Question

1. Use the model of the particle in the ring (rotation in two dimensions) of quantum mechanics to describe the movement of electrons in the conjugate system of the benzene molecule. Presume that the circumference of the ring is equal to 8.40A. a. ((3 pts) Make a diagram of the energy levels of the electrons Pi in the benzene molecule, clearly identifying the HOMO and the LUMO. b. (3 pts) Calculate the energy of the HOMO and LUMO C. (3 pts) Calculate the wavelength corresponding to the transition HOMO- LUMO 2. Use the model of the particle in the box for the electrons in the conjugated system of the 1,3,5-hexatriene molecule. Presume that the length of the box (five C-C links) is equal to 7.18 А. a. (3 pts) Make a diagram of the energy levels of the electrons Pi in the 1,3,5-hexatriene molecule, clearly identifying the HOMO and the LUMO b. (3 pts) Calculate the energy of the HOMO and LUMO C. (3 pts) Calculate the wavelength corresponding to the transition HOMO-> LUMO 3. Suppose that a certain system is in a state described by the wave function Av. Suppose that this function is a linear combination of the functions AV1 and AV2. where the functions Av1 and AV2 are functions proper to operator A (corresponding to the property physical A), with proper values â€á€.al and a2, so that AAV1=a1 and AAV2-a2 Presume that all these functions are to. (4 pts) From the general expression for the expected value of A, <Aintegral(Aväe CAAVdt) prove that <A IC1A2ãca1 +IC2|A2aca2 b. (2 pts) What is the physical meaning of the numbers |C12 and C2 A2?

Answer 1