In introductory quantum mechanics it is always said that $∣ \rangle$ is nothing but a notation. For example, we can denote the state $\vec \psi$ as $∣\psi \rangle$. In other words, the little arrow has transformed into a ket.
But when you look up material online, it seems that the usage of the bra-ket is much more free. Example of this usage: http://physics.gu.se/~klavs/FYP310/braket.pdf pg 17
A harmonic oscillator with precisely three quanta of vibrations is described as $|3\rangle$., where it is understood that in this case we are looking at a harmonic oscillator with some given frequency ω, say.
Because the state is specified with respect to the energy we can easily find the energy by application of the Hamiltonian operator on this state, H$|3\rangle$. = (3 + 1/2)$\omega h/2\pi |3 \rangle$.
What is the meaning of 3 in this case? Is 3 a vector? A scalar? If we treat the ket symbol as a vector, then $\vec 3$ is something that does not make sense.
Can someone clarify what it means for a scalar to be in a ket?
Answer
What is the meaning of 3 in this case?
In this case, the character "3" is a convenient, descriptive label for the state with three quanta present.
It is often the case that an eigenstate is labelled with its associated eigenvalue.
In the harmonic oscillator case, the number operator commutes with the energy operator (Hamiltonian) so a number eigenstate is also an energy eigenstate.
Thus, the state with three quanta present satisfies
$$\hat N |3\rangle = 3\,|3\rangle$$
But, it also satisfies
$$\hat H |3\rangle = (3 + \frac{1}{2})\hbar \omega\, |3\rangle = \frac{7}{2} \hbar \omega\,|3\rangle$$
So we would be justified in labelling this state as
$$|\frac{7}{2} \hbar \omega\rangle $$
though that's not typical.
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