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Harmonic Oscillator States

  1. Sep 10, 2006 #1
    Suppose I have 4 bosons in a one-dimensional harmonic oscillator potential and that the total energy is [tex]E_\text{tot} = 8 \hbar \omega[/tex]. Recall, [tex]E_n = (n+1/2)\hbar\omega[/tex].

    Question: How many quantum states exist? (assume no spin degeneracy)

    After accounting for the ground state, we have 6 quanta of energy to distribute. The harmonic oscillators are indistinguishable, so we do not care about the ordering. Here is my count:

    (6,0,0,0)
    (5,1,0,0)
    (4,2,0,0)
    (4,1,1,0)
    (3,3,0,0)
    (3,2,1,0)
    (3,1,1,1)
    (2,2,2,0)
    (2,2,1,1)

    where, for example, (5,1,0,0) means we give one boson 5 quanta and another boson 1 quanta...the other two bosons are in the ground state.

    So, there are 9 possible quantum states.

    My question: What is the general formula that determines the number of quantum states?

    Typical answer:

    This situation is akin to have 6 balls and 4-1 walls, and then asking how many ways can you permute those objects. To which, the answer is:

    [tex]
    \binom{6+(4-1)}{4-1} = \frac{[6+(4-1)]!}{(4-1)!6!} = 84
    [/tex]

    and is _clearly_ wrong. This formula comes from the question of asking how many ways can you put N indistinguishable bosons into d distinguishable degenerate energy levels. Essentially, it cares about the order. That is,

    (6,0,0,0) 4!/3! permutations
    (5,1,0,0) 4!/2! permutations
    (4,2,0,0) 4!/2! permutations
    (4,1,1,0) 4!/2! permutations
    (3,3,0,0) 4!/2!/2! permutations
    (3,2,1,0) 4! permutations
    (3,1,1,1) 4!/3! permutations
    (2,2,2,0) 4!/3! permutations
    (2,2,1,1) 4!/2!/2! permutations

    (yes that adds to 84)

    For this problem, we are asking: How many ways can you put 6 indistinguishable quanta into 4 indistinguishable bosons? So the questions are not the same and the above formula does not apply. Here is my question once again:

    How many ways can we divide M indistinquishable quanta into N indistinguishable bosons?

    Notice, if N >= M, then the answer is given by p(M), the partition function of M.
    http://en.wikipedia.org/wiki/Integer_partition
     
    Last edited: Sep 10, 2006
  2. jcsd
  3. Sep 12, 2006 #2
    Anyone? I know there are smart enough people here.
     
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