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Homework Help: Identical particles and degenrate energy levels

  1. Dec 3, 2007 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant equations


    3. 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]
     
  2. jcsd
  3. Dec 3, 2007 #2

    nrqed

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    of course. As long as they don't all have the same quantum numbers.
    Wait. Why aren't you starting with the n=0 states??
     
  4. Dec 3, 2007 #3

    Avodyne

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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.
     
  5. Dec 3, 2007 #4
    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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