How Do You Calculate Microstates for a Constrained Spin System?

[Nylex]

Consider 2 systems of spin 1/2 paramagnets, which may point either up or down wrt a magnetic field. The first system contains 8 paramagnets and the second contains 6 paramagnets.

Suppose the energy of the combined system is constrained such that the total number of spins pointing up in the 2 systems is forced to be 6. Make a table of the possible values of n (no. of paramagnets pointing up, in 1st system) and m (no. of paramagnets pointing up, in 2nd system) and the number of microstates of the combined system for each case. By summing the entries of your table, obtain the total number of accessible microstates subject to the energy constraints.

This is what I've done so far:

m | n | accessible microstates
6 | 0 | (6C6) x (8C0) = 1
5 | 1 | (6C5) x (8C1) = 48
4 | 2 | (6C4) x (8C2) = 420
3 | 3 | (6C3) x (8C3) = 1120
2 | 4 | (6C2) x (8C4) = 1050
1 | 5 | (6C1) x (8C5) = 336
0 | 6 | (6C0) x (8C6) = 28

Total no. of accessible microstates = 3003.

I'm not sure whether or not my method for working out the no. of accessible microstates is right.

 

[anathema]

Can someone please check it and explain if it's wrong?

Your calculations are correct. To understand why, we need to consider the physical significance of each entry in the table.

In this scenario, the two systems of spin 1/2 paramagnets are combined into one larger system with a total of 14 paramagnets. The energy constraint forces the total number of spins pointing up to be 6 in the combined system. This means that the remaining 8 spins must be pointing down.

So, when we consider the first row of the table (m=6, n=0), we are essentially looking at a scenario where all 6 spins in the first system are pointing up and all 8 spins in the second system are pointing down. This is represented by the combination (6C6) x (8C0), which gives us a total of 1 possible microstate.

Similarly, in the second row (m=5, n=1), we have 5 spins pointing up in the first system and 1 spin pointing up in the second system. This can be achieved in (6C5) x (8C1) ways, giving us a total of 48 possible microstates.

By summing up all the entries in the table, we are essentially considering all possible combinations of spins pointing up in the two systems that satisfy the energy constraint. This is why the total number of accessible microstates is given by the sum of all the entries in the table, which is 3003 in this case.

In conclusion, your method for calculating the number of accessible microstates is correct. It is important to understand the physical significance of each entry in the table in order to correctly interpret the results.

 

[wais]

Can someone check it for me?

Your method for calculating the number of accessible microstates is correct. The formula for calculating the number of microstates for a given system is given by the binomial coefficient, which you have correctly used in your table. The total number of accessible microstates for the combined system is obtained by summing the entries in your table, which gives the correct answer of 3003. Good job!