections and show that for l = 0 they yield - E (0) n n 2 Z 2 α 2 ( 3 4 - n ) while for l 6 = 0 we obtain - E (0) n n 2 Z 2 α 2 3 4 - n j + 1 / 2 . Note that this gives the same result for j = l - 1 / 2 and that the degeneracy in l is restored. (c) For levels n = 1, 2, and 3, evaluate the combined effect of all fine structure corrections and sketch the resulting levels. 2. This problem is designed to highlight the importance of basis choice (individual or combined), and how rotation operators depend upon this choice. We will consider two spin-1/2 objects. They may be described in either the individual basis or the combined basis. We will consider a rotation about the ˆ y axis by an amount θ . (a) Prove that the operator that performs this rotation for a single spin-1/2 particle, e - iθσ y / 2 , can be written as 1cos( θ/ 2) - y sin( θ/ 2), where 1 is the 2x2 unit matrix. (b) Form a matrix representation of the rotation operator in the individual basis by finding all the matrix elements h m s 1 m s 2 | e - iθσ y 1 / 2 e - iθσ y 2 / 2 | m 0 s 1 m 0 s 2 i , where σ y 1 affects only the m s 1 quantum number and σ y 2 affects only the m s 2 quantum number. (The state | m s 1 m s 2 i
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