If I read him correctly, in his book Quantum Theory, David Bohm argues that the reason an electron doesn't collapse into the nucleus is there is essentially a pressure due to the uncertainty principle opposing the Coulomb attraction. His book is the only place I've read about this. Is it written about elsewhere? Is it at all related to Degeneracy Pressure?
It seems plausible. While he doesn't say how to quantify this "force" , I made a guess with interesting results.
If $\Delta p \Delta x=\hbar$ and we use unit analysis, we have a force of some kind, $F=\frac{\Delta p^2}{m \Delta x}$. If you equate that to the negative of the Coulomb attraction between electron and proton in the hydrogen atom, also equating momentum with uncertainty in momentum and position with uncertainty in position:
$p=\Delta p. x=\Delta x.$
$ \frac{p^2}{mx}=\frac{e^2}{4\pi\epsilon_0 x^2}.$
Swap in $p=\hbar/x$ or $x=\hbar/p$ into that equation, you get the Bohr Radius for x and a p,x combination that gives you the correct total energy for the ground state of the electron.
Alternatively, one gets the same results if the Hamiltonian $H=\frac{p^2}{2m}+V(x), \frac{\partial H}{\partial x}=\frac{\partial V}{\partial x}$ , with the constraint $xp=\hbar. $ Applying Lagrange Multipliers:
$\frac{\partial H}{\partial x}=-\dot{p}=\lambda p$
$\frac{\partial H}{\partial p}=\lambda x$. Also using the Euler-Lagrange equations.
Solving for lambda, one gets the equation:
$$-\dot{p}=p\frac{\dot x}{x}=\frac{p^2}{mx}$$
This "Heisenberg Force" term shows up again, equal and opposite to $\dot{p}$.
Is this "Force" just a mathematical artifact? Is it related to the idea that the position of a particle is at least as uncertain as it's Compton Wavelength?
UPDATE
I'm tempted to answer the question in the affirmative, but I'm not sure if I can prove it rigorously:
$$xp=\hbar$$ $$pdx+xdp=0$$ $$p\frac{dx}{dt}+x\frac{dp}{dt}=0$$ $$p^2+mx\frac{dp}{dt}=0$$ $$\frac{p^2}{mx}=-\frac{dp}{dt}$$ $$\frac{2(p^2/2m)}{x}=-\frac{dp}{dt}$$ $$\frac{\hbar^2}{mx^3}=-\frac{dp}{dt}$$ So as Feynman says in the cited lecture, a reduced uncertainty in the distance from center implies a higher kinetic energy which has a direct correspondence with this Heisenberg Force. That principle and the idea of a "Heisenberg Force" appear mathematically equivalent.
