I have few questions, but any elaboration that help me to understand is welcome . Please, do not hesitate in made "silly" clarifications or corrections, as I am not a physicist.
I am trying to understand the term $- \vec \mu \cdot \vec B$ included in hamiltonians (e.g.) used to describe charged particles in magnetic fields. Where $\vec \mu$ represents the magnetic momentum of the particle (due to spin) and $\vec B$ some external magnetic field.
I've seen many books where is just stated that its inclusion is necessary. Others makes analogyes with the energy of a classical magnetic dipole in a magnetic field. It is confusing to me. I followed the equations describing the force excerted on charged particles moving in a loop (e.g):
$\vec F = \nabla(\vec \mu \cdot \vec B) $
From there, many sources deduce a change in some potential energy $E = -\vec \mu \cdot \vec B$
What does $E$ exactly mean?
My thoughts: It is not as clear to me the meaning of that. As Lorentz's force cannot do work, it can not be net work at microscopic level due to $\vec B$. So, $E$ may represent some macroscopically perceptible energy change, but there must be an equal and opposite change at microscopic level (in order to have energy conservation). A reasonable mechanism could be the following: As the loop is orienting, the flux of $\vec B$ through the loop is changing in time giving rise to a $emf$ due to Faraday's law of induction. This could change the speed of the charge carriers producing the compensation in the total energy through the change in the kinetic energy. The macroscopically perceptible force would be due to the change in potential energy because of the $emf$. ¿Is this reasonable?.
The term $- \vec \mu \cdot \vec B$ can (and must) be included in the quantum Hamiltonian and, in close analogy to the above, does not make changes in the total energy of an electron through its orbital contribution (while $\vec B$ is not time dependent). This does not seem apply to the spin contribution: being that its magnetic moment is fixed, the $emf$ argument seems to be invalid. How does it work?
If I understand correctly, the energy of a non zero spin particle go down when an uniform $\vec B$ (in comparison with $\vec B=\vec 0$) is present. As the constant magnetic field cannot change the total energy, the energy change took place while increasing the field from zero to a constant value. Is this correct?