Some sample questions				15 October, 2013

These questions are meant to help you study for the PHY 356 midterm to be held next week.  Obviously, you should review all the material we have covered, but these are meant as an additional aid.

(1) Consider a state a|up> + b|down>.  
a. Taking advantage of your knowledge of the Sz operator and the identities Sx|up> = |down> and Sx|down>=|up>, calculate <Sx> and Delta Sx using Dirac notation (do not explicitly write out the matrix representations).
b.  Using any method you like, calculate the probabilities of Sx = +1 or -1 respectively, and from these, find <Sx> and Delta Sx explicitly.
[Note: Throughout these questions, I am using Sx as a synonym for sigma_x, i.e., omitting the factor of hbar/2.]

(2) Suppose in some state <X^2> = 4.  
a. What can you conclude about <X>?
b. If you are also told that the state is an eigenstate of X, what more can you conclude about <X>?
c.  Extra credit: what can you conclude about the state if told that it is not an eigenstate of X, but is an eigenstate of X^2?

(3) Consider the two-particle state |-z>|-z> + |+x>|+x>.
Use the projection postulate to find what state is left if you measure particle 1 to be in the state |-z>.

(4) Consider a general two-particle spin state a|up>|up> + b|up>|down> + c|down>|up> + d |down>|down>.
a.  If you are told that whenever the spins are measured along z, they are the same, what can you conclude about the coefficients?
b.  If you are additionally told that whenever the spins are measured along x, they are the same, what more can you conclude?
c.  What, if anything, can you determine about the correlations of the two spins if both are measured along y?

(5) State the Feynman rules for interference

(6) Explain in a brief paragraph what principle(s) of quantum mechanics makes it promising for secure cryptography.  (You don't need to explain the entire protocol, but the fundamental difference between classical and quantum physics which underlies it.)

(7) I have a two-path (Mach-Zehnder) interferometer.  If I block path 1, the probability of a photon following path 2 and hitting the detector is 25%.  If I block path 2, the probability of a photon following path 1 and hitting the detector is 16%.
a.  If I open both paths and there is no information to tell which path a photon takes, what is the maximum possible detection probability?
b.  If I open both paths and there is no information to tell which path a photon takes, what is the minimum possible detection probability?
c.  If I open both paths and an eavesdropper possesses perfect information about which path the photons take, what is the maximum possible detection probability?

(8) Two operators which are important in quantum mechanics are J_z and J^2.  It happens that these two operators commute.  Explain what the significance of this fact is.

(9) For a spin-1 particle, the operator S_y has eigenvalues +1, 0, and -1.  
a.  Write a possible matrix representation of this operator, and the eigenvectors corresponding to each eigenvalue.
b.  Write down the general matrix representation for any state which has equal probabilities to be found with any of the three values of S_y.
c.  Write down two orthogonal examples of such states.
d.  Write down the projection operators onto the states from c.
e.  Calculate the product of these projection operators.
f.  Comment on the result in (e).

(10) It turns out that for any operator A and state |Psi>, A|Psi> may be written as x|Psi> + y|Phi>, where <Phi|Psi> = 0.  You may assume that |Psi> and |Phi> are unit vectors.
a.  Calculate <A> in terms of x and y.  
b.  Calculate <A^2> in terms of x and y.  [Hint: although I have not told you A|Phi>, it will be useful to consider the Hermitian adjoint of A|Psi>.]
c.  Calculate the uncertainty in A, and comment on its dependence on y.