Spin Hilbert space

Question

I'm currently doing some quantum mechanics and was able to transform my Hamilton operator to something that basically looks like this:

$$ H = H_{xy} + \frac{p_z}{2M} + \alpha S_z, $$

where $H_{xy}$ is the Hamilton operator of a harmonic oscillator in the $x,y$ plane.

The exercise states that I should find the eigenfunctions.

I know that for two operators that commute I can find a base of common eigenfunctions.

I can proof that $H_{xy}$ and $p_z$ commute and I'm also pretty sure that their eigenfunctions "live" in different Hilbert spaces so their common eigenfunction would be a product of the eigenfunction of a two dimensional harmonic oscillator in the $x,y$ plane and a plane wave in $z$ direction.

Now I'm not quite sure what to do about the Spin $S_z$. I couldn't find some neat definition like for the orbital angular momentum $\boldsymbol{L} = \boldsymbol{x} \times \boldsymbol{p}$. I did some research and I think it's because there is no classical analogon. From the definition of $\boldsymbol{L}$ I conclude that it "lives" in the same Hilbert space as $p_x, x$ $p_y, y$ and $p_z, z$.

Now is $S_z$ completely separated from $x,y,z$? That $z$ in $S_z$ confuses me.

Are the eigenfunctions simply a product of the eigenfunctions of all three parts?

Answer 1

Unlike in QFT where you can derive spin from more basic principles, in ordinary non-relativistic QM spin is essentially defined into existence as a group of operators $S^i = (\hbar/2) \sigma^i$ that satisfy the algebra $$[\frac {\sigma^i}{2}, \frac{\sigma^j}{2}] = i \epsilon^{i j}_{\,\,k} \frac{\sigma^k}{2}.$$

The dimensions in the Hilbert space on which the Pauli operators act are completely separate from the more familiar $x^i$ and $p^i$ operators, as you suspected. It's just two extra dimensions added to the Hilbert space. The algebra they satisfy is loosely inspired by the commutation relationships that the $L^i$ operators satisfy, the latter of which are composed of the $x$'s and $p$'s, but the spin operators are defined completely on their own; they're as fundamental as the $x$'s and $p$'s. It's just a weird convention that we give them the same indices (at least in this problem; I'm used to referring to them as $S_1$, $S_2$, and $S_3$; you could do the same to avoid confusion).

So commute away, my friend!