Spin One-Half Particle Polarisation Direction

Question:
A spin one-half particle is prepared in the state:

a) Is the vector |ψ0 > normalized? If it is, then justify your answer. If it is not, then compute the normalized state vector.

b) What is the spin polarisation direction, i.e., the direction along which a measurement of the particle’s spin will certainly yield +1 or spin up?

c) Suppose we set up a Stern-Gerlach (SG) apparatus to measure the spin of the particle along the positive y direction. What are the probabilities that the particle exits the SG device with spin up or down?

d) If the particle exits the SG device in part (c) with its spin up, calculate the subsequent expectation value of the observable σφ = σx cos φ + σy sin φ. What is the physical significance of the observable σφ?

e) Now suppose that σφ is the Hamiltonian of a system (in some appropriately chosen units of energy), whose initial state at time t=0 is . Compute the energy eigenvalues and eigenstates of this system.

f) Write down an expression for the time dependent state vector of the system described in part (e).

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