# Question about an “empty ket” and Dirac's notation

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 $$\sum c_n |n\rangle$$ ?

Dirac is a brilliant writer and this is a nice paper. But, in modern physics (at least in my experience), it is not particularly common to use $$|\rangle$$ or $$|\rangle_\Psi$$ to refer to a state.
Looking through the paper, I think that in the language of the time (pre-Dirac notation), one would use $$\Psi$$ or $$\psi$$ as special symbols referring to the state. So instead of $$|a\rangle$$, one would write $$\Psi_a$$ or $$\psi_a$$.
In more modern notation, the symbols $$\Psi$$ or $$\psi$$ have no special meaning, and what appears in the ket is the label of the state. For example, one would use $$|a\rangle$$ to refer to a state $$a$$. One could also use $$|\Psi\rangle$$ or $$|\psi\rangle$$ to refer to a state. More often than not, $$|\Psi\rangle$$ or $$|\psi\rangle$$ are used to refer to "generic" states (arbitrary superpositions of eigenstates), whilst other symbols appearing in the ket like $$|a\rangle$$ tend to refer to special states. For example, perhaps $$a$$ is an eigenvalue of some operator $$A$$, and $$|a\rangle$$ is the corresponding eigenstate. Of course, your mileage may vary since notation is flexible, and it's important to be aware of how notation is being used in context.