Question

2. Spin-1/2 system: (20 points) The Pauli matrices are, 0 -1 from which we can define the spin matrices, s.-슬&z, Šv = , S.-출.. We'll use the eigenkets of S that, for the spin half system, they can be represented by the spinors, a) Show, by matrix multiplication that |+) and |-) are eigenstates of the S operator and determine the eigenvalues. Show that they are not eigenstates of S and Sy b) Show that the matrix squares s , , and S are equal what does this tell us about the possible eigenvalues of S2 and S2, and hence about possible measurement results? c) Show that, where i is the identity matrix. d) Knowing the eigenvalues of S, and Sy from question b), what combination of |+) and |-) are their eigenkets? e) Let n be a unit vector (not an operator) in the direction specified by the polar angles θ such that nz-: sin θ cos φ nu-sin θ sin φ n': cos θ. Find the normalized eigenkets of the operator SS nySy + naS, with eigenvalues

Answer 1

(a) since s_z = h/2 sigma_z therefore Eigen value calculation of s_z s_z^^= h/2[1 0 0 -1] h/2h/2 - lambda 0 0 -h/2 - lambda = 0 lambda = h/2, -h/2 eigen value of s_z^^since A^^psi > = lambdapsi > psi > = [a b] = components = number of spin component i.e z s_z^^psi > = + h/2 psi > h/2[1 0 0 -1][a b] = h/2[a b] solving we get a = unknow & b = 0 therefore psi > = [a 0] Using Normalization condition psi > = [1 0] \r\n similarly for lambda = -h/2 psi > = [0 1] = - > (b) from property of pauli matrices we know that sigma_x^2 = sigma_y^2 = sigma_z^2 = 1 because s_x^^= h/2 sigma_x, s_y^^= h/2 sigma_y, s_z = h/2 sigma_z therefore s_x^2 = s_x s^x^dot = h/2 sigma_x h/2 sigma_x = h^2/4 sigma_x^2 = h^2/4 times 1 = h^2/4 because sigma_x^2 = similarly s_y^2 = s_y^^s_y^^= h/2 sigma_y h/2