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GoLightly said: "A particle with negative potential energy that you're talking about is making no kind of sense to me. What am I missing?"

I mis-spoke a little. Let me try to re-phrase before going further.

I previously wrote: "The potential energy of the bound particle is then negative. The bound particle will, I believe, have a mass which exactly accounts for this missing energy."

I should have written: "The potential energy of the bound particle is then negative. The bound system of particles will, I believe, have a mass which exactly accounts for this missing energy."

It is the entire system which will reveal the change in mass, not necessarily the "bound particle".

Let's talk a little about potential energy. One of the simplifications in physics involves treating particle interactions using the concept of a "field".

The laws of electromagnetics tell us that a proton and an electron will attract each other. There is a force law which describes how the amount of force changes with distance.

A simplification of the interaction of these two particles consists of ignoring the proton, for example, treating it as if it is at a fixed point in space. (Because the proton is so much heavier than the electron, it can often be treated as if it is stationary.) Then one imagines that there exists a "field" in space and the electron exists in this field. The value of the force in that field at any point can be calculated using the known electromagnetic force law.

Knowing the force which is acting on the electron at any position, one can then calculate the amount of energy needed to move the electron from any one position to another position. The net change in energy represents how much work must be done to move the electron from the first position to the second. If the amount of energy required is positive, then the "potential energy" of the second point is said to be greater than at the first point.

If the amount of net energy needed to move from the first point to the second point is negative, then the second point has a lesser potential energy than the first point.

Finally, there is a singularity at the location of the proton. That is, the force between a proton and an electron becomes infinite, according to the force law, if the distance from proton to electron is zero. For this reason, the reference position used for zero potential energy is at an infinite distance from the proton, rather than at zero distance.

Calculus allows one to calculate the energy required to move the electron from an infinite distance to any given finite distance. Since the force is attractive, this energy is always negative. That is, the electron accelerates as it moves from infinity to any given finite distance.

The potential energy at infinity is being converted into kinetic energy at some finite distance from the proton by virtue of the force acting on the electron.

At some finite distance from the proton, the positive kinetic energy observed is equal to the change in potential energy. That means that all finite positions have negative potential energy and that the potential energy gets more negative as the electron approaches the proton. At the same time, the kinetic energy increases. Total energy is conserved.

There isn't any absolute potential energy quantity. It can be defined any way we want, but we choose the way that is most convenient. Once a reference is chosen, then we are only interested in changes in potential energy.

For example, if we start with an electron at rest one meter from the proton and then allow that electron to "fall" toward the proton until it is one-half meter away, then we know by the conservation of energy that the kinetic energy of the electron must be equal to the difference between the potential energy at which it ended and the potential energy at which it started.

We only end up interested in the potential energy change. Where we chose the reference becomes irrelevant. The "potential energy" at any given position is simply a number which tells us how the electron will behave in moving to some other position. The difference in potential energy tells us whether we have to expend "work" to move the electron or whether the electron can do "work" in moving to the new position.

In summary, then, when we say some particle has a negative potential energy, we are describing the fact that we must expend energy to move it to the zero reference position. The magnitude of the negative potential energy is the amount of energy to move the electron to the zero potential energy position.