Download S operator( ). 2) Magnetic field is applied along positive Z axis. Find

Homework #13, 7310 (2012 fall)
Problem #1
10.3 Sethna. (You may want to take a look at the Appendix on Fourier Transforms in the end the
textbook).
Problem #1
Consider a beam of light which is propagating in the +z direction. An arbitrary pure polarization state can
be written as a linear combination a ↑ + b ↓ , where ↑ represents the state which polarized in xdirection and ↓ the state polarized in y-direction.
(a) Calculate the density matrix for (i) the state polarized at 45o and (ii) the state polarized at 135o.
(b) What is the density matrix for a mixed state in which 50% of the light is polarized along 45o and 50%
along the 135o?
(c) What is the density matrix for a mixed state in which 50% of the light has clockwise circular
polarization and 50% has counterclockwise circular polarization?
Problem #3
At initial time t=0 a ½-spin has 75% probability to be aligned in positive Y-direction and 25% probability
to be aligned in negative Y direction (so the spin is in a mixed state).
1) Write down the density matrix of the spin at t=0 in the basis of S Z operator( Z + , Z − ).
2) Magnetic field is applied along positive Z axis. Find time evolution of the density matrix (in
Z + , Z − basis) starting from the initial conditions at t=0 that you found in question (1).
3) Find as a function of time average projection of the spin on Y axis.