This question is related to this other one and it's about Bra-Kets formalism. Hope I'm not bothering you but the truth is I'm very confused.
Reading 1939 Dirac's publication on Bra-kets notation "A new notation for Quantum Mechanics" (pdf) he says that we can understand the wave function $\Psi$ as an empty ket.
$$\Psi \rightarrow |\rangle \equiv |\rangle_{\Psi}$$
As the same time a state $a$ in a wavefunction adopts the form $\Psi_a \rightarrow |a\rangle$. With column vector wavefunctions (complex transposed) we can write $\Psi_a^\dagger \rightarrow \langle a|$.
I understand the "simplicity" behind this and the adventage of having only one way to denote what before admitted two representations.
So, getting to the point: if I have an harmonic oscillator and I want to represent:
$$\Psi = \sum c_n \psi_n e^{-iE_n t/\hbar}$$
in which the wavefunction is composed of the first two states equiprobably:
$$\Psi = \frac{1}{\sqrt{2}}\left[ \psi_0 e^{-iE_0 t /\hbar} + \psi_1 e^{-iE_1 t /\hbar}\right]$$
in Dirac's notation I know that
$$\psi_0 \rightarrow |0\rangle$$ $$\psi_1 \rightarrow |1\rangle$$ $$\Psi \rightarrow |\rangle$$
so following the above:
$$|\rangle = \frac{1}{\sqrt{2}} [|0\rangle e^{-i\omega_0 t}+ |1\rangle e^{-i\omega_1 t}]$$
Is this correct? What's the difference between $|\rangle$, $|\rangle_{\Psi}$ and $|\Psi\rangle$?
Can $\psi$ be written as $\sum c_n |n\rangle$ ?