Difference between minimal coupling $(p-qA)$ and $B cdot L$,In Hamiltonian

Question

It seems like when coupling an electron to a magnetic field we always use $B cdot S$ to couple its spin magnetic moment to the magnetic field.
However sometimes we couple its motion to the magnetic field with $(p-qA)$ and sometimes with $B cdot L $.
I guess that the former form is more general and can be used even when the electron will not have well defined L, but, other than that, what are the differences?
I would also like to know what is ‘minimal’ about the minimal coupling. What other forms are ‘less minimal’ than the minimal coupling and why?

GiorgioP · Accepted Answer

The meaning of minimal in minimal coupling refers to the possibility that a system interacts with an electromagnetic field through different multiples. If only the monopole interaction is present, that is called minimal coupling.
In the case of the electron, the minimal coupling only takes into account the orbital motion, via the $p-qA$ term in the Hamiltonian. However, that full interaction should also include the presence of the spin. In that case, the non-relativistic Hamiltonian contains an $S cdot B$ term.

Difference between minimal coupling $(p-qA)$ and $B cdot L$,In Hamiltonian

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