Compute the total energy of the N electrons in the
oscillator of frequency 
in terms of N and 
by
summing the energies of the occupied states.
Now, estimate this value using the electron gas picture,
evaluating the energy of a gas of N electrons in a box at density
. What is the ratio of your electron gas result
to the exact result you calculated in the previous section?
Hint: To compute the energy E of the gas of electrons in the SHO,
you will find the following sum useful,
Background note: The good agreement here is partly fortuitous.
We ignored two things. First, the actual density profile in the SHO
is not constant, and second, we ignored the potential energy of the
gas in the SHO.