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Statistical mechanics problem of spin

  1. Jul 26, 2013 #1
    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 [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.


    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 [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?
     
  2. jcsd
  3. Jul 27, 2013 #2
    Hi athrun200!

    --+ -+ would have total moment [itex]-\mu_0[/itex], not [itex]+\mu_0[/itex], as is required by conservation of energy.
     
  4. Jul 27, 2013 #3
    I know --+ -+ produces total moment [itex] - {\mu _0}[/itex]
    But how does it affect the answer?
     
  5. Jul 27, 2013 #4
    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].
     
  6. Jul 27, 2013 #5
    Oh! Now I get it, thanks a lot!
     
  7. Jul 27, 2013 #6
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