# Energy levels of a system with just two electrons?

## 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.

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

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.

nrqed
Homework Helper
Gold Member

## 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.

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?