Limits of General Relativity and Quantum physics I have often heard people say that you cannot use quantum mechanics and general relativity in particular regimes. Consider if we were to predict the orbit of the Earth around the Sun using quantum mechanics. Similarly if we were to predict the orbit of an electron about a nucleus using relativity. But why won't this work? Can anyone perhaps show me a derivation of something like I mentioned that shows how these theories do not work in these regimes?
 A: "Quantum" and "relativistic" are properties of a theory.
To describe the orbit of an electron around a nucleus, you need a theory of electrodynamics. It doesn't have to be relativistic but if it is you get a better answer. It also doesn't have to be quantum but if it is you again get a better answer.
To describe the orbit of the Earth around the Sun, you need a theory of gravitation. 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 Scott Aaronson, gravity has not yet been ported to the quantum OS.
A: 
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.
John Doty 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 Schrodinger equation with the gravitational potential. 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 Planck's constant. You may have seen confusing material in pop culture, diagrams in chemistry textbooks, historical discussions of the Bohr 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 Mauro Giliberti 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.
Connor Behan 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 Scott Aaronson, gravity has not yet been ported to the quantum OS.

Again, totally wrong. The problems arise only for strong-field gravity.
A: You have a misunderstanding with the word "regime".
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. In the same way, using GR to predict the motion of an electron fails to consider all the electromagnetic (and weak) properties of the electron, and therefore isn't a good description.$^1$
What people usually mean when they say that "you cannot use quantum mechanics and relativity in particular regimes" is that in most complex situations$^2$ the gravitational effects don't matter when the quantum effects do, and vice versa. The electron is a good example: its tiny mass means that its gravitational effects will be tiny in comparison to the quantum effects that explain how the electron moves. This is a "good regime" because QM works (and we have no way to see if GR works for the electron). The planetary motion is another "good regime" because GR works (and we have no way to check QM there).
The "bad regimes" are the ones where GR and QM effect both play important roles to solve the problem, like what happens around black holes or what happened instants after the Big Bang. Lastly, to answer your question, in these regimes, you can't use QM because it isn't a theory of gravitation, and you can't use GR because it isn't a quantum theory.


*

*as the comment pointed out, I've used "QM" improperly, when I really meant quantum theories like QED, QCD, or the Standard Model, because that's what is usually meant in the infamous "QM vs GR debate"


*with "complex" I mean where both the gravitational and strong-electro-weak forces are present
A: 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.
The clash between GR and QM comes from attempting to interpret spacetime curvature as a consequence of a quantum phenomenon.
A: 
Similarly if we were to predict the orbit of an electron about a
nucleus using relativity. But why wont this work?

Because it works. This is called QED. Try Dirac equation.
The limit of all of physics is measurement on one side (experimental) and mathematics on the other side (theoretical).
A: 
"Consider if we were to predict the orbit of the Earth around the Sun
using quantum mechanics. Similarly if we were to predict the orbit of
an electron about a nucleus using relativity. But why won't this work?
Can anyone perhaps show me a derivation of something like I mentioned
that shows how these theories do not work in these regimes?"

As for the orbit of Earth, if you believe that Newton's second law is able to describe this accurately, then you might be very happy to learn that Newton's second law can be derived from the Schroedinger equation (i.e. from a quantum mechanical theory).
As for an electron's orbit around a nucleus, special relativity is indeed used when describing an electron's interaction with a nucleus, if you want the most accurate results (compare for example the relativistic Dirac equation in quantum mechanics, versus just the plain non-relativistic Schroedinger equation).
As for general relativity, it's a theory of gravitation, not a theory of everything. General relativity wouldn't be used to describe other non-quantum-mechanical things too, for example you would not use general relativity to calculate the current in an electric circuit that has a certain voltage and resistance, you'd use Ohm's law which is $V=IR$. Similarly, general relativity isn't used to describe the orbit of an electron around a proton, largely because gravitational effects are not needed to describe this phenomenon in any experimentally relevant regime (gravity between two extremely light particles, is extremely weak, and its effect is expected to be dozens of orders of magnitude smaller for this interaction, than anything we can detect in an experiment).
The problem is not with describing things like the Earth's orbit with quantum mechanics or an electron's orbit with relativity: nothing quantum mechanical is needed for describing the Earth's orbit unless you want to know some extremely minor details about the orbit that are orders of magnitude smaller than anything we can possibly measure, and nothing general relativistic is needed to describe the electron's motion around a nucleus since gravity's effect would be orders of magnitude smaller than anything we can possibly measure. The problem is when you encounter a situation where you need both quantum mechanics and general relativity at the same time such as when describing Hawking radiation from a black hole. 
