# Difficulties with overlap rule

• Bemused
In summary: Well, the variational principle can be used to calculate PSI. I'm afraid I can't give you a more definitive answer here.
Bemused
First post and not used to the symbols that I have seen used in posting formulae, so please be patient!

The probability of obtaining the ith energy eigenvalue Ei is given by the square modulus of the integral psi_i*(x)PSI(x,t)dx where PSI(x,t) is the wave function of the system.

How can this be calculated since the wavefunction is a product of the superposition of all the possible energy eigenfunctions, psi_i(x)? I guess what I am trying to ask is - how can we know PSI(x,t)? Every text I have read typically starts problems in this area with "Suppose we have a wavefunction,..."

Thanks to all that help

As noted above, I'm new to the board and just checked the forums to see my question moved to this Homework section. Just to note that this is not part of any assignment - I'm trying to clear up gaps in my understanding.

Thanks

To actually calculate a probability, they'll need to give you some more or less explicit form for PSI. Right?

Dick,

Well, this is how I have always encountered the overlap integral - you're given a wavefunction and can use it to e.g. calculate the probability of finding the system in a particular eigenenergy. I'm trying to broaden my horizons though - where does PSI come from? I seem to remember something about the variational principle being used to produce a "best guess" but would like to hear from someone with a definitive answer here.
John

Ok. A 'useful' example of this sort of problem would be the following. Suppose an electron is in a particular energy state for an atom. The atom undergoes radioactive decay so the charge on the nucleus suddenly changes. What is the probability the electron transitions to some given state in the new atom. Then PSI is the old wave function and the P_i are the new energy eigenfunctions.

That PSI is the solution to the SE

$$\frac{d\left|\Psi\right\rangle}{dt}=\frac{1}{i\hbar}\hat{H}\left|\Psi\right\rangle$$

Still not feeling enlightened as to my straightforward question - how can we know PSI so as to evaluate the overlap integral?

## 1. What is the overlap rule in science?

The overlap rule in science refers to the concept that when two or more substances or elements occupy the same space, they will interact and potentially create new properties or reactions. This can occur in various fields of science, including chemistry, biology, and physics.

## 2. Why is the overlap rule important in scientific research?

The overlap rule is important in scientific research because it helps scientists understand how different substances or elements interact and how these interactions can affect their properties and behaviors. This knowledge is crucial in developing new technologies, medicines, and materials.

## 3. What are some common difficulties with the overlap rule?

One common difficulty with the overlap rule is identifying all the substances or elements present in a system and understanding how they interact with each other. This can be challenging when dealing with complex systems or when substances are present in very small amounts.

Another difficulty is accurately predicting the outcomes of interactions between substances or elements. While the overlap rule provides a framework for understanding these interactions, there are often unexpected results that can be difficult to explain or control.

## 4. How do scientists address difficulties with the overlap rule?

Scientists address difficulties with the overlap rule by using various research techniques and technologies to analyze and understand interactions between substances or elements. This can include laboratory experiments, computer simulations, and mathematical models.

Collaboration and peer review also play a crucial role in addressing difficulties with the overlap rule. By sharing their findings and discussing them with other scientists, researchers can gain new insights and perspectives on complex systems.

## 5. How does the overlap rule impact our daily lives?

The overlap rule impacts our daily lives in numerous ways. For example, understanding the interactions between different medications is crucial in preventing harmful drug interactions. In the food industry, the overlap rule is used to develop new and improved flavors and textures. Additionally, the overlap rule is essential in fields such as environmental science, where understanding how substances interact in the natural world can help mitigate pollution and other environmental issues.

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