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A black hole has a radius of $R = \frac {2Gm}{c^2}$, in this context, if we take a single proton and neutron as a black hole, its Schwarzschild radius will be near about $4.8 \times 10^{-52} \mathrm{m}$. 

Now quantum mechanics says that both particles can stay together if they satisfy the uncertainty relation as $$\Delta P \Delta R \geq \hbar$$ by taking mass as the mass of the proton and space as the Schwarzschild radius; this uncertainty relation is definitely violated.

However, in this scenario, a black hole is not consistent with the quantum theory. Please answer my question.

Níckolas Alves
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Flynn Ryder
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    Do anyone claim that black holes are compatible with quantum mechanics? – Marius Ladegård Meyer Mar 21 '23 at 13:54
  • What did you plug in as delta P? You wrote "taking the mass of the proton and space as the Swarzschild radius" and this sentence doesn't make sense to me. – AXensen Mar 21 '23 at 14:08
  • @AndrewChristensen P is momentum, P = m.c – Flynn Ryder Mar 21 '23 at 14:16
  • @MariusLadegårdMeyer my question is in context of quantum theory both particle can't stay in a black hole. – Flynn Ryder Mar 21 '23 at 14:18
  • P=mc is the order of magnitude of the momentum a proton would have if it was relativistic (near the speed of light). The momentum in special relativity is $\gamma mc$, so in case you thought $mc$ was the maximum possible momentum for a proton, it isn't.

    Your equation just implies "a proton with a relativistic uncertainty will not be localized well enough for it to be inside a black hole". With this particular uncertainty, it won't be in a black hole. The first commenter is right; nobody believes black holes and quantum mechanics are fully compatible, but this isn't actually a contradiction.

     – AXensen Mar 21 '23 at 14:27
  • PBS SpaceTime has some relevant videos. See Hawking Radiation or The Black Hole Information Paradox – mmesser314 Mar 21 '23 at 14:36
  • An interesting article that relates directly to the quantum singularity in a black hole: https://bigthink.com/starts-with-a-bang/singularity-black-hole/ – foolishmuse Mar 21 '23 at 16:30

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Black holes are a prediction of classical general relativity, which completely ignores any sort of quantum effects. At small scales, this is a very bad approximation and most physicists (perhaps all of them) believe general relativity will fail to provide an accurate description and we will need something else, such as a full theory of quantum gravity.

A possible solution would be the quantization of the gravitational field. In this case, the uncertainty on the particle's position would also lead to an uncertainty on the gravitational field itself, and even on the black hole itself. Through some mechanism like this, you might be able to recover Heisenberg's principle. (This paragraph is speculative: I'm merely illustrating a possible solution, not claiming it is the correct one. No one knows what happens with gravity at such small scales).

Shortly, black holes are classical solutions to general relativity. They do not take quantum effects into consideration.

Níckolas Alves
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    Both Quantum Mechanics and General Relativity are based on the same principle that strictly holds in all physical theories, the principle of least (stationary) action. Per this principle, time and energy are a Fourier pair, as well as space and momentum. In other words, spacetime and matter are Fourier conjugates in both QM and GR. The foundation is the same in both theories. The Heisenberg uncertainty principle also is a direct expression of the very same Fourier relation. Same with the Noether theorem. So chances are the uncertainty principle, in some form, will also hold in Quantum Gravity. – safesphere Mar 23 '23 at 21:11
  • @safesphere Agreed. The speculative aspect I meant in my answer concerns the example of how the uncertainty principle would hold in QG (superposition of the gravitational field, etc). In full QG it might even happen that it doesn't make sense to speak of space and time at the scales mentioned by OP, so we can't say much about the particular form in which the HUP holds. – Níckolas Alves Mar 26 '23 at 03:55