We now impose the Bohr-Sommerfeld quantization condition
Here, the motion is along x and so
.
Also, the momentum is just
, and finally the
symbol means we are to integrate over one complete
period 
. Putting this together, we have
where we have used the fact that the average of 
over one
full
period is 
. We now have our condition on E:
which is in perfect accord with our expectations in (15)! The spectrum we predict then appears as in Figure 2.2.
As we learned from the uncertainty principle the lowest energy is
actually
. It will turn out that the actual
spectrum of the harmonic oscillator is just our prediction (17) with a
simple shift to bring our result in accord with the
uncertainty principle
