Is the Hydrogen Atom Stable for a Potential Behaving as -1/rs?

[Kelly Lin]

Homework Statement

An electron in a hydrogen atom does not fall to the proton because of quantum motion (which may be accounted for by the Heisenberg uncertainty relation for an electron localized in the volume with size r). This is true because the absolute value of the Coulomb potential energy goes to minus infinity with decreasing distance to the center r relatively slowly, like -1/r. Is such an ''atom'' stable for any potential behaving as -1/r^s? If not, find the range of values of s at which the ''atom'' is stable, so that ''the electron'' does not fall to center.

Homework Equations

The Attempt at a Solution

Based on Bertrand’s Theorem, the closed and stable motion will be that s equals to -2,1,2. However, I don't know how to solve this problem by uncertainty principle. Moreover, I can't figure out why electron not falling to the proton is related to quantum motion. Can someone give me some hints or correct my opinion? Thanks!

 

[zenmaster99]

Some questions to consider:
(1) What does it mean for a system to be "stable?" Think in terms of energy as a function of distance.
(2) Given the Heisenberg Unc. Principle, if you confine the electron to a small space, $\Delta x$, what will be its $\Delta p$? Can you come up with its average kinetic energy from this?
(3) What will be the electron's average potential energy in this region?

Take your answers to (2) and (3) as a function of position and see if you can come up with the values of $s$ where the energy doesn't fall to zero as $\Delta x$ goes to zero.