# Probability density, expectancy value for extended Pi-network

• wanderingturtl
In summary, we discussed the position-dependent probability density for finding an electron in the pi-network of beta-carotene and the question of whether the total probability density would also be strongly position-dependent. We also discussed the calculation of Delta P-total(x) / <P-total(x)> for an electron in the highest occupied energy level, which involves solving the Schrödinger equation and taking the integral of the wavefunction and its complex conjugate.
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
Pn(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 Ptotal (x) = Sum[|Psin(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?

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.

## What is probability density?

Probability density is a measure of the likelihood of a continuous random variable taking on a specific value within a given range. It is represented by a probability density function, which maps the probability of a variable falling within a certain range to the range itself.

## What is the expectancy value for an extended Pi-network?

The expectancy value for an extended Pi-network is a measure of the average value or outcome that is expected from a system. In this context, it refers to the average voltage or current that can be expected from an extended Pi-network circuit.

## How is probability density related to expectancy value?

Probability density and expectancy value are related in that the expectancy value can be calculated by taking the integral of the product of the probability density function and the variable over its entire range. This means that the probability density function can be used to determine the likelihood of different outcomes and their expected values.

## What factors can affect the probability density and expectancy value for an extended Pi-network?

The probability density and expectancy value for an extended Pi-network can be affected by various factors such as the values of the circuit components, the input voltage or current, and external factors such as temperature and noise. Additionally, the type of probability density function used can also impact these values.

## Why is understanding probability density and expectancy value important for studying extended Pi-networks?

Understanding probability density and expectancy value is important for studying extended Pi-networks because it allows us to make predictions about the behavior of the circuit and its components. By knowing the likelihood of different outcomes and their expected values, we can design and analyze circuits more accurately and efficiently.

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