Why does the mass of a hydrogen atom have less mass than the sum of its parts?

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Why does the mass of a hydrogen atom have less mass than the sum of its parts?

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This is how binding energy works. By combining the parts together, you get something in a lower energy state than if the parts separated again.

Think of it this way: Why do balls roll down hills and stay in holes? Because a hole is a low energy point. If you have a ball on a surface, and it *can* right away physically roll to a lower point…it will! That’s just what balls do on flat, inclined surfaces. Same reason why heat flows from hot to cold, why gases expand to fill empty spaces, etc. It is what physics does.

But things get a bit more complicated with *lumpy* surfaces. What happens if there’s a small hole in the hillside? The ball might get stuck in that hole. In order to get out, to get to the lowest *absolute* energy state, the ball would have to first spontaneously *rise* over the lip of the hole. For most physical things, that cannot happen. (Ignore quantum effects for now. This is ELI5, after all.) As a result, the ball can be stuck, stable, at an energy higher than the true minimum it could achieve. But if it’s given a little push, it might roll up over the lip of the hole, and continue down to the bottom of the hill.

An electron and a proton, by themselves, are balls stuck in holes. Those holes aren’t physically visible (they’re electromagnetic potentials), but the metaphor still works. Once you push the two of them close enough together, they bind, forming a stable atom. If the total energy went *up* when this happened, we would expect it to be *unstable* and fly apart. But since it’s more stable for them to stick together than to separate, they have lower total energy.

Another way to look at it is: start with already having a hydrogen atom. If you want to split them apart, you would need to *add* energy until the electron escapes. But this process is perfectly reversible, meaning, a proton can *pull in* a free-floating electron. Since you had to *add* energy to make the electron escape, that must mean that you *get* energy out when the electron is pulled in. But the only place that energy can come from is the net momentum (rest mass plus motion) of the particles in question. Hence, they must have lower mass than the components, because the energy had to come from *somewhere.*