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I know that the Coulomb potential in a hydrogen atom is $V(x)=\frac{-e^2}{4 \pi \epsilon_{0} r}$. Usually the potential is given by the potential energy divided by the charge, so I would expect $V(r)=\frac{-e}{4 \pi \epsilon_{0} r}$. I am not sure whether "potential" is just a shorthand for "potential energy" in this case, but it does not make sense otherwise and I was wondering if anyone could enlighten me.

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Strictly speaking, the Coulomb potential is proportional to $e$ and the Coulomb potential energy to $e^2$. Sometimes people get lazy and use term "potential" when talking about potential energy and vice versa.

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I would never choose potential for potential energy (as shorthand for potential energy). Cause It's totally misleading.

Potential energy is work. While potential is work per unit charge.

Math version : $$U=-\int \vec F\cdot d\vec r$$ $$V=\frac{U}{e}$$

Here $e$ is charge of electron. And some people uses $V$ for potential energy, which leads to confusion. That's why when I work on electrodynamics I never use $V$ for potential energy.

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