**The Heisenberg uncertainty
principle**

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 at the boundary to a finite value, then fall back to zero
at the opposite boundary. Small volumes necessarily imply larger derivatives and higher
momenta. This is analogous to the DeBroglie wavelength being inversely related to the
momentum. One consequence of the Schroedinger equation (derived in a different way by
Werner Heisenberg) is the **Heisenberg uncertainty principle: **

**D**p
**D**x > h / (4
**p**)
.

where
**D**** x** and

The more confined a particle is to a given position, the more uncertain is its momentum. Thus a particle in its lowest energy state in a box has a range of momentum and therefore a non-zero kinetic energy. Unlike in classical physics, it cannot simply "lie still" with zero velocity and zero 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 would collapse to a point.

Another profound change from the classical viewpoint follows directly from the uncertainty principle: Position and momentum (hence velocity) cannot be precisely known simultaneously; so particles do not move according to any kind of well-defined paths, such as the circular or elliptical "orbits" that are often drawn in cartoon style to represent the motion of electrons in atoms.