Conceptual understanding of quantum harmonic oscillators The way I understand it is that we have the time-independent Schrödinger equation for a particle described by a wave function $\psi$ in a potential V(x)
$$-\frac{\hbar}{2m}\frac{d^2}{dx^2} \psi + V(x)\psi(x) = E\psi(x)\tag{1} \, .$$
If I were to approximate the potential $V(x)$ to be quadratic in $x$ (for example $V(x) = \frac{1}{2} m \omega x^2$) we know that this describes a harmonic oscillator.
If we then solve equation (1) for $\psi$ and $E(x)$ we will find that 
$$E_n = \hbar\omega \left(n+\frac{1}{2} \right)$$ and the corresponding $\psi_n$.
Now it is mostly the sentences I can't really formulate.
Like for example which of these statements is correct:


*

*Our particle described by $\psi$ is a harmonic oscillator.

*Our particle described by $\psi$ is modeled by a harmonic oscillator.

*Let us consider a particle described as a harmonic oscillator...
As you might see, I have the basic mathematical framework, but I can't really grasp the words/concepts surrounding it.
 A: There is no prior difference about naming the concept. IT depends on how you call things, this is just one more.


*

*A quantum free particle = a wavefunction that obeys the Schrödinger's equation with no potential energy.


*

*A particle enclosed in an infinite-well = a wavefunction that obeys the Schrödinger's equation with such infinite-well potential.

*A quantum Harmonic Oscillator = a wavefunction that obeys the Schrödinger's equation with the harmonic potential.
But there are two "problems" here. The first one is general: talking about "particle " at this level is "risky". How do you define "particle"? If you want to be clear, talk in terms of "wavefunctions" or "systems".
And now, here comes the problem. The quantum harmonic oscillator behaves completely differently from the classical one. In fact, the states with defined energies are stationary, so they do not change in time, so it cannot oscillate. Yep, states with a single energy do not change in time. So we've got here an oscillator which does not oscillate!
This means that, if your wavefunction if, for example, the one with $n=0$, it has only one possible energy, $E_0=½\hbar\omega$. Since it has only one possible energy, that wavefunction stays the same all the time. So it is not oscillating. 
That's why saying "it's an harmonic oscillator" can be a little confusing. When we say "a quantum harmonic oscillator", we mean " a particle under the influence of a potential that classically would yield a harmonic oscillator", but we are lazy and we just say "harmonic oscillator". The thing is that quantum particles behave really differently from classical mechanics.
Your sentences are right, because everybody understands that it means "a wavefunction that obeys Schrödinger's equation with a quadratic potential". That quadratic potential would give a classical harmonic oscillator if we used classical mechanics and the scalar equation for energy, but this is quantum mechanics and it works differently. The name has been kept the same.
A: The same problem arises classically. If you have a mass on a spring we have a harmonic oscillator. But do we say the mass is a harmonic oscillator? Perhaps. If so then we would say for your question that the particle is a harmonic oscillator
But the spring is an important part of the system too. If we really want to be picky I would say that that the spring mass system constitutes a harmonic oscillator system. In that case we would say the particle in the given potential can be treated as a harmonic oscillator system.
In that sense anything which has a Hamiltonian that looks like
$$
H = ax^2+bp^2
$$
can be called a harmonic oscillator.
In any case, all of the sentences you wrote down are reasonable and would be understood by anyone.
