Homework 10 (due Wednesday, Dec. 5) This is the last homework assignment! There is one problem on spin and one problem on addition of angular momentum. 1. Spin: The Hamiltonian of an electron in a magnetic field of magnitude B and direction n̂ is H = µB B n̂ · ~σ . (1) In spherical coordinates the unit vectors n̂ is n̂ = sin(θ) cos(φ)x̂ + sin(θ) sin(φ)ŷ + cos(θ)ẑ. (2) (a) Verify that the general spin eigenstates given in Eq. 4.155 in the book are indeed eigenstates of the Hamiltonian in Eq. (1) with the unit vector given in Eq. (2). (Just multiply the vector times the Hamiltonian.) (b) Next suppose that the magnetic field is B x̂. Let the spin initially be pointing in the +ẑ direction, ! 1 |ψ(t = 0)i = . (3) 0 Determine |ψ(t)i. (c) We will now interpret this |ψ(t)i in two ways. First, compute the expectation value of Sx , Sy , and Sz for |ψ(t)i in part (b). Sketch how the expectation is rotating in three dimensions. (d) Next using the exact eigenstates (Eq. 4.155) determine the θ and φ time dependence for |ψ(t)i in part (b). Compare the θ and φ time dependence to the expectation value time dependence of part (c). (e) Repeat (b)-(d) for a magnetic field of B ŷ with the spin sill initially pointing in the +ẑ direction. 2. Addition of angular momentum: In this problem we will compute the Clebsch-Gordon coefficients for adding spin 1 and spin 21 angular momentum. The results will be compared to those in Table 4.8 of the book. (a) List the (six) possible pairs of m1 and m2 for j1 = 1 and j2 = m = m1 + m2 values for these two states. 1 . 2 Give the (b) The state with m1 = 1 and m2 = 12 has total angular momentum 23 . To avoid confusion use the notation of the book and call this state |1, 1i| 12 , 12 i. Apply the lowering operator to this state to get the the three other states with j = 3/2. Again following the notation of the book denote these states at | 23 , mi for m = 1 , − 12 , − 32 . 2 (c) Find a state which is orthogonal to | 32 , 21 i, but still has an m value of 12 . Normalize it and make the coefficient of |1, 1i| 12 , − 21 i positive. Verify that this state has total angular momentum 12 by acting with S 2 on it. (d) Apply the lowering operator, S− , to the state of (c) to obtain the j = 21 , m = − 21 state. (e) Compare your results in (a)-(d) to those in Table 4.8.