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Degenerate energy levels

  1. Oct 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the quantum numbers of the three lowest states that have the same energy. (Enter the quantum numbers for the three states in increasing order of n1, using the format n1,n2.)

    2. Relevant equations

    En1n2=[(hbar)^2/(2m)]*[(pi)^2/(L^2)]*[(n1)^2+(n2)^2]
    - Sorry about the formula; I tried entering it using LaTeX but that failed

    3. The attempt at a solution

    I am confused about how to find the three lowest energy states when I only have two quantum numbers. For example, the question I answered before this wanted the two lowest energy states that were degenerate, so I entered E1,2=E2,1 and it was correct. How am I supposed to come up with three different degenerate levels with only two quantum numbers? Any help would be much appreciated.
     
  2. jcsd
  3. Oct 8, 2009 #2

    lanedance

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    here's some tex
    [tex] E_{n_1,n_2}=\frac{\hbar^2}{2m} \frac{\pi^2}{L^2(n_1^2+n_2^2)} [/tex]

    so only n1 & n2 change, i would start by listing out some of the energies, or equivalently teh first few values fro different copmbinatinos of n1 & n2 & see if anything pops out:
    [ext] (n_1^2+n_2^2) [/tex]
     
    Last edited: Oct 8, 2009
  4. Oct 8, 2009 #3

    Dick

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    Kind of a funny question, but you want find a number N that can be written in three different ways, as (n1)^2+(n2)^2, (n2)^2+(n1)^2 where n1 and n2 are different, and as (n3)^2+(n3)^2. From the last one it follows that N is two times a perfect square. So possibilities for N are 2, 8, 18, 32, 50, 72, 98,... One of those works. Can you find it?
     
  5. Oct 11, 2009 #4
    Okay I understand what you're saying, but at the same time I don't. Will I end up with two combinations that are just opposite and one that is different than the first two.
     
  6. Oct 11, 2009 #5
    Okay, nevermind I figured that out.
     
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