Spring 2017 Homework Assignments
1. Consider the 3x3 spin-1 Hilbert Space. (a) What are the spin operators Sx, Sy, and Sz? Find their eigenvectors. (b) Find the spin operator in the n direction, where n = (i+j+k)/√3. (c) Rotate the eigenvector of the Sz operator with eigenvalue +ħ around the direction n by an angle 2π/3. What physical object does the rotated vector correspond to? (d) If the particle spin has been measured to be zero in the z-direction, what are the probabilities of possible results when the measurement is then repeated in the y-direction? What would these probabilities be after the rotation in part (c)? 2. Two electrons are interacting with the potential α S1⋅S2 + β (S1 + S2)⋅B where S1 and S2 are the spin opreators of the two electrons and B = kBo is a uniform magnetic field in the z-direction. α, β and Bo are constants. Find the energy eigenvalues and eigenvectors.
Use the result of problem 5.11 of the textbook to determine the first order perturbation to the ground state energy of the He atom due to electron-electron interaction.
Problems 7.13 and 7.14 in the textbook.
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