# Electrons on a ring - too easy.

• Fallen Seraph
In summary, the problem involves three electrons moving on a ring of radius R and the task is to find their equilibrium positions in terms of the angles between them. Some hints with differentials and the potential between two electrons are given, but they seem unnecessary. The question is straightforward and the answer is likely to be symmetric with the particles separated by 120 degrees due to their identical nature. While the solution may seem obvious, it may not be computationally easy to prove by hand, similar to the classic problem of proving the shortest distance between two points in a Euclidean geometry.
Fallen Seraph

## Homework Statement

Three electrons are confined to move on a ring of radius R. Find the equilibrium positions of the electrons in terms of the angles between them.

## Homework Equations

Some hints with differentials are given to us, as is the potential between two of the electrons , but as far as I can tell these are completely redundant.

## The Attempt at a Solution

Well my problem is this: we've been given this question in a second year comp lab and the answer is so glaringly obvious to me that I can only come to the conclusion that I'm misunderstanding something. I don't get how one could possibly need a computer to get this.

As I see it, the three electrons being identical particles implies that the final state must be symmetric under interchanging the particles. And on a ring the only way to do this is to have them all separated by 120 degrees.

Surely it can't be that simple? What am I missing?

The answer might be obvious but computationally it's not the simplest thing in the world to prove by hand

EDIT: It's kinda like the classic prove the shortest distance between two points in a euclidean geometry is a straight line. Duh, right? So go ahead and do it

There's an INFINITE number of other possible paths to take(just like here there are infinite configurations) you're going to have a helluva time going through all infinite cases and showing they don't work

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It is correct that the three electrons on a ring will naturally arrange themselves at 120 degree intervals due to their identical nature. However, it is important to also consider the energy of the system and the potential between the electrons. The equilibrium positions will not only depend on the symmetry of the system, but also on the potential energy of the electrons at different angles. In addition, the use of differentials and potential equations can help to further analyze the stability and behavior of the system. Therefore, while the solution may seem obvious, it is important to consider all factors and use mathematical techniques to fully understand the system.

## What is the concept of "Electrons on a ring - too easy"?

Electrons on a ring - too easy refers to a simplified model used in physics to study the behavior of electrons in a circular path, as if they were moving on a ring. It is often used as an introductory topic in quantum mechanics.

## How do electrons behave on a ring?

Electrons on a ring exhibit wave-like behavior, meaning they can exist in a range of energies and positions at the same time. This is known as the Heisenberg uncertainty principle. The exact location and momentum of the electrons cannot be known simultaneously.

## What is the difference between electrons on a ring and electrons in an atom?

Electrons on a ring are a simplified model and do not account for the attractive force between the positively charged nucleus and the negatively charged electrons. In an atom, the electrons do not move in a circular path, but rather exist in specific energy levels around the nucleus.

## Are there any real-life applications of the concept of electrons on a ring?

The concept of electrons on a ring is used in the design and operation of electron storage rings, which are used in particle accelerators and synchrotron radiation facilities. It is also used in the study of superconductivity and quantum computing.

## How is the concept of electrons on a ring related to quantum mechanics?

The concept of electrons on a ring is used to introduce the principles of quantum mechanics, such as the wave-particle duality and the uncertainty principle. It helps to explain the behavior of particles at the atomic and subatomic level.

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