$3.($a$)$ An electron in a one$-$dimensional box requires the absorption of a photon with a wavelength of $8080$ nm to excite it from the n $=2$ to the n $=3$ energy level. Calculate the length of this box. Show your work and express your final answer in nanometers.

$($b$)$ An electron in a $10\mathrm{.}0$ nm one$-$dimensional box is excited from the ground state into a higher energy state by absorbing a photon of electromagnetic radiation with a wavelength of $1\mathrm{.}374105$ m$.$ Determine the final energy state $($i$.$e$.,$ quantum number$)$ for this electronic transition.

$($c$)$ For a particle in the n $=3$ state of a one$-$dimensional box of length L$,$ what is the total probability of finding the particle between x $=0$ and x $=$ L$/6.$ Show your work and rationalize your approach to receive full credit.