# Arranging 3 Identical Electrons

1. Mar 9, 2014

### kashokjayaram

I have seen this question somewhere.. I diidnt get the answer. So plz help me.

In how many ways three identical electrons can be arranged in two energy levels..??

Is it 2 or 4??

Here will you consider the spin of the electrons??

2. Mar 9, 2014

### ChrisVer

If you don't take in consideration the spin, you cannot arrange the 3 electrons into 2 energy states...
I don't understand what you mean by "is it 2 or 4?"

3. Mar 9, 2014

### craigi

All electrons are identical apart from their state which can never be identical, so this term is misleading.

Presuming that one of the energy levels is the lowest energy level of a system, I would say the most rational answer is 3, correspnding to 0, 1 or 2 electrons in the lowest energy level, each giving a different total energy for the system.

2 electrons can share the lowest energy level only because they can have opposing spin.

If both are higher energy levels then we can have all 3 in the same energy level also.

Should we consider spin and orientation? To some extent we already have in determining how many can be in any particular energy level. Beyond that, it really depends upon the context, but the problem gets much more complicated if we do. In the case of an isolated system there is no reason to.

Last edited: Mar 9, 2014
4. Mar 9, 2014

### Bill_K

The answer is 4. With two energy levels, counting spin, there are 4 states. Three of these states are occupied, one is not. You get to pick which one is not, and you have 4 choices.

5. Mar 9, 2014

### craigi

...but in multiple dimensions, with the exception of the ground state, more than 2 states corrrespond to the same energy level through degeneracy, right?

Last edited: Mar 9, 2014
6. Mar 9, 2014

### Bill_K

Yes, and the degeneracy will be different for electrons in a hydrogen atom as opposed to ones in a harmonic oscillator. And the degeneracy might be broken if you consider spin-orbit coupling.

But most questions are designed to have answers, and it's pretty far-fetched to imagine that this question is about harmonic oscillators and hydrogen atoms. It's only intended to illustrate a point about Fermi-Dirac statistics, and I think that's all you should read into it.

7. May 19, 2014

### kashokjayaram

I didnt get what are all the states occupied and what is not..! "Not occupied states" are also included...???!!!

Ok.. Let the states be 0 and ε with $\uparrow\downarrow$ in lower state 0 and $\uparrow$ or $\downarrow$ in higher state.
Thus, two arrangements. What are the other two..?
Is $\uparrow\downarrow$ in upper state ε and $\uparrow$ or $\downarrow$ in lower state 0 the other two?

Is it good to think like that?