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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?

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    $\begingroup$ You have to equate $E = \hbar\omega/2$ with the zero point energy of the particle in the 1D potential box, i.e. $E = \frac{h^2}{8mL^2}$. From here, find $\omega$! And once you have $\omega$, insert it to the equation $k = m\omega^2$. $\endgroup$ – rnels12 Dec 13 '18 at 13:12
  • $\begingroup$ @rnels12 That is exactly what I did and was asking a confirmation for. Thank you! $\endgroup$ – physicist Dec 14 '18 at 18:04

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