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The Brillouin function, defined as

$$B_j(x) = \frac{j+1/2}{j} \coth\left(\frac{j+1/2}{j} x\right) - \frac{1}{2j}\coth\left(\frac{1}{2j} x\right),$$ tends to the Langevin funcion

$$ \mathcal{L}(x) = \coth(x) - \frac{1}{x}, $$

in the limit when $j \rightarrow\infty$. My lecture notes refer to this limit as the classical limit.

This function comes up in the study of spin interaction in the presence of a magnetic field, and is of particular interest in Statistical Mechanics. Here, $j$ is the spin number of a particle.

My question is: how does $j \rightarrow\infty$ make this system classic?

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The total angular momentum $j$ is quantized. If you take the limit $j \rightarrow \infty$ then a huge amount of states will become accesible to the system (more specifically 2$j$ + 1 states). Due to this, $j$ (a vector), can now point in almost every direction just like a normal "classical" vector can take on every value. So you can treat the system in a classical way.

In addition, this is justified by something which is known as the "correspondence principle". It states that in the limit of large quantum numbers every quantum system converges to a classical system.

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