Thermodynamics: Building Blocks


Quantum Basis

Summary Quantum Basis

Calling the system binary means that each magnet can be oriented either in the "up" position or the "down" position, and no other. If a magnet is in the down position, then we say that its magnetic moment is - m, if up, it is + m. The magnets do not interact with each other; i.e. the position of a magnet's neighbors does not influence its position. A sample collection of such magnets can be seen in .

Figure %: Binary System of Magnets

Magnetic moments add together just as vectors do. Therefore, we can ask, how many ways are there to have a total magnetic moment M of M = Nm? Such a state would require all of the magnets to be in the up position, so there is only one way to achieve this state. How many ways are there to have a total magnetic moment of M = (N - 2)m? Such a state requires one magnet to be in the down position. Since there are N magnets, there are N such ways.

Letting C represent the up position and D represent the down, we can use a shorthand notation for representing all of the possible states of the system:

(C + D)N

Using a binomial expansion, and writing in summation notation, we can write:

(C + D)N = CN-iDi

The Multiplicity Function

Usually we are interested not in writing out a general form for all states, but are more focused on one particular state. As we saw above, sometimes there are multiple states with the same number of spins in the up position. Let Nup be the number of particles in the "up" state, and Ndown be the number of particles in the "down" state (then N = Nup + Ndown). We refer to the number of states with the same values of N and Nup by the function g(N, Nup), called the multiplicity function. For our system, g(N, Nup) is given by the coefficient in the preceding sum:

g(N, Nup) =

Notice that for very large and very small values of Nup, g is small, but for Nup = Ndown, g is a maximum.