Why the atom does not collapse

A mystery solved by uncertainty

We have seen that, according to classical physics, the electron should spiral toward the nucleus and crash into it. The Bohr model settles this by decree (permitted orbits). But the uncertainty principle gives a much deeper and more natural explanation.

The argument

Suppose the electron falls onto the nucleus. It would then be extremely well localized: Δx would become tiny (the size of the nucleus, ~10^-15 m).

But according to Heisenberg, a tiny Δx forces an enormous Δp:

   Δx tres petit   ->   Δp = ħ/(2Δx) tres grand   ->   grande energie cinetique

A large uncertainty on the momentum means that the electron has, on average, a high kinetic energy. This energy pushes it to move away from the nucleus. There is therefore a tension:

   attraction electrique   ->  veut confiner l'electron (Δx petit)
   principe d'incertitude   ->  s'y oppose : confiner coute cher en energie

An equilibrium, and a size for the atom

The atom stabilizes at the distance where these two effects balance:

   trop pres du noyau :  Δx petit -> energie cinetique enorme -> l'electron repart
   trop loin :           attraction l'emporte -> il se rapproche
   equilibre :           une taille caracteristique ~10^-10 m

This trade-off fixes the size of the atom (of the order of the ångström, 10^-10 m) and forbids collapse. The electron cannot "fall" onto the nucleus, because confining it to that point would require an energy that nothing can supply in a stable way.

A general lesson

This reasoning is very fruitful: confining a particle costs energy. The more you enclose a particle in a small region (Δx small), the greater its minimum kinetic energy. This "confinement energy" explains a host of phenomena, from the stability of atoms to the pressure that prevents certain stars from collapsing.

In summary

The uncertainty principle explains the stability of the atom: confining the electron near the nucleus (Δx very small) would impose an enormous Δp, hence a kinetic energy that pushes it away. The atom stabilizes at the size where attraction and uncertainty balance (~10^-10 m). General rule: enclosing a particle in a small space costs it energy.