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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 (0) − 3 En Z 2 α2 ( − n) 2 n 4 while for l 6= 0 we obtain (0) En − 2 Z 2 α2 n 3 n − 4 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 ŷ axis by an amount θ. (a) Prove that the operator that performs this rotation for a single spin-1/2 particle, e−iθσy /2 , can be written as 1cos(θ/2) − iσy 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 hms1 ms2 |e−iθσy1 /2 e−iθσy2 /2 |m0s1 m0s2 i , where σy1 affects only the ms1 quantum number and σy2 affects only the ms2 quantum number. (The state |ms1 ms2 i signifies that particle one has projection h̄ms1 along the ẑ axis and particle two has projection h̄ms2 along the ẑ axis.) Collect these terms into a 4x4 matrix. Specify your basis For example, the choice. 1 0 familiar Pauli matrices are expressed in a basis where |+i = and |−i = . 0 1 (c) Verify that the above matrix does what you expect it to do by using the test case of rotating |ms1 ms2 i = | + +i by an amount θ = π around the ŷ axis. (d) Find the matrix elements of J˜y = S̃1y + S̃2y in the |ms1 ms2 i basis, where the ˜ symbol signifies that the h̄ factor has been removed. Place these matrix elements in a 4x4 matrix using the same ordering of states as you did in (b) above. ˜ (e) Prove that e−iθJy may be written as 1 − iJ˜y sinθ + J˜y2 (cosθ − 1) , where Jy is the 4x4 matrix you found above. Here 1 is the 4x4 unit matrix. Find this 4x4 matrix and confirm that it is identical to that found in (b) above. (f) Now we will consider the combined basis. Write the four “combined” basis states in terms of the four “individual” basis states by finding the relevant C-G coefficients. (g) Find the 3x3 matrix L̃y for the l = 1 states. Show that the rotation operator e−iθL̃y may be written as 1 − iL̃y sinθ + L̃2y (cosθ − 1) , where 1 is the 3x3 unit matrix. Find that rotation matrix. 1 (h) Form a new 4x4 rotation operator by placing “1” in the upper left hand corner and the 3x3 rotation matrix from (g) in the lower right hand corner. All other elements are zero. This is the block diagonal rotation matrix valid in the combined basis. What role does the “1” in the upper left hand corner play? Specify the basis choice. (i) Verify that both of your 4x4 rotation matrices perform the same operation by acting upon the |l = 1, ml = 0i state (specified in a basis consistent with that of the rotation matrix). 3. Problem Modified from Exam 1 Consider two spin-1/2 objects that have a relative angular momentum of l = 1. If particle one has its spin aligned along the positive x̂ axis and particle two has it spin along the positive ẑ axis and the relative angular momentum has a projection of 0 along the ŷ axis, find the probability that the total angular momentum of the system is found to have a projection of one unit along the positive ẑ direction. 4. Use addition of angular momentum (“addition” of spherical harmonics in this case) to find the selection rules for transitions between the hydrogenic levels |lml ms i and |l0 m0l m0s i caused by the operator H 0 = αX̂ 2 Ẑ Ŝy . 5. The Zeeman Effect: Intermediate Field Region Consider a hydrogen atom in a uniform external magnetic field B = Bz ẑ, directed along the ẑ axis. Our goal is to find the first-order corrections to the energies of the 2p (n = 2, ` = 1) states of atomic hydrogen due to the combined effects of the spin-orbit interaction and the interaction of the magnetic moment of the atom and the (0) magnetic field B. The zeroth-order Hamiltonian H(0) , has the zeroth-order eigenfunctions Ψn`jmj (r), for n = 2, ` = 1. (a) The combined perturbations may be written ~ = Hs−o − µz Bz , H(1) = Hs−o − µ ~ ·B ~ + gs S) ~ and where µ ~ = −µB (L Hs−o = 1 ~ ~ e2 S·L. 2 2 2me c r3 Demonstrate which of the operators L2 , S 2 , J 2 , and Jz commute with H(1) and hence with H = H(0) +H(1) . What does this tell you about the nature of the eigenfunctions of H, as compared with the zeroth-order (0) eigenfunctions Ψn`jmj (r)? (b) Write expressions for all the matrix elements of H = H(0) + H(1) in terms of the zeroth-order energy eigenvalues, the “spin-orbit” energy, and the matrix elements (0) (0) (0) (0) hΨn`jmj |Sz |Ψn`j 0 m0 i and hΨn`jmj |Lz |Ψn`j 0 m0 i. j j Calculate the required matrix elements of Sz and Lz . Order the states so that the matrix has the simplest block diagonal structure. (c) Solve for all the roots, thereby obtaining the energy eigenvalues of H correct to first order. Based on your results, draw an energy level diagram that shows how the energy levels change as the perturbations Hs−o and −µz Bz are included and the strength of the magnetic field changes. (d) Taking the weak field limit, show that your answer recaptures the results obtained in class. Do the same with the strong field limit. 2