# Meaning of "vacuum state"?

I just learned about $|0\rangle$ and siblings $|0_\gamma\rangle$ and $|0_\infty\rangle$ while studying coherent and squeezed states in a QM class, and I have a question about the meaning of "$|0\rangle$".

I get that it is the ground state of the QHO, with energy $\frac{1}{2}\hbar\omega$, no problem there. But the Prof in class proceeds to calling it the vacuum state", he "squeezes the vacuum state with the squeezing operator" (lovely!), but I fail to see that this "vacuum" is "empty" at all: there is a particle or a QHO there.

Why does the Prof call it the "vacuum" state? Is it because the lowest possible energy level of any field anywhere is in fact $\frac{\hbar}{2}$ via Heisenberg uncertainty? (that was not in the class, so I'm probably wrong here)

It is defined starting from the Fock space, that is the sum of spaces with any number $n\in \mathbb{N}$ of (identical particles). The space with zero particles is, roughly speaking, one-dimensional, and its basis vector is the vacuum.
In quantum mechanics, one can use the Fock space formalism but with a slightly different meaning, because the subspaces become the subspaces corresponding to a number $n$ of energy excitations of the harmonic oscillator. The vacuum is then the ground state, i.e. the state with minimal energy and no excitations.