Statistical mechanics problem of spin

In summary, the system consists of two subsystems A and A' with 3 and 2 spins respectively. The initial separation shows a total magnetic moment of -3{\mu _0} in A and +4{\mu _0} in A'. After being placed in thermal contact and allowed to exchange energy, the final equilibrium situation results in a probability of P({\mu _0}) = \frac{6}{7} and P( - 3{\mu _0}) = \frac{1}{7} for the total magnetic moment of A. This is due to the original configuration of --- ++ having a total moment of +{\mu _0} and thus all possible configurations should have a total moment of +{\
  • #1
athrun200
277
0

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 anyone 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?
 
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  • #2
Hi athrun200!

--+ -+ would have total moment [itex]-\mu_0[/itex], not [itex]+\mu_0[/itex], as is required by conservation of energy.
 
  • #3
I know --+ -+ produces total moment [itex] - {\mu _0}[/itex]
But how does it affect the answer?
 
  • #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].
 
  • #5
Oxvillian said:
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
np :smile:
 

1. What is the statistical mechanics problem of spin?

The statistical mechanics problem of spin is a fundamental problem in physics that seeks to understand the behavior of particles with intrinsic angular momentum, or spin, at the microscopic level. It involves understanding the statistical properties of spin states and their interactions with other particles.

2. What is the significance of studying the statistical mechanics problem of spin?

Studying the statistical mechanics problem of spin allows us to better understand the behavior of matter at the atomic and subatomic level. It has applications in diverse fields such as condensed matter physics, quantum computing, and materials science.

3. What are some key concepts in the statistical mechanics problem of spin?

Some key concepts in the statistical mechanics problem of spin include the spin Hamiltonian, the Ising model, the Heisenberg model, and the concept of spin waves. These concepts help us model and understand the behavior of spin systems.

4. How is the statistical mechanics problem of spin related to thermodynamics?

The statistical mechanics problem of spin is closely related to thermodynamics as it seeks to understand the macroscopic properties of a system based on its microscopic properties. The concept of entropy, which is central to thermodynamics, is also important in the study of spin systems.

5. What are some current research topics in the statistical mechanics problem of spin?

Some current research topics in the statistical mechanics problem of spin include the study of spin glasses, topological phases of matter, and quantum spin liquids. There is also ongoing research into the use of spin systems for quantum computing and information processing.

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