A particle with a spin 1/2 is placed in a time-independent magnetic field. The direction of the magnetic field is along the z-axis. The Hamiltonian system is given by H = -μ•B, where μ is the magnetic moment and B is the effective magnetic field. We have included all the coefficients in the effective magnetic field and as a result, the dimension of B is inverse of time.
A) Find the energy eigenvalues and the corresponding eigenstates for the particle.
B) Assume that the particle is in the lowest energy state and in an experiment, the spin component is measured along the x-axis (M1 test). What is the probability that the result in the M1 measurement is equal to +1/2?
C) Just after measuring M1, another measurement is made (M2). This time the spin of the particle is measured in the direction of the z-axis. What is the probability that the measurement of M2 gives the result -1/2?
D) After the measurement of M2, let time pass and perform the third measurement (M3) by measuring the spin component along the x-axis. What are the two possible results of the M3 test? Write their probabilities as a function of time, assuming that the output of M2 is +1/2.
F) Repeat section (D) without making any assumptions about the result of the M2 measurement. Note that the M2 measurement is performed.
G) Repeat section (D) assuming that M2 was not measured at all, that is, assuming that M1 was measured first and then M3 was measured after time t. What are the possible results of the M3 test and what are their probabilities?
H) Assume that the output of the M2 test is -1/2. The M3 test this time, however, measures the spin component along the z-axis. For this arrangement, repeat section (D) and find two possible results of M3 and their probabilities as a function of time.