hilbert space - Conceptual understanding of the Quantum Harmonic oscillator

First: When we consider a quantum particle in a harmonic (quadratic) potential we say that this particle is a harmonic oscillator, because it behaves like one. Is this correct?

Now let us assume our particle is in the groundstate $\mid \psi_0 \rangle$. This would mean that in this picture it would be described by the lowest wave with energy $E_0$.

I have a bit of a hard time to understand what happens when we increase the energy of said particle. The old wave function disappears and we find our particle in a new state $\mid \psi_n\rangle$ with some new energy $E_n$ described by a new wave in the picture. Is that correct?

Honestly, I'm giving a presentation on this topic soon, and want to visualize such a 'jump' from a lower energy state to a higher energy state of a particle in the harmonic potential, in the correct way.

Any help is greatly appreciated!