4. Spin Systems. Consider an experiment in which a cloud of silver atoms emitted from a hot oven is passed through a collimator to create a beam of spin 1/2 particles. This beam is then fed into two different experimental setups, as follows: i) The experimentalist tosses a coin and then the atom beam passes through a Stern-Gerlach device oriented in the +z direction which splits the beam into two. Depending on the results of the coin toss, one of the beams is randomly discarded and the other is directed into the next room. ii) The atom beam passes through a Stern-Gerlach device oriented in such a way as to generate a beam of particles in the state 1/2 |) + 1/2 |) that is likewise directed into the next room while the other beam is discarded. (a) Which direction would the Stern-Gerlach device have to be oriented in to pro- duce a beam of particles as described in ii.? Note that there are no random choices being made in this setup, the beam is simply created by taking one of the two beams from a properly oriented device. (b) Explain whether or not it is possible to distinguish between situations i. and ii. by examining the outcome of passing the beam through another Stern-Gerlach device oriented in the +z direction, rationalizing your answer. Do the same for a second Stern-Gerlach device oriented in the +x and in the +y directions. (c) Repeat the analysis in (b) for situations iii. and iv. below. You should get an answer that is not identical to your answer for (b). In particular, situation iii. has an interesting property that you may notice after analyzing its behavior. (iii) In this case the beam is narrowed to allow only a few particles per minute to pass. The atoms are sent through a Stern-Gerlach device oriented in the +z direction. The atoms are emitted so rarely that a coin can be tossed after each atom and the setup is modified based on the outcome so as to randomly block one of the beamlines and direct the other through the wall. (iv) The arrangement of the apparatus is once again adjusted so that it produces a stream of particles in the 1/2 |) + 1/2 |) state but with the same rate of emission as in case iii. H P-1/2 or P. 1/2 @ 24+ Figure 1: Stern-Gerlach experiment with a stream of silver atoms.