Identical particles and degenrate energy levels

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  • #1
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Homework Statement


Five electrons (with mass m) whose interaction can be neglected, are in the same 3-dim harmonic oscillatorpotential
[tex]V(x,y,z) = \frac k2 (x^2 + y^2 + z^2)[/tex]

What is the ground state energy?

Homework Equations




The Attempt at a Solution



I have the energy for the potential. It is:
[tex] E_{n_x,n_y,n_z} = (n_x+n_y+n_z + \frac 32)\frac{\hbar\omega}2[/tex]

My question is about the degeneracy.Since it's electrons, at most 2 of them can be in the same state, but can more than 2 electrons have the same energy?

Relating to this question: Should the ground state energy for this system be

[tex]E=2E_{111}+2E_{211}+E_{121}[/tex]

or are the two states (211) and (121) not allowed to have more than 2 electrons totally, ie

[tex]E=2E_{111}+2E_{211}+E_{221}[/tex]
 

Answers and Replies

  • #2
nrqed
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Homework Statement


Five electrons (with mass m) whose interaction can be neglected, are in the same 3-dim harmonic oscillatorpotential
[tex]V(x,y,z) = \frac k2 (x^2 + y^2 + z^2)[/tex]

What is the ground state energy?

Homework Equations




The Attempt at a Solution



I have the energy for the potential. It is:
[tex] E_{n_x,n_y,n_z} = (n_x+n_y+n_z + \frac 32)\frac{\hbar\omega}2[/tex]

My question is about the degeneracy.Since it's electrons, at most 2 of them can be in the same state, but can more than 2 electrons have the same energy?
of course. As long as they don't all have the same quantum numbers.
Relating to this question: Should the ground state energy for this system be

[tex]E=2E_{111}+2E_{211}+E_{121}[/tex]

or are the two states (211) and (121) not allowed to have more than 2 electrons totally, ie

[tex]E=2E_{111}+2E_{211}+E_{221}[/tex]

Wait. Why aren't you starting with the n=0 states??
 
  • #3
Avodyne
Science Advisor
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Well, first of all, the n's can be zero ...

Yes, more than two electrons can have the same energy if they are in a different state.
The states are labeled by the values of nx, ny, and nz, so the states (211) and (121) are each allowed to have two electrons, for a total maximum of four.
 
  • #4
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Ok, didn't know that n could be 0. So the energy should be

[tex]2E_{000}+2E_{100}+E_{010}[/tex]?
 

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