See the pictures below:

In this problem we will see how to apply the grand canonical ensemble to a simplied version of the two-level paramagnet. Though really what we will be doing is laying the groundwork for quantum statistical mechanics! Suppose we have an \"open\" paramagnet system where the number of spins is variable. Our system lives in a reservoir of temperature T and a gas of free Spins with chemical potential ,a.1 Suppose our Open system can contain at most one Spin. Since we will be using u as our chemical potential, let the two different spin states of a single spin be: ETE/2, E:+E/2. Our Open system thus contains three possible states: 0 No spins. 0 One spin in the spin-up state. 0 One spin in the spin-down state. (a) Find the grand canonical partition function E for this system. Use this to nd the average number of particles in the system (N). [Note: As a check, we should reasonably ea'pect that our answer will strictly be between 0 and 1 since those are the minimum and maa'imum possible number of spins in our system given our setup] For a specic energy state we dene the occupation number of the state in a multiple-particle system to be the average number of particles in that energy state. For example, in our system we have two occupation numbers, (\"Tl and (ni), the average number of particles in the Spin-up state and spin-down state, respectively. Mathematically, ni,max (\"2): Z mPmi), Tli=0 where P(i; n,) is the probability that there are it, particles in the i-th energy level. Note that even though part (b) won't be graded you should still give it a shot before moving on to part (c)! (b) Extra Part (Not for Credit) Without explicit calculation, what do you predict the occupation number of the spin-up state will be in the two limits T > 0 and T > 00? (c) Using your E from part (a), nd P(l;1), P(2; 1), (n1), and (n2). [Note: As a check, you should have (n1)+(n2) : (N). You can also check your predictions/intuitions from part (15)] We call the number of N ow that we have some of the basics of working with the grand canonical ensemble let's make our system a little more interesting. Suppose instead of allowing at most one spin in our open system, we loosen the restriction to only allowing at most one spin in each state in our Open system. That is, each state's occupancy must be either 0 or 1. This introduces a fourth microstate, 1. No Spins (both occupancies are {7%, 711,} : {0, 0}) 2. One Spin in the Spin-up state (occupancies {1, 0}) 3. One spin in the spin-down state (occupancies {0, 1}). 4. One Spin each in the Spin-up state and the Spin-down state (occupancies {1, 1}). ((1) Find E and (n1) for this new setup. There is something really interesting going on with E in this case which we will now demonstrate. Suppose that we consider each energy level of the system as its own subsystem. That is, we consider just the \"spin-up\" state as an independent subsystem. This subsystem now has only two possible microstates: o No spins are in the Spin-up state (this is realized by states 1 and 3 in part (d)). 0 One Spin is in the Spin-up state (this iS realized by states 2 and 4 in part (d)). We can find the grand canonical partition function E? for this \"single-state\" subsystem. Similarly, we consider just the \"spin-down\" state as a different independent subsystem and find E L- Using this we can determine the occupation numbers directly from the Single-state grand canonical partition functions, e.g. alnE (n ) = kBT( T) . (1) l 8:\" T,V (e) Find the grand canonical partition functions E} and E L for these two subsystems. Show that your answer for E from part (d) factors into ETE 1,. Finally, calculate (RT) using Eq. 1 and Show it is equivalent to what you found in (d)