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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?

John Eastmond
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    No. The metric expansion of space doesn't apply on such small scales. It even doesn't apply on such macroscopic scales as galaxies and galaxy groups, where gravity binds objects together. Only on scales of several Mpc does space expand. – pela Apr 08 '15 at 16:18
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    See Why does space expansion not expand matter? – ACuriousMind Apr 08 '15 at 16:20
  • Plus, this has only described spacetime. To describe the energy of atoms, you'd need to use a perturbed metric, choose a gauge, and then watch as the perturbations that correspond to the size of atoms and their energy is unaffected by having space expand. – Jim Apr 08 '15 at 16:32
  • I would have thought that one can apply the FRW metric at all scales - provided masses are small. Imagine a hydrogen atom sitting alone in space. I don't assume that it expands. In fact I explicitly state that it maintains a constant proper size. – John Eastmond Apr 08 '15 at 16:33
  • If mass/energy is inversely proportional to size and you maintain a constant proper size, then you'd maintain a constant mass/energy. But this still doesn't recognize the fact that the unperturbed FRW metric is for an empty universe only. You need to add in perturbations even if there is only one atom in it – Jim Apr 08 '15 at 16:38

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