Are atomic energies increasing as the Universe expands?

[jcap]

Starting from the FRW metric (for simplicity flat space, radial direction only):
$$ds^2=-c^2dt^2+a(t)^2dr^2$$
If we take $dt=0$ then the proper distance $ds(t)$ between two spatially separated points at cosmological time $t$ is given by:
$$ds(t)=a(t)dr$$
Now at the present time $t_0$ we can define $a(t_0)=1$ so that we also have:
$$ds(t_0)=dr$$
Therefore by eliminating $dr$ in the above equations we find:
$$ds(t)=a(t)\ ds(t_0)$$
If we define $ds(t)=1$ so that a hydrogen atom has a unit proper diameter, at any time $t$, then the equivalent diameter at the present time $t_0$ is given by:
$$ds(t_0)=\frac{1}{a(t)}$$
According to quantum mechanics the mass/energy of a quantum system is inversely proportional to its size.

Therefore if the mass/energy of the hydrogen atom at time $t$ is one unit then the mass/energy of an equivalent atomic system at the present time $t_0$ is $a(t)$ units.

Thus can one infer that hydrogen atoms at time $t$ in the future have an energy that is a factor $a(t)$ higher relative to the energy of hydrogen atoms today?

 

[QuasiParticle]

The FRW metric applies only to a homogeneus and isotropic system. You cannot use it inside an atom, for example. And the expansion of the universe doesn't change the laws of nature, so the same energy levels exist also in the expanded universe.

 

[PeterDonis]

can one infer that hydrogen atoms at time

t in the future have an energy that is a factor

a(t) higher relative to the energy of hydrogen atoms today?

No, because a hydrogen atom is a bound system, and your reasoning is not valid for bound systems. It's only valid for objects which are "comoving", i.e., their relative motion is determined by the expansion of the universe. The individual parts of bound systems do not meet that criterion; their relative motion (if any) is determined by the forces binding them together (in the case of the hydrogen atom, the electromagnetic force between the electron and proton).