I'll try to keep this short and concise. What made me start thinking about this question was the material we have been covering regarding PIB wave functions in my quantum chemistry class, and material we have been covering regarding ozone absorption in my climate class. I would like to try to understand the relationship between the increased wave function domain that delocalization yields for conjugated pi electrons, and the effect that this has on the wavelengths of light that they can absorb. Like ozone, I understand that most other conjugated pi systems have strong UV absorption, so I would assume there is some correlation there.
Specifically though, I would like to know if this can be explained (at least to some extent) by a simple QM model like the 1D PIB, and if how electrons' wave functions are altered when their spacial domain is extended across the whole molecule by delocalization can explain UV absorption in conjugated pi systems.