First course in QM, so understand that I know next to nothing. I have been reading and watching videos from a number of sources but I am confused a bit in certain aspects of the Dirac notation due to various sources giving me seemingly contradictory information.
For example, I saw the polarization state as $$ \left|\theta\right> = \cos\theta\left|x\right> + \sin\theta\left|y\right> $$ and the spin state as $$ \left|s\right> = \Psi_u \left|u\right> + \Psi_d \left|d\right> . $$
To me it implies that some observable's state is written as a linear combination of basis kets, where basis ket has a 1 in a designated, unique place and zeros everywhere else. And the inner product of some state with the other should give the probability amplitude of the given state collapsing (?) into another: for example, $ \Psi_u = \left<u\middle|s\right> $ gives the probability amplitude of the system being detected as spin up.
But this seems to get confusing if we extend the same thing to things like position, momentum, or energy. For a particle in a one-dimensional space, shouldn't the overall state be $$ \left|x\right> = \sum_i \Psi(x_i) \left|x_i\right>, $$ where $\left|x_i\right>$ is a Dirac delta function i.e. a position eigenfunction, and $\Psi(x_i)$ is the probability amplitude at that particular position eigenvalue $x_i$?
Why do so many texts write this summation as $\left|\Psi\right>$? It's confusing: $\Psi$ is for wavefunction. Why don't we represent a state of an observable by writing its name in the ket? What is $\psi$ supposed to mean here? I thought $\Psi$ was the probability amplitude.
Confusion about Dirac notation. When represent basis kets of position state by dirac delta function, when by matrices? These are supposed to be eigen-function right?