# Energy increase due to the Zeeman effect

The Zeeman effect is the change in energy of a system with a permanent magnetic moment in the presence of an external magnetic field. Consider electrons for example. In general, in $$B \neq 0$$ electrons with $$\mu$$ aligned with $$B$$ will experience an increase in energy, while electrons with $$\mu$$ antiparallel to $$B$$ will experience a decrease in energy.

My question is why aren't electrons with $$\mu$$ antiparallel to $$B$$ forced to change their spin orientation, since a decrease in energy is favourable? Is it because of the Pauli exclusion principle? And does this imply that energy increase due to the Zeeman effect is only observed in systems where all energetically favourable states are already occupied?

$$\mu$$ parallel to $$B$$ is the lower energy state (the one with a negative energy). In some conventions this might mean spin is anti-aligned with $$B$$ if $$\mu$$ points the opposite direction of spin because the electron is negatively charged. Consider this classical current loop. See that the forces on it make it want to align its magnetic field with the background field
This is a little odd - if the magnetic field's potential is $$\int B^2$$ (ignoring some factors of $$\mu_0$$ and $$2$$), then doesn't that mean a stronger magnetic field has more potential? Well remember that for charged particles, magnetic fields do no work. In my classical current loop example above, the loop will turn and the current will decrease, and $$\int B^2$$ won't change. The current might remain constant if the loop was hooked up to a power supply, in which case the power supply will supply the energy that makes $$\int B^2$$ increase.
The resolution for the permanent quantum dipole is that actually the the potential of the magnetic field is $$\int H\cdot B=\int B\cdot B/\mu_0-M\cdot B$$, so magnetization actually does want to point in the same direction as B.