Using linear algebra operations find the expectation value

Question

Q1. (a) Show by direct evaluation that the eigenvectors of Pauli's σ_y matrix (in the representation of the eigenvectors |+) and |-) of σ_z)

is the y-component of the ½-spin operator in the same representation. That is, show, that |+_y) and |-_y) are eigenvectors of S_y with eigenvectors ± h/2.

(b) Give the numerical value of h/2 in J·s and in J·s/mol units, and (c) show that the SI units of position times linear momentum are also J·s.

Q2. Find the probability to measure S_s = + h/2 in:

(a) a ½-spin beam of particles polarized along the y direction with quantum state

|+_y) =  1/√2 |+) + i/√2 |-)

and (b) in ½-spin beam of particles polarized in xz plane along an axis at an angle θ = 45^o relative to the z-axis, with quantum state

|ψθ) = cos θ/2 |+) + sin θ/2 |-)

Express probabilities in percent units.

(c) Using linear algebra operations, find the expectation value of the spin along the z axis for the quantum state in (b)

(S_x)_θ = (ψ_θ|S_x|ψ_θ)

where

Express the results in SI units.

(d) Confirm that the result in (c) is consistent with the weighted average of the spin up and spin down states along the z axis:

(S_x)_θ = h/2 p₊ + (-(h/2))p_-

where p₊ and p_- are, respectively, the probabilities to measure S_x = + h/2 and S_z = - h/2 in the quantum state in (b).