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My question is fairly simple and straightforward. I'm studying Quantum Mechanics, specifically the spinor formalism. I understand that one can define a generator of rotations, say around axis $z$ by an angle $\phi$, by considering the exponential $$ \mathcal{D}_z(\phi)=\exp\left(-\frac{iS_z}{\hbar}\phi \right) $$ I also understand that this notion can be further generalized to a rotation around a general direction given by $\hat{n}$ such that $$ \mathcal{D}(\hat{n},\phi)=\exp\left(-\frac{i\vec{S}\cdot\hat{n}}{\hbar}\phi \right) $$ Where the inner product $\vec{S}\cdot\hat{n}$ can be written in terms of the Pauli matrices $\sum_k \mathbb{\sigma}_kn_k$. Now, my problem really starts when trying to construct the eigenstates of the operator $\vec{S}\cdot\hat{n}$. In my text book the eigenstates are defined by $$ \vec{S}\cdot\hat{n}|\vec{S}\cdot\hat{n}, \pm\rangle=\pm\frac{\hbar}{2}|\vec{S}\cdot\hat{n}, \pm\rangle $$ but I can't really wrap my head around this ket $|\vec{S}\cdot\hat{n}, \pm\rangle$, what does it mean exactly to have that inner product inside the ket? Is it just spin up or down in the $\hat{n}$ direction? If so, how can we express it in the $S_z$ basis?

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I generally dislike this type of ket notation as it leads to this sort of confusion. Understand this, when someone writes an object $|symbols\rangle$, this is a vector in a vector space which is labeled by $symbols$. In more standard math notation, one might refer to this as the vector $v_{symbols}$. There is nothing more to it.

So, whenever we are working with bra-ket notation, we have to decide what symbols will be convenient for keeping track of which vector we would like to think about. The particular notation you are using here is of the form $|A,a\rangle$. The idea is that we first list an operator, and then an eigenvalue of that operator. The implication then, is that the vector $|A,a\rangle$ is the eigenvector of $A$ with eigenvalue $a$, so it will satisfy $A|A,a\rangle=a|A,a\rangle$ as a matter of definition of notation.

So the only question you could still have which isn't a matter of notation here is how one might actually show that the eigenvalues of $\vec S\cdot\hat n$ are actually $\pm\hbar/2$. But you can convince yourself of this by writing out the Pauli matrices, writing down the matrix $\vec S\cdot\hat n$ explicitly, and computing its eigenvalues (it's only $2\times 2$, so not too bad).

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Is it just spin up or down in the $\hat n$ direction?

Yes, exactly!

If so, how can we expresse it in the $S_z$ basis?

You can write it like this, $$ |\psi\rangle = \cos\left(\tfrac{\theta}{2}\right)|\uparrow\rangle + \text{e}^{\text{i}\varphi} \sin\left(\tfrac{\theta}{2}\right)|\downarrow\rangle $$ which is often used in the context of the Bloch sphere (Wiki link). The angles $\theta$ and $\varphi$ are defined as the usual angles of spherical coordinates, depending on which direction $\hat n$ points to. You can find a derivation of this state e.g. here (YouTube link).

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