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human
Why can't gravity be reconciled into quantum mechanics via Schrodinger's equation like how electromagnetism did?

I am actually a high school student with lots of interest in QM. Don't worry, i can understand the stuff behind QM, not into pseudosciences.

So the formal description of the Time Invariant Schrodinger's Equation is as follows.

EΨ = [-ħ^2 /2m ∇^2 + V]Ψ

Where ∇^2 refers to the Laplacian, and V refers to potential energy.

Switching from Cartesian into spherical coordinates, gives us 2 Equations, one being the Spherical harmonics and the Other being another but simpler 1 Dimensional Schrodinger's Equation that varies with R from the origin.

Defining a new function U = R*r, one gets the 1D SE as

EU = -ħ^2 /2m ∇^2 U + [V + (ħ^2 /2m)*(l(l+1)/r^2 )]U

Where [V + (ħ^2 /2m)*(l(l+1)/r^2 )] has become the new "potential term" for distance r

So after so much math, Now comes the question. People have substituted V for the classical Coulomb potential and gotten the hydrogen wave-functions. One of the most almost earth-shattering work of century In my honest opinion.

Why hasn't there yet been any work on substituting Gravitational potential into V.

Coulomb potential = -K Ze^2 /r ; K being coulomb's constant.

Gravitational potential = -GMm /r

If we could find evidence for quantization of mass, i believe its almost trivial to just replace K everywhere in the hydrogen wavefunctions for G, and Ze^2 for Zm^2 (where m represents quantized mass) and call it a day for Quantum Gravity.

Besides the 20 orders of magnitude between K and G and therefore the impossibly small scale of quantization that man may never be able to measure.

What is actually stopping us from doing that? 

Why can't gravity be reconciled into quantum mechanics via Schrodinger's equation like how electromagnetism did?
gpt
This is a great question!

So the problem isn't putting in a potential into the Schrodinger equation.  When you do that, you are treating the associated field as a background which is unchanging.

The problem arises when you recognize that that external field is actually dynamical, and has equations of motion it must satisfy.  So instead of just having the Coulomb potential, you have electric and magnetic fields that are dynamical.  When you do this with electromagnetic fields, extending the classical equations for those fields to the quantum case, it turns out you can do it in a mathematically self-consistent way that also agrees with experiment.  But if you try the analogous thing with gravitational fields, you run into a problem, because the classical equations you start with for general relativity are sufficiently different from those of electromagnetism.