# Energy levels of a system with just two electrons?

• It's me
In summary: The two electron atom would have the same energies as positronium because they are two identical particles.
It's me

## Homework Statement

If a system comprised only of two electrons was physically possible (such as positronium but with two electrons), what would its energy levels be and how would they relate to the energy levels of Helium?

## Homework Equations

##E_{Helium} = E_{n1}+E_{n2}=-\frac{\mu Z^2 e^4}{2\hbar^2}\frac{1}{n_1^2}-\frac{\mu Z^2 e^4}{2\hbar^2}\frac{1}{n_2^2}## with ##Z=2##

##E_{Positronium} = E_{n}=-\frac{\mu Z^2 e^4}{2\hbar^2}\frac{1}{n^2}## with ##Z=1##

## The Attempt at a Solution

I'm really not sure how to approach this question since it's not a physically possible system. Is it similar to helium (because the particles are identical) or to positronium (because there is no nucleus)? Is ##Z=1## or ##Z=2##? I'm just confused.

Why do you feel that it is not a physically possible system?

Chandra Prayaga said:
Why do you feel that it is not a physically possible system?

Sorry, I forgot to mention that the electrons interact through a coulombic potential ##V=-e^2/r##. Two electrons can't be in a bound state attracting each other through this potential, as far as I understand.

It's me said:

## Homework Statement

If a system comprised only of two electrons was physically possible (such as positronium but with two electrons), what would its energy levels be and how would they relate to the energy levels of Helium?

## Homework Equations

##E_{Helium} = E_{n1}+E_{n2}=-\frac{\mu Z^2 e^4}{2\hbar^2}\frac{1}{n_1^2}-\frac{\mu Z^2 e^4}{2\hbar^2}\frac{1}{n_2^2}## with ##Z=2##

##E_{Positronium} = E_{n}=-\frac{\mu Z^2 e^4}{2\hbar^2}\frac{1}{n^2}## with ##Z=1##

## The Attempt at a Solution

I'm really not sure how to approach this question since it's not a physically possible system. Is it similar to helium (because the particles are identical) or to positronium (because there is no nucleus)? Is ##Z=1## or ##Z=2##? I'm just confused.
You are correct about the impossibility of a bound state with two electrons, the question should have been about positronium.

That said, the equation you wrote for positronium is actually valid for any hydrogen-like atom: an atom made of a nucleus with a charge Z and a single electron orbiting the nucleus. In the case of positronium, we can think of the nucleus as being simply the positron. In any case, just use Z^2=1 for positronium. All you have left to do is to figure out the reduced mass ##\mu## for your two electron atom (or, which is the same, for positronium) and for helium, and relate the energies of Helium to the energies of positronium.

nrqed said:
for your two electron atom (or, which is the same, for positronium)

But wouldn't the two electron atom have different energies than the positronium because they are two identical particles?

## 1. What is the significance of energy levels in a system with two electrons?

The energy levels of a system with two electrons determine the possible states and configurations of the two electrons within the system. This has important implications for the behavior and properties of the system.

## 2. How are the energy levels of a system with two electrons calculated?

The energy levels of a system with two electrons are typically calculated using the Schrödinger equation, which takes into account the potential energy of the system, as well as the kinetic energy of the electrons.

## 3. What is the difference between the ground state and excited states in a system with two electrons?

The ground state of a system with two electrons is the lowest energy state that the system can occupy, while the excited states are higher energy states that the system can transition to by absorbing energy.

## 4. How does the number of energy levels in a system with two electrons affect its properties?

The number of energy levels in a system with two electrons can greatly influence its properties, such as its stability, reactivity, and electronic configuration. A system with more energy levels may have a more complex behavior and a larger range of possible configurations.

## 5. Can the energy levels of a system with two electrons be manipulated?

Yes, the energy levels of a system with two electrons can be altered by changing the external conditions, such as applying an electric or magnetic field. This can result in changes in the system's properties and behavior.

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