For the π-network of β-carotene modeled using the particle in the box, the position-dependent probability density of finding 1 of the 22 electrons is given by

The quantum number n in this equation is determined by the energy level of the electron under consideration. As we saw in Chapter 15, this function is strongly position dependent. The question addressed in this problem is as follows: Would you also expect the total probability density defined by P_total (x) = Σ_nˆ£Ïˆ_n(x)ˆ£² to be strongly position dependent? The sum is over all the electrons in the Ï€-network.

a. Calculate the total probability density P_total x = Σnψ ˆ£n(x)ˆ£² using the box length a = 0.29 nm and plot your results as a function of x. Does P_total (x) have the same value near the ends and at the middle of the molecule?

b. Determine ΔP_total (x) / Œ©Ptotal (x)Œª , where ΔP_total (x) is the peak-to-peak amplitude of P_total (x) in the interval between 1.2 and 1.6. nm.

c. Compare the result of part (b) with what you would obtain for an electron in the highest occupied energy level.

d. What value would you expect for P_total (x) if the electrons were uniformly distributed over the molecule? How does this value compare with your result from part (a)?

Transcribed Image Text:

P,(x) = |w„(x) %3D -sin %3!