1. The “electron in a box” model!
Hi! I’m Erica the electron ☺

2. The “electron in a box” model!
Imagine an electron is confined within a linear box length L.
According to de Broglie, it has an associated wavelength λ = h/p ☺ L

3. The “electron in a box” model!
Imagine then the electron wave forming a stationary wave in the box.
Therefore we have a stationary wave with nodes at x = 0 and at x = L (boundary conditions) L

4. The “electron in a box” model!
Write the wavelength of in terms of the length of the box, L
i.e.; λ = ? L L

5. The “electron in a box” model!
The wavelength therefore of any stationary wave must be
λ = 2L/n
where n is an integer. L

6. The “electron in a box” model!
Using the following equations:
λ = 2L/n
p = h/λ
Ek = ½ mv2
Show that Ek of an electron = n2h2/8mL2

8. Energy States
This can be thought of like the allowed frequencies of a standing wave on a string (but this is a crude analogy).

9. Energy Levels The red curve is the ground energy level
The dark purple line is fifth energy level. 
With the a higher energy level, the wavelength is reduced by 1/2  
(remember the higher frequency means higher energy) 
The idea of an "energy level" is if a particle is quantized then all of the properties of that particle is quantized including energy.  The infinite square well is a way to show these energy levels.  
Why does it have to be infinite?  Well the idea is that wave cannot make it beyond the "walls" of the "well", the infinite square well means it would take an infinite amount of energy to penetrate the walls