Exploring the Mystery of Electron Magnetic Moments in Quantum Mechanics

[calinvass]

If electrons have a magnetic moment (or behave like they have) why don't all orient North Up in a magnetic field (pointing down)? Plus, if thought as magnetic dipoles, like poles repel, therefore all should point the same direction. In a Stern-Gerlach apparattus shouldn't the magnetic moment align in the same direction regardless of their deflection direction?

 

[vanhees71]

In a constant magnetic field the spin precesses around the direction of the magnetic field. It's a nice exercise to solve the corresponding initial-value problem. The corresponding Hamiltonian is
$$\hat{H}=-\frac{q}{2m} g_s \hat{\vec{S}} \cdot \vec{B}.$$
You can go into the spin representation and use $\hat{\vec{S}}=\frac{1}{2} \hat{\vec{\sigma}}$, where $\hat{\vec{\sigma}}$ are the Pauli matrices. If you put the $\vec{B}$-field in the $z$ direction, it's very easy to solve the initial-value problem,
$$\mathrm{i} \frac{\mathrm{d}}{\mathrm{d} t} |\psi(t) \rangle=\hat{H} |\psi(t) \rangle, \quad |\psi(0) \rangle=|\psi_0 \rangle.$$

In the Stern-Gerlach experiment you have an inhomogeneous magnetic field, and you consider the full problem of the (electrically neutral!) particle including position. It turns out that then there's also a force acting on the particle, and thus the particle will be deflected such that you get an entanglement between position and value of the spin component along the direction of the magnetic field, i.e., an unpolarized particle beam will split up into partial beams all with well prepared spin components in direction of the magnetic field.

 

[calinvass]

That means the electrons that go up will have magnetic moment pointing up and there should be a force pointing down because the north pole repells the north pole of the electron. I understand you get the entanglement as spin up goes up, spin down goes down, but why the force is in opposite direction that it would normally be?

 

[Jilang]

My understanding is that they are almost equally happy prescessing around either north or south. Weird that!

 

[calinvass]

Yes, but in a non uniform field pointing down (N-S) why would they precess to point up?
QM uses these mathematical concepts to model systems so that the results matches the experimetal results but also can give answers to more complex situations before you will actually perform the experiment. I'm still not sure if it answers my simple question.