Energy of molecular vibrations I have just read that the energy of a molecular vibration with frequency $\omega$ has eigenvalues of $(n+\frac12)\hbar\omega$, where $n$ is the quantum number. However, this equation really surprises me, because it does not contain any term related to the masses of the atoms involved. For instance, it seems obvious that the vibration of a bond between two Fluorine atoms should be much greater than the vibration at the same frequency of two Hydrogen atoms, since Fluorine has almost 15 times as much mass. Is there something I'm missing here, or is the energy of vibration really independent of the masses of the molecules involved?
 A: The frequency depends not only on the (reduced) mass of the system but also on the broad shape of the potential.  If the atoms interact via $V(r)$, where $r$ is the relative distance between the atoms, then near the minimum
$r_0$ of this potential the Schrodinger equation is (assuming no rotation):
$$
-\frac{\hbar^2}{2\mu}\frac{d^2}{dR^2}\psi(R)+ \frac{1}{2}V^{\prime\prime}(r_0)R^2\psi(R)=(E-V(r_0))\psi(R)\tag{1}
$$
where $R=r-r_0$ is the departure from the equilibrium position, and
$V^{\prime\prime}(r_0)$ is the 2nd derivative of $V$ evaluated at $r=r_0$.
Eq.(1) is nothing but the Schrodinger equation for a harmonic oscillator of frequency
$$
\omega=\sqrt{\frac{V^{\prime\prime}(r_0)}{\mu}}
$$
so $V^{\prime\prime}(r_0)$ plays a role analogous to the spring constant in the usual SHO.
In this way the vibrational frequencies of various molecules can be estimated assuming some form for the potential.  An example is the Morse potential, initially introduced by Morse in

Morse, Philip M. "Diatomic molecules according to the wave mechanics. II. Vibrational levels." Physical review 34.1 (1929): 57.

The original paper actually contains all kinds of data for molecules known at that time, data which in turns enter into a potential from which one can obtain vibrational spectra.
A: It does depend on the mass, but mass it is not explicit in the equation $$E_n=(n+\frac12)\hbar\omega$$ but it is implicit.
In the equation above we have the vibrational frequency $$\omega=\sqrt{\frac{k}{\mu}}$$ that includes $\mu$ which is the reduced mass. So it is true that the value of $E_n$ depends on the masses involved and your concerns are justified.
Note that the reduced mass for diatomic molecules is given by $$\frac{1}{\mu}=\frac{1}{m_1}+\frac{1}{m_2}$$ and in your examples of molecules with two similar atoms, $$\mu=\frac{m}{2}$$ so the equation explicit in mass for the energy of each mode can be written $$E_n=(n+\frac12)\hbar\left(\frac{2k}{m}\right)^\frac 12$$ So, using your examples, for the $H_2$ molecule, $m$ would be the mass of a hydrogen atom and for $F_2$ it would be the mass of a flourine atom.
Clearly, the energies of each mode for $H_2$ and $F_2$ would not be the same.
