State Ket Interpretations I have a question that came from lecture earlier in the semester, that I never fully understood:
Let's suppose that we have an electron, that at $t=0$ is in an eigenstate of $\hat{S}\bullet\hat{n}$, with an eigenvalue of $\hbar/2$. My question is... is it more correct to write the eigenvalue equation for this situation as $$\hat{S}\bullet\hat{n}\rvert \psi \rangle = \hbar/2\rvert\psi\rangle$$ OR as $$\hat{S}\bullet\hat{n}\rvert \hat{S}\bullet\hat{n} \rangle = \hbar/2\rvert\hat{S}\bullet\hat{n}\rangle~?$$ Here $\bullet$ is the dot product in 3D space. Someone please explain the difference between these two setups.  
 A: Fundamentally this is just a difference in labelling, and, as long as you are clear about what you mean, you can label things however you like.
Having said that in the second case you seem to be labelling your state with operator of which it is an eigenket. This is probably not a great choice in most situation because one operator will generally have many different eigenket, which this notation will not distinguish between. A very common choice, however, is to label the eigenket of an operator by their eigenvalue, so in your example the state would be lablled $|+\hbar/2\rangle$ (In your spesific case you might also want to include the vector $\hat{n}$ if you are going to be considering many different directions). This is often enough to uniquely identify the state and has the upshot of telling us something useful about the state.
$|\psi\rangle$ is often used to signify a 'generic' state in whatever context we are considering, but it is also used as the go to name when we can't think of anything more interesting. 
A: 
Let's suppose that we have an electron, that at $t=0$ is in an eigenstate of $\hat{S}\bullet\hat{n}$, with an eigenvalue of $\hbar/2$. My question is... is it more correct to write the eigenvalue equation for this situation as $\hat{S}\bullet\hat{n}\rvert \psi \rangle = \hbar/2\rvert\psi\rangle$ OR as $\hat{S}\bullet\hat{n}\rvert \hat{S}\bullet\hat{n} \rangle = \hbar/2\rvert\hat{S}\bullet\hat{n}\rangle$? 

An eigenvalue equation that produces a real value  is composed of a Hermitian operator,  a eigenvector it operates on and  the eigenvalue itself.
The correct notation is: 
$$\hat{S}\bullet\hat{n}\rvert \psi \rangle = \hbar/2\rvert\psi\rangle$$
This produces an operator, and eigenvalue and and the same vector you started with.
Usually, with a spin operator, we use $|+ \rangle$ or $|- \rangle$ than $|\psi\rangle$  as up and down directions, or with up and down arrows instead of + and -. 
