Hamiltonian for a particle with spin and in 3 dimensions For a particle in a box, I'm fairly confident with how to work with it. It's equation is given by:
$$i\hbar \frac{d}{dt}=-\hbar ^2 \frac{1}{2m} \nabla ^2$$
And it's solution gives the wave function, which describes the position of the particle if written in terms of the position basis. This; however, doesn't describe the spin. So to describe the spin and position of the particle, should I take the tensor product:
$$|s\rangle\otimes |\psi\rangle$$
Such that $|s\rangle$ is the spin state, and $|\psi\rangle$ is the wave function?
If so, how does the Dirac equation describe both the spin and position state? It seems the Dirac equation only describes the evolution for the spin of both particles and antiparticles.
 A: A particle with spin is not a tensor product of the spin part and the space part. At least, not in general. I suppose it could be in some very special conditions, but it is not the usual thing.
A particle with spin is operated on by a particular matrix representation of the group group SU(2), modified by the need to be relativistic. For electrons (and positrons) in the Dirac equation you want to look at the "gamma matrices."  A very brief intro can be found in the wikipedia entry on the subject.
In very brief terms (even briefer than the wiki article) it goes like so. A spin-half particle would be in a 2x2 rep of SU(2) based on the Pauli matrices. Adding in the need for time-reveral and relativistic covariance, you have to move to the gamma matrices, and add a fourth one. The result is, they generate the changes that build up SU(2), plus the changes involved in special relativity.
Basically, each particle will "sit" in a particular representation of the symmetry group of rotations and relativistic invariance. Spin 0 will sit in a scalar rep. Spin 1 (photons for example) in a vector rep, where a 360 degree rotation brings you back to your start. And spin 1/2 will sit in a spinor rep that requires 720 degrees to get back to identity, 360 only providing a minus sign.
A particle has to sit in a rep. Otherwise if you did a rotation or a change of velocity, the particle would transform onto something besides the type of thing you thought you started with. If it was a photon and an electron near each other, then a 360 degree rotation will change the sign of only the electron.
I am leaving out just huge amounts of mathematics here. Group theory and relativistic theory and representations and Lie algebras and other stuff that I'm not recalling just off. To understand this in detail you would have to learn all of that stuff, which is at least a couple years of university.
