Question about an "empty ket" and Dirac's notation

Question

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 $\sum c_n |n\rangle$ ?

Answer 1

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.