# Statistical mechanics problem of spin

1. Jul 26, 2013

### athrun200

1. The problem statement, all variables and given/known data
Consider a system consists of two subsystem A and A' in which A contains 3 spins and A' contains 2 spins. Suppose that, when the systems A and A' are initially separated from each other, measurements show the total magnetic moment of A to be $- 3{\mu _0}$ and the total magnetic moment of A' to be $+ 4{\mu _0}$. The systems are now placed in thermal contact with each other and are allowed to exchange energy until the final equilibrium situation has been reached. Under these conditions calculate:
(a) The probability P(M) that the total magnetic moment of A assumes any one of its possible values M.

2. Relevant equations
Basic statistic and probability tools.

3. The attempt at a solution
The question implies that each of the spin in A has magnetic moment ${\mu _0}$ while A' has 2${\mu _0}$.
Originally, the system has the spin arrangement like this:
--- ++
where - means spin down and vice versa.

After they are in contact, they can have the following possibility.
--- ++ $- 3{\mu _0}$
--+ -+ $- {\mu _0}$
-+- -+ $- {\mu _0}$
+-- -+ $- {\mu _0}$
+-- +- $- {\mu _0}$
+-+ -- ${\mu _0}$
++- -- ${\mu _0}$
--+ +- $- {\mu _0}$
-++ -- ${\mu _0}$
-+- +- $- {\mu _0}$

So $P( - 3{\mu _0}) = \frac{1}{{10}}$, $P( - {\mu _0}) = \frac{6}{{10}}$, $P({\mu _0}) = \frac{3}{{10}}$.

But the answer is $P({\mu _0}) = \frac{6}{7}$, $P( - 3{\mu _0}) = \frac{1}{7}$.

What's wrong?

2. Jul 27, 2013

### Oxvillian

Hi athrun200!

--+ -+ would have total moment $-\mu_0$, not $+\mu_0$, as is required by conservation of energy.

3. Jul 27, 2013

### athrun200

I know --+ -+ produces total moment $- {\mu _0}$
But how does it affect the answer?

4. Jul 27, 2013

### Oxvillian

The original configuration --- ++ had total moment $+\mu_0$, yes?

So all the possible configurations that you're counting should have total moment $+\mu_0$, not $-\mu_0$.

5. Jul 27, 2013

### athrun200

Oh! Now I get it, thanks a lot!

6. Jul 27, 2013

np