Arranging 3 Identical Electrons

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Discussion Overview

The discussion revolves around the arrangement of three identical electrons in two energy levels, considering the implications of electron spin. Participants explore the number of possible arrangements and the role of spin in determining these configurations.

Discussion Character

  • Debate/contested
  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that if spin is not considered, arranging three electrons in two energy states is not feasible.
  • Others propose that considering spin allows for multiple arrangements, with one participant stating that the most rational answer is 3, based on the distribution of electrons in energy levels.
  • One participant asserts that the answer is 4, explaining that with two energy levels and counting spin, there are 4 states, and one can choose which state is unoccupied.
  • Another participant questions the concept of occupied and unoccupied states, seeking clarification on how to represent the arrangements mathematically.
  • There is mention of degeneracy in multiple dimensions and how it may affect the arrangement of electrons, particularly in different systems like hydrogen atoms versus harmonic oscillators.
  • Some participants express uncertainty about the context of the question, suggesting it is primarily illustrative of Fermi-Dirac statistics rather than a specific physical system.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the number of arrangements or the significance of spin, with multiple competing views presented throughout the discussion.

Contextual Notes

There are unresolved assumptions regarding the treatment of spin and the specific context of the energy levels, which may affect the interpretation of the arrangements. The discussion also highlights the complexity introduced by considering degeneracy and different physical systems.

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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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?"
 
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.
 
Last edited:
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.
 
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?
 
Last edited:
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.
 
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 \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?
 

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