We thus see, that a system of harmonic oscillators cannot be made with
zero average energy. The non-zero minimum average energy 
is known as the ground state energy or sometimes the
zero-point motion energy. This is a purely quantum effect. A system
of classical oscillators could all be left at rest P=0 in their
lowest energy configurations X=0. But quantum mechanically, this
would violate the uncertainty principle because then we would have
.
We again verify that our result is in accord with the classical expectation in the limit
. One could plug ``classical'' values into
(3) and find that an oscillator with frequency 
has a minimum energy limit of 
which is indeed a
negligible energy on classical scales. Or, one could simply and
efficiently take the formally limit 
.
Finally, in this case we found that at 
the particles are
confined to a region 
and that for this configuration 
in accord with our
expectation for the energy of confinement, even when the confinement
is the result of an attraction to the origin rather than from an
explicit hard-box constraint.