I'm confused by my books treatment of the Schrödinger equation. In steado f listing my questions at the end of my post, I'll add them as questions in parentheses after the line in question.
For a free particle:
$$i \hbar \left| \dot{\Psi} \right \rangle = H\left| \Psi \right \rangle = \frac{P^2}{2m}\left| \Psi \right \rangle$$ (where did the potential go?)
The normal mode solutions are of the form: $\left| \Psi \right \rangle = \left| E \right \rangle e^{-iEt/ \hbar}$
Feeding this into the equation above (the schrödinger equation written above), we get the time-independent equation for $\left| E \right \rangle$ :
$$H \left| E \right \rangle = \frac{P^2}{2m}\left| E \right \rangle = E \left| E \right \rangle$$
(this follows from the eigen-equation, where the eigenvalue must be equal to $E$?)
This problem can be solved without going into any basis. First note that any eigenstateof $P$ is also an eigenstate of $P^2$. So we feed the trial solution for $\left| p \right \rangle$ into the equation (the equation directly above).
$$\frac{P^2}{2m}\left| p \right \rangle = E\left| p \right \rangle $$
(why are we all of a sudden talking about the ket $p$?
Which means that $\left| p \right \rangle = \pm \sqrt{2mE}$
This gives us two eigenkets of $E$, which span an eigenspace.
The next steps are to set up the propagator (which I have questions about, but understanding the previous steps may help me clear them up myself).