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Statistical mechanics problem of spin

  • Thread starter athrun200
  • Start date
  • #1
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Homework Statement


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 [itex] - 3{\mu _0}[/itex] and the total magnetic moment of A' to be [itex] + 4{\mu _0}[/itex]. 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.


Homework Equations


Basic statistic and probability tools.


The Attempt at a Solution


The question implies that each of the spin in A has magnetic moment [itex]{\mu _0}[/itex] while A' has 2[itex]{\mu _0}[/itex].
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.
--- ++ [itex] - 3{\mu _0}[/itex]
--+ -+ [itex] - {\mu _0}[/itex]
-+- -+ [itex] - {\mu _0}[/itex]
+-- -+ [itex] - {\mu _0}[/itex]
+-- +- [itex] - {\mu _0}[/itex]
+-+ -- [itex] {\mu _0}[/itex]
++- -- [itex] {\mu _0}[/itex]
--+ +- [itex] - {\mu _0}[/itex]
-++ -- [itex] {\mu _0}[/itex]
-+- +- [itex] - {\mu _0}[/itex]

So [itex]P( - 3{\mu _0}) = \frac{1}{{10}}[/itex], [itex]P( - {\mu _0}) = \frac{6}{{10}}[/itex], [itex]P({\mu _0}) = \frac{3}{{10}}[/itex].

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

What's wrong?
 

Answers and Replies

  • #2
196
22
Hi athrun200!

--+ -+ would have total moment [itex]-\mu_0[/itex], not [itex]+\mu_0[/itex], as is required by conservation of energy.
 
  • #3
277
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I know --+ -+ produces total moment [itex] - {\mu _0}[/itex]
But how does it affect the answer?
 
  • #4
196
22
The original configuration --- ++ had total moment [itex]+\mu_0[/itex], yes?

So all the possible configurations that you're counting should have total moment [itex]+\mu_0[/itex], not [itex]-\mu_0[/itex].
 
  • #5
277
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The original configuration --- ++ had total moment [itex]+\mu_0[/itex], yes?

So all the possible configurations that you're counting should have total moment [itex]+\mu_0[/itex], not [itex]-\mu_0[/itex].
Oh! Now I get it, thanks a lot!
 
  • #6
196
22
np :smile:
 

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