A Classical intr. ang mom and quantum intr ang mom relation

[fisher garry]

Here is a link to a derivation of classical intrinsic angular momentum:

https://www.scribd.com/document/349675642/Potential-energy-Prop-to-SL

In 2.1 in the image above they define: $m=2\mu S$ and say that $\mu$ is the bohr magneton. By using the defintion of the bohr magneton I get $m=2\frac{e \hbar}{2m_e}S=\frac{e \hbar}{m_e}S$. But that is the same as the relation in the classical intrinsic angular momentum derivation except that it has an extra $\hbar$ in the numerator. Anyone who knows how to derive an relation between the classical intrinsic angular momentum relation to magnetic moment and the one with an extra $\hbar$?

 

[thephystudent]

This is only a matter of units, $S$ as defined on the LHS is dimensionless, where it has the dimension of angular momentum (length*momentum/time) on the RHS. Typically, one includes the $\hbar$ in the definition of quantum spin, as in $S_z=\frac{\hbar}{2}\sigma_z$ with $\sigma_z$ the pauli matrix.
Interestingly, $\hbar$ can count both as a quantum of action and of angular momentum (probably there's a deep connection, though I can't think of it right now).