In order to calculate the dispersion relation (i.e $w(k)$) for the electrons and protons, I used the following relations:
$ E = ℏω$, $p = ℏk$, and I substituted them in this formula for energy:
$E = p^2/ 2m$.
Therefore I got that $ω(k) = ℏk^2/ 2m$. However, here it is another exercise:
A particle of mass $m$ moves in the potential of a $1D$ harmonic oscillator. Calculate the spring constant $k$ of the oscillator if the zero point energy of the particle equals the zero point energy of the same particle in a $1D$ potential box of width $a$.
In order to solve this, I equated the expressions for zero-point energies, respectively. But for the formula for the zero-point energy of a harmonic oscillator, I have that:
$E = ℏω/2$, according to my book.
Now, should I substitute $ω$ = $(k/m)$^$0.5$ or the $w(k)$ I derived for the dispersion relation?