How Many States Exist Between E and E+δE in a Spin 1/2 System?

[Salterium]

Homework Statement

Consider an isolated system consisting of a large number N of very weakly interacting localized particles of spin 1/2. Each particle has a magnetic moment $$\mu$$ which can point either parallel or antiparallel to an applied field H. The energy E of the system is then E = -(n₁-n₂)$$\mu$$H, where n₁ is the number of spins aligned parallel to H, and n₂ is the number of spins aligned antiparallel to H.

(a) Consider the energy range between E and E+$$\delta$$E where $$\delta$$ is very small compared to E, but is microscopically large so that $$\delta$$E>>$$\mu$$H What is the total number of states $$\Omega$$(E) lying in this energy range?

Homework Equations

I really have no clue.

The Attempt at a Solution

I've been sitting with my small study group talking about this for an hour, and we're no closer to a solution than when we started. We've looked at the answer and it reminds us of the classical "drunken sailor" problem. Trust me when I say we've attempted this solution from every angle we can think of.

 

[chrispb]

How many states are there with a given/fixed energy, say, 0 or mu_H or mu*H or whatever it is?

After you know that, you'll want to sum up that number for every energy between E and delta E. Since the gap is big enough (delta E >> mu H), you can transform the sum into an integral.

 

[sr6622]

Think about the possible number of states that exist between E and delta E. Like your drunken sailor problem, in which different ways could they move? How is that similar to your particles? Could you use a common formula for this situation as you might have used in the drunken sailor?