Now we consider a particle of mass M ``confined'' to the vicinity of the origin X=0 with a spring of spring constant k. The energy of the particle is then given by the usual energy for a simple harmonic oscillator (SHO)
Again, by the uncertainty principle, no matter how we prepare our individual oscillators, there will be some spread in the distributions of the position and momentum X and P of the particles. Again, by symmetry, we expect <X>=<P>=0.
The average energy of our systems under the distributions set up by our preparation of them is given by
Here we have used the facts that the average of a sum is always the
sum of the averages that the average of a constant times a variable is
always the constant times the average of the variable. (If you feel
uncomfortable with any of these manipulations, you may read about them
in the supplemental notes on statistics.) At the final step, we used
the same trick as in our discussion of the confinement to a box to
conclude 
and 
.
The uncertainty principle puts an absolute lower limit on this energy
as well. We know that 
and so we must
have 
.
Although we need not at this
point, we will be precise in our analysis of the SHO
because we will later revisit this result when we have a
full, formal theory. Also note that we can give an exact analysis in
this case because the SHO just happens to have
an energy quadratic in both X and P so that we can make the exact
replacements 
and 
.
Now, because 
, we know
There is a strict lower bound on the value which <E> may have. This
lower bound is given by the minimum of the function 
.
We locate this minimum by setting the derivative equal to zero,
(2) gives the localization for the particle in the SHO in its lowest energy state
One way to prepare this state would be to give the particles some
mechanism to slowly loose energy (giving the particles charge so that
they radiate electromagnetic energy as they oscillate, for example),
and then waiting a long time. We now find the value of the energy in
this this lowest energy state 
,
In the last step we used the familiar result that the angular frequency
associated with the angular frequency of the SHO is
.
