Arranging 3 Identical Electrons

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In summary: Some people might find it helpful to think of the states as being occupied and unoccupied, and to consider only the arrangements that have at least one electron in the unoccupied state. This can help you to focus on the question at hand.
  • #1
kashokjayaram
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I have seen this question somewhere.. I diidnt get the answer. So please 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??
 
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  • #2
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
kashokjayaram said:
I have seen this question somewhere.. I diidnt get the answer. So please 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??

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.
 
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  • #4
kashokjayaram said:
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??
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
Bill_K said:
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.

...but in multiple dimensions, with the exception of the ground state, more than 2 states corrrespond to the same energy level through degeneracy, right?
 
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  • #6
craigi said:
...but in multiple dimensions, with the exception of the ground state, more than 2 states corrrespond to the same energy level through degeneracy, right?
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
Bill_K said:
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.

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 [itex]\uparrow\downarrow[/itex] in lower state 0 and [itex]\uparrow[/itex] or [itex]\downarrow[/itex] in higher state.
Thus, two arrangements. What are the other two..?
Is [itex]\uparrow\downarrow[/itex] in upper state ε and [itex]\uparrow[/itex] or [itex]\downarrow[/itex] in lower state 0 the other two?

Is it good to think like that?
 

FAQ: Arranging 3 Identical Electrons

How do I arrange 3 identical electrons in an atom?

In an atom, the 3 identical electrons will occupy the lowest energy level possible, known as the ground state. This means they will fill the 1s orbital, with 2 electrons in the first sublevel and 1 electron in the second sublevel.

Can I arrange 3 identical electrons in the same orbital?

No, according to the Pauli exclusion principle, no two electrons in an atom can have the same set of quantum numbers. This means that the 3 identical electrons must have different spin states and therefore cannot occupy the same orbital.

How does electron spin affect the arrangement of 3 identical electrons?

Electron spin refers to the intrinsic magnetic property of an electron. The arrangement of 3 identical electrons is affected by electron spin because the Pauli exclusion principle states that no two electrons can have the same spin state. This means that the 3 electrons must have different spin states and cannot occupy the same orbital.

Do the 3 identical electrons have to be in the same energy level?

No, the 3 identical electrons can be in different energy levels as long as they follow the rules of electron configuration. This means that the first 2 electrons will fill the 1s orbital, and the third electron will fill the next available energy level, which is the 2s orbital.

Can the arrangement of 3 identical electrons change in an excited state?

Yes, in an excited state, the electrons can rearrange themselves and occupy different orbitals. However, they must still follow the rules of electron configuration, meaning no two electrons can have the same set of quantum numbers. This means that the 3 identical electrons will still occupy different orbitals in the excited state.

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