In
Schroedinger's equation the momentum was replaced by a
derivative. This means that if a wave function is
confined to a small volume, it must rise from zero
quickly. Small volumes necessarily mean high momenta.
This is analagous to the DeBroglie wavelength being
inversely related to the momentum. Werner Heisenberg
showed that one consequence of the Schroedinger equation
is the
where D are the
uncertainty in the momentum and position. The more
confined a particle is to a given position, the more
uncertain is the momentum. Thus a particle in its lowest
energy state in a box of finite size has a range of
momentum and therefore a finite kinetic energy.
Similarly, an electron in a attractive Coulomb potential
(e.g. a Hydrogen atom) must have more kinetic energy for
smaller orbits. If it were not for this aspect of
Schroedinger's equation, atoms could collapse to a point.p |