How can I derive the energy of an electron using the electron in a box model?

The electron in a box model uses the idea that the energy of the electron and that of a standing wave in a box are analogous. So using your knowledge of standing waves you can derive an equation for the electrons energy.

The equation you need to derive is KE = (n²h²)/(8mL²)

It looks a little daunting and is not something I would memorise. Thankfully the derivation just requires 3 main steps and 3 main equations.

Treat the electron in a box as you would treat a standing wave on a string length L. The boundary conditions are that the wave has nodes at either end of the string. If you start drawing out the possible wavelengths on a string length L you will start to see a relationship emerging between the number of anti-nodes, n, and the wavelength, λ. Thus the allowed wavelengths are λ = 2L / n where n = 1, 2, 3, ...

Now that we have this relation we can use our equations from quantum physics, look for one that might includes λ, in this case λ = h / p.

Now substitue p = mv we have λ = 2L / n  = h / (mv). Which rearranges to v = (hn) / (2Lm). So now we have an equation for the velocity of the electron in terms of L.

Remember we needed to find an expression for the energy, but there is the equation KE = mv² / 2. So by substitution:

KE = (n²h²)/(8mL²)

This equation can model the energies of an electron according to it's energy level in the atom. So the n in the equation represents its energy level, the h is Planck's constant, m is the mass of the electron and L is the the length of the box or the space to which the electron is confined.