Probability density, expectancy value for extended Pi-network

[wanderingturtl]

1. For the \pi-network of \beta-carotene modeled using the particle in the box, the position-dependent probability density of finding 1 of the 22 electrons is given by
P_n(x) = |$\Psi$$_{}n$(x)|^2 = (2/a)Sin^2 (n Pi x / a)
The quantum number n in this equation is determined by the energy level of the electron under consideration. As we saw in Chapter 15 in the textbook, 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) = Sum[|Psi_n(x)|^2,n,22] to be strongly position dependent? The sum is over all the electrons (22) in the pi-network.

a = 2.9 nm

2. It proceeds to ask about the Delta P-total(x) / <P-total(x)> for the interval 1.2 to 1.6 nm
And then it asks for the Delta P-total(x) / <P-total(x)> for an electron in the highest occupied energy level3. Delta P-total(x) I have found to be Sum[2/(2.9*(10^9) ) (Sin[n Pi 1.6/2.9])^2, {n, 22}] -
Sum[2/(2.9*(10^9) ) (Sin[n Pi 1.2/2.9])^2, {n, 22}] = 7.6 * 10^7 (I hope mathematica code is spoken here)

Now, I believe the chevrons < > mean average value or expectancy value, per postulate 4 of quantum mechanics. To find average value, you take ∫Psi*Operator Psi . My first problem is, what is the operator here? We are talking about probability density, so is there an operator? There must be something, because on an answer sheet I see <P-total (x)> to be 0.79 * 10^9 m^-1 .

Then on the next question, how do I even begin to calculate delta <P-total(x)> for one electron without boundaries? Am I reading the question wrong?

 

[Nate810]

No, you are reading the question correctly. To find the expectation value of the total probability density for an electron in the highest occupied energy level, you need to first calculate the wavefunction for that electron. This is done by solving the Schrödinger equation for the Hamiltonian of the pi-network. The wavefunction for the electron in the highest occupied energy level will be a superposition of all the wavefunctions for the individual electrons in the network. Once you have the wavefunction for the electron in the highest occupied energy level, you can then calculate the expectation value of the total probability density by taking the integral of the product of the wavefunction and its complex conjugate over all space. This integral should give you the expectation value of the total probability density at any point in space. You can then find the difference between the expectation value of the total probability density at two points in space, 1.2 nm and 1.6 nm, to get the Delta P-total(x) / <P-total(x)> for the interval 1.2 to 1.6 nm.