513.001 Molecular and Solid State Physics

Estimate the size of an atom using the uncertainty principle

Use the uncertainty principle to estimate the size of an atom. The potential energy of the Coulomb attraction goes to minus infinity at a radius of zero so it would seem that the radius should be as small as possible to minimize the energy. However, according to the uncertainty principle, a small Δx implies a large Δp = √<p²>-<p>². This means that a small radius implies a large kinetic energy. If you minimize the total energy (potential + kinetic), a condition for the size of the atom can be found.

Since this is an estimate, more than one solution is possible depending on the approximations made. One solution is given.