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Atoms are a bound state of electrons and protons held together by the EM force, in the same way a quarks are bound by the strong force etc.

All these particles being massive though, gravitational attraction should make a (albeit small) contribution to the binding energy.

Has any experiment every measured this discrepancy? What precision would be necessary to test this?

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The mass of the electron is roughly $10^{-30}$ kg. The mass of the proton iis roughly $10^{-27}$ kg. The distance between them is roughly $10^{-10}$ m. Inserting these into the formula for gravitational binding energy:

$$U=\frac{Gm_pm_e}{r}=\frac{10^{-11}\times 10^{-27}\times 10^{-30}}{10^{-10}}=10^{-58}\textrm{ J}=10^{-40}\textrm{ eV}$$

The most sensitive energy measurement we have ever made (i.e. the Harvard Tower version of the Pound-Rebka experiment) measured an energy difference of $10^{-11}$ eV.

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    $\begingroup$ I should probably add that we don't even know if Newtonian gravity even applies at quantum scales. $\endgroup$
    – probably_someone
    Commented Jan 27, 2018 at 19:47
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    $\begingroup$ Even though this shows that this measurement is not feasible, one might want to mention that it is not even possible in principle since we cannot make the proton heavier, nor switch off gravity. To make this measurement feasible in principle, one may have to do something crazy like trapping a particle that interacts only gravitational (clearly, the neutron does not qualify) at the position of the proton. $\endgroup$
    – user178876
    Commented Jan 27, 2018 at 20:46
  • $\begingroup$ @marmot Why do we need to make the proton heavier or switch off gravity to measure this? There are definitely possible experiments that involve neither of those things. For example, gravitational attraction perturbs the energy levels of hydrogen very slightly; just look for tiny, tiny offsets to the wavelengths of the hydrogen emission spectrum that match what theory would predict for the action of gravity. $\endgroup$
    – probably_someone
    Commented Jan 28, 2018 at 7:05
  • $\begingroup$ I thought you want to measure the effect of gravity on the binding energies in atoms. This requires a comparison between the binding energies with and without gravity. $\endgroup$
    – user178876
    Commented Jan 28, 2018 at 13:49
  • $\begingroup$ @marmot By measuring how much the hydrogen spectrum deviates from what you would predict using QM without gravity, we are measuring the effect of gravity on the binding energy of hydrogen. $\endgroup$
    – probably_someone
    Commented Jan 28, 2018 at 17:53

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