$($a$)$ Regarding the mental exercise of describing a particle in a box, answer the following with

supporting diagrams and explanations.

$($i$)$ Why is a particle in a $1-$dimensional box used to allude to molecular orbital theory in

chemistry?

$($ii$)$ Describe the evolution of a particle in a $1-$dimensional box, into a particle in a $3-$

dimensional box, and further to description of bonding in the context of molecular

orbital theory. Justify your answer using well$-$labelled diagrams.

$($b$)$ In a parallel universe, electrons and nuclei have identical masses, yet with all other aspects

equivalent $($energetics$,$ equations, etc.$).$ Predict the structure of benzene in this parallel

universe. Your answer should include detailed schematics and diagrams and$/$or equations to

support its conclusions.