What are the energy eigenvalues of a harmonic oscillator?

[Lotto]

TL;DR Summary

I have this formula $E_n=hf\left(n+\frac 12 \right)$. I don't understand what energy it describes.

Is it a total energy of a vibrating molecule? So is it a sum of potential and kinetic energy? Or it is only a total energy of a vibrational motion of the molecule? Or is it only a potencial energy, when it is related to a dissociation curve? I am confused.

 

[Nugatory]

TL;DR Summary: I have this formula $E_n=hf\left(n+\frac 12 \right)$. I don't understand what energy it describes.

Is it a total energy of a vibrating molecule? So is it a sum of potential and kinetic energy? Or it is only a total energy of a vibrational motion of the molecule? Or is it only a potencial energy, when it is related to a dissociation curve? I am confused.

It looks the energy levels of an ideal harmonic oscillator, and will be the sum of the potential and kinetic energy of the oscillator. How this relates to a vibrating molecule depends on how accurately the molecule can be modeled as an ideal harmonic oscillator.

In general, we know what energies are involved by looking at the Hamiltonian that we started with. In the case of the ideal harmonic oscillator, that Hamiltonian contains a kinetic energy term and a potential energy term.

 

[vanhees71]

Indeed, these are the energy eigenvalues of a harmonic oscillator. It describes the conserved total energy of the oscillator, when it is prepared in a state of determined energy. The possible values of this total energy are the eigenvalues of the Hamilton operator,
$$\hat{H}=\frac{1}{2m} \hat{p}^2 + \frac{m \omega^2}{2} \hat{x}^2.$$
The energy eigenvalues are
$$E_n=h f \left (n+\frac{1}{2} \right) = \hbar \omega \left (n+\frac{1}{2} \right), \quad n \in \{0,1,2,3,\ldots \}=\mathbb{N}_0,$$
where $\hbar=h/(2 \pi)$ is the "modified quantum of action/Planck's constant)". Nowadays almost nobody uses the original $h$ anymore.