Consider if we were to predict the orbit of the Earth around the Sun using quantum mechanics.
Contrary to some of the other answers, there is no fundamental objection to doing this. Quantum mechanics has been tested in situations where gravity is involved. For example, people have observed two-path interference patterns for neutrons that have traveled through a gravitational field. QM actually doesn't have any problem dealing with gravity in the weak-field regime, i.e., Newtonian gravity. If quantum gravity were needed in such situations, then we would have decades ago been in the possession of the empirical data needed in order to start roughing out a theory of quantum gravity. The neutron interference experiments, for example, would have shown us something exciting and incompatible with standard QM. Wouldn't that have been nice!
For the problem you pose, the solution is known and is totally unproblematic. People have already studied semi-classical motion in the hydrogen atom. The experiments and theory involve very specially prepared coherent superpositions of states with large quantum numbers. (At a more elementary level, a pretty standard textbook example is to show how high-n states of the one-dimensional harmonic oscillator can be used to recover the motion of a classical harmonic oscillator.) So if you take the quantum-mechanical theory of these semi-classical states in hydrogen and just change the coupling constant, you have a theory that correctly predicts the orbit of the earth.
<PERSON> says:
There's no particular problem with using GR to determine spacetime curvature, and then using QM to determine the effect of that on matter waves. Do it for the Earth, and you'll trivially get the same result as a pure GR calculation because the wavelength is negligible.
But this is not even necessary, e.g., in analyzing the neutron interference experiments, nobody did anything even remotely this sophisticated. They simply used the <PERSON> equation with the gravitational potential. 800
It's freshman physics.
Similarly if we were to predict the orbit of an electron about a nucleus using relativity.
So now let's go back to states that are not semi-classical states but just the kind of states we normally deal with in solar spectra, etc -- states with definite energies and values of n up to about 5 or 10. For these states, there is no such thing as an orbit. This is because $\Delta p\Delta x$ is not large compared to <PERSON>'s constant. You may have seen confusing material in pop culture, diagrams in chemistry textbooks, historical discussions of the <PERSON> model, and so on in which electrons are depicted as traveling in circular orbits, but the circles just happen to be restricted to certain radii. This is just wrong.
You can use QM plus special relativity to describe these states, and that works great (but there's no such thing as an orbit). You can't use general relativity to predict atomic orbits, both because there are no orbits and because GR is a theory that is very specifically limited to the gravitational force. GR describes the motion of a particle in a gravitational field as a geodesic, which is the equivalent of a straight line in curved spacetime. This trick of geometrizing the problem only works for gravity, because it's only for gravity that we have the equivalence principle.
The accepted answer by <PERSON> says:
To predict the orbit of the Earth around the Sun you can't use Quantum Mechanics (QM) or its modern versions, because it would be like trying to measure the mass of an apple with a thermometer. The orbit of the planets is a gravitational phenomenon, and you, therefore, need to use a theory of gravitation to solve it, be it Newtonian Gravity, General Relativity (GR), or any other gravitational theory that you want. [...] QM isn't a theory of gravitation, it doesn't explain how gravity works.
This is just completely wrong, for the reasons given above. This is a good example of why it's a bad idea on SE to accept an answer without waiting for a while.
<PERSON> says:
Accounting for relativity leads to a more accurate prediction but this time it is not known how to account for quantum mechanics. In the words of <PERSON>, gravity has not yet been ported to the quantum OS.
Again, totally wrong. The problems arise only for strong-field gravity. 800