5. Now let's think about the bounds of variable . Show that applying the boundaryconditions on forces our quantum number (m) to be equal to 0, ±1, ±2, ±3, ±4 ... Whatdid applying the boundary condition do to our general solution from question 4?(Hint: to the apply the boundary condition, ask yourself: how is (p) related to(+2π)? Once you have the boundary condition, check the appendix for somepotentially useful mathematical expressions)6. Normalize (d).7. Prove that a particle-on-a-ring will have discrete energy levels described by the followingequation,Emi=m²ħ²218. Now we will apply these solutions to a Benzene molecule. Assuming benzene is a circle,estimate the circumference of benzene - assume each carbon-carbon bond in benzene is1.4 Å. Calculate the radius of benzene using that circumference. There are six л-electronsin benzene, which are free to move around the ring due to the conjugation of the doublebonds. Make an energy level diagram using the energy expression from question 7, andcalculate the wavelength for the lowest energy electronic transition. The experimentallyobserved transition for benzene is at 200 nm, does this agree with your prediction? (Hint:the ground state will be my = 0, and the rest of the energy levels are going to be doublydegenerate).

Answered: Image /qna-images/answer/26ac2e36-398b-4502-9da3-b46db97fedf9.jpg

5. Now let's think about the bounds of variable p. Show that applying the boundary conditions on forces our quantum number (mi) to be equal to 0, ±1, ±2, ±3, ±4 ... What did applying the boundary condition do to our general solution from question 4? (Hint: to the apply the boundary condition, ask yourself: how is (p) related to (+2π)? Once you have the boundary condition, check the appendix for some potentially useful mathematical expressions) 6. Normalize (d). 7. Prove that a particle-on-a-ring will have discrete energy levels described by the following equation, Emi = m²ħ² 21

5. Now let's think about the bounds of variable p. Show that applying the boundary conditions on forces our quantum number (mi) to be equal to 0, ±1, ±2, ±3, ±4 ... What did applying the boundary condition do to our general solution from question 4? (Hint: to the apply the boundary condition, ask yourself: how is (p) related to (+2π)? Once you have the boundary condition, check the appendix for some potentially useful mathematical expressions) 6. Normalize (d). 7. Prove that a particle-on-a-ring will have discrete energy levels described by the following equation, Emi = m²ħ² 21