Consider an electron in a hydrogen atom with -1. there are three possible eigen- states of the operator l2, given by |l, m) 11, 1), |1,0), and |1,-1). (a) recall that the raising/lower operators are given by ll ily. also recall the rela- tionship use this relationship to determine the 3x3 matrix representation of ly (using a basis com- prised of eigenstates of lz). (b) what are the eigenvalues and associated eigenstates of the operator ly? (c) if the quantum state is initially |1,1), what is the probability that a measurement of l yields 0?

The particle in a two-dimensional well is a useful model for the motion of electrons around the indole ring (3), the conjugated cycle found in the side chain of tryptophan. we may regard indole as a rectangle with sides of length 280 pm and 450 pm, with 10 electrons in the conjugated p system. as in case study 9.1, we assume that in the ground state of the molecule each quantized level is occupied by two electrons. (a) calculate the energy of an electron in the highest occupied level. (b) calculate the frequency of radiation that can induce a transition between the highest occupied and lowest unoccupied levels. 9.27 electrons around the porphine ring (4), the conjugated macrocycle that forms the structural basis of the heme group and the chlorophylls. we may treat the group as a circular ring of radius 440 pm, with 20 electrons in the conjugated system moving along the perimeter of the ring. as in exercise 9.26, assume that in the ground state of the molecule quantized each level is occupied by two electrons. (a) calculate the energy and angular momentum of an electron in the highest occupied level. (b) calculate the frequency of radiation that can induce a transition between the highest occupied and lowest unoccupied levels.