1. Write the Schrodinger equation for particle on a ring, andrearrange it until you have the following: ? 2? ??2 = − 2?? ℠2 ?…
a) Assuming that ?? 2 = 2?? ℠2 , (where ml is a quantum numberand has nothing to do with mass), show that the following is asolution for the Schrodinger equation you obtained: ?(?) = ? ?????…
b)Now think about bounds of variable ?. Using that argue that?(?) = ? ( ?+2pi), and prove that ml can be 0, ±1, ±2, ±3, ±4,Normalize that ?(?)
c)Prove that particle on the ring will have discrete energylevels described by the following equation, ??? = ?? 2℠2 2? …
d) Now we will apply these solutions to Benzene molecule. Eachdouble bond in benzene is 1.4Ã…, so you know the circumference ofbenzene. Now calculate the radius of benzene using thecircumference. There are six ï°-electrons on benzene, which are freeto move around the ring due to conjugated double bonds. Make anenergy level diagram using equation 6, and calculate the wavelengthfor the lowest energy electronic transition. Experimentallyobserved transition for benzene is at 200 nm. (Hint: The groundstate will be ml = 0, and then rest of the energy levels are goingto be doubly degenerate.)
