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Your mistake is that the other particle has energy too. In the hydrogen atom, the nucleus is just sitting there, but in positronium, the positron has an equal amount of kinetic energy. (You are also cutting the potential energy in half.) Doubling all energies will give the right result. As other people have said, it's much much easier to do this with ...


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Your proposal will work as long as mass and charge enter into the expression in a ratio 1:-2. Find an expression with a different ratio (like electric potential, or radiated power ...) and you're sunk. The fine structure constant $$ \alpha = \frac{e^2}{\hbar c} \tag{Gaussian units}$$ is an example of such an expression and controls the fine structure of ...



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