# Spin in magnetic field: work?

1. May 17, 2007

### da_willem

from the Dirac equation one can straightforwardly derive that in the nonrelativistic limit, the Hamiltonian of a charged particle in a magnetic field acquires a term

$$-\vec{\mu} \cdot \vec{B}$$

which shows the particle has an intrinsic magnetic moment (by analogy with the classical expression of the energy of a magnetic dipole in a magnetic field). Contrary to the classical case however, this moment is not sustained by a current (at least that is not the mainstream thought).

Now, looking at this classically the energy of an electron in a magnetic field causes a force if the field is inhomogeneous

$$\vec{F}=-\nabla(-\vec{\mu} \cdot \vec{B})=\nabla(\vec{\mu} \cdot \vec{B})$$

So the electron starts moving right? Now what force is responsible for this movement, who does the work?

2. May 17, 2007

### cesiumfrog

Regards force, I was content with the answer that we cannot expect results from classical EM to hold in the limit where classical EM fails.

Regards work, I presume it is energetically favourable (for whatever apparatus that provides the eternal magnetic field) if any stray dipoles are shifted aside, and the exact mechanism in that macroscopic apparatus (as per the previous thread) will turn out not to be directly magnetic.

3. May 18, 2007

### da_willem

So you think that in the process of moving the spin, the magnetic field decreases (field energy converted to kinetic energy) and that the source of the magnetic field is what's doing the work?

But then what if this magnetic field is also quantum mechanical in origin, like in the case of the hyperfine splitting. Here the (inhomogeneous) magnetic field of one spin interacts with the 'dipole moment' of another spin to form an interaction energy.

Suppose we can 'switch of' other interactions, I suppose these spins start moving. Where does this kinetic energy come from, as demanded by energy conservation? Classically the dipole moment can diminish, converting the kinetic energy of the charges generating the moment into translational motion. Qunatum mechaniclly however, the spins of a particle can't change, so where does this kinetic energy come from?

If the answer is, from the magnetic field energy, then isn't this exactly what can't happen classically: magnetic field energy > kinetic energy = magnetic work?!

Or does this (even classically) only hold for charges and the magnetic Lorentz force, and can one think of instances where classical magnetic moments start moving by converting magnetic field energy into kinetic energy?!

4. May 18, 2007

### cesiumfrog

I think that from the perspective of whatever creates the external force, there is no difference between intrinsic magnetic dipole and a classical current loop. As such, in accelerating the dipole, the external apparatus should do work and (per the previous thread) this loss of energy may be attributed to something entirely other than the magnetic field itself.

But from the perspective of the elementary point particle, to the extent that we can speak about force and the like, work must be done by the magnetic field because there are apparently no other internal components through which to attribute the increase in energy. Anyway, that's my take on it.

5. May 18, 2007

### da_willem

But I'm not talking about any external force, just two spins in a vacuum. I didn't do the classical calculation, so I'm not sure if they even start to move due to the inhomogeneous fields they posses, but if they do, where does the energy come from?!

6. May 18, 2007

### cesiumfrog

This is hand-wavy, but.. If the two magnetic fields add linearly, and the energy of the field is proportional to the square, would that suffice?

7. May 19, 2007

### da_willem

So, I guess my point is, magnetic forces can do work, maybe not on charged particles by the magnetic part of the Lorentz force, but on magnetic dipoles they can!