Hypothetical 'black hole'/electron atom

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SUMMARY

The discussion focuses on calculating the mass of a hypothetical black hole (Mh) that would yield energy levels identical to those of a hydrogen atom, specifically the n=1 orbital energy of 13.6 eV. The participants clarify that while the gravitational field can replace the electromagnetic field in this scenario, the quantization pertains only to the electron, not the field itself. The conversation emphasizes that solving Schrödinger's equation for the hydrogen atom does not necessitate quantum gravity considerations, simplifying the analysis of quantized gravitational orbits.

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Homework Statement


I have been asked to consider an atom formed between an electron and 'black hole' i.e. a point singularity with mass and no charge. I should find the mass of the black hole Mh such that the energy levels of this atom would be identical to the energy levels of a hydrogen atom

I'm fairly sure I can calculate a mass that would give the same energy of the n=1 orbital (13.6eV) at the Bohr radius a0 by considering gravitational rather than EM interactions, but I'm having difficulty grasping the notion of quantised gravitational orbits. Are all 'energy levels' quantised?

Cheers guys
 
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You are maybe thinking there are complications here when there aren't. When you solve schrodingers equation for the hydrogen atom, you aren't quantizing the EM field. You are only quantizing the electron. The field itself is a classical electric field. There isn't any problem replacing that with a gravitational field that generates the same inverse square potential. No need to do quantum gravity.
 
I always end up doing that! It's the 'levels' aspect of it that threw me. Got a reasonable value out of it now, thanks Dick
 
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