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Homework Help: Probabilities for system of two bosons

  1. Dec 6, 2008 #1
    If there are two indistinguishable bosons that can either be in the |0> or |1> state, what is the probability that both will be in the |0> state? (ie the system will be in the |0>|0> state)

    I know there are only three possibilities for the total state of the system: |0>|0>, |1>|1>, and (1/sqrt2)(|0>|1>+|1>|0>), but are these states equally probable? (I'm assuming that a single boson has an equal probability of being in either |0> or |1>) Would the probability of being in |0>|0> just be 1/3? or is it more complicated than that?
     
  2. jcsd
  3. Dec 6, 2008 #2

    Avodyne

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    Call the three states

    [tex]|00\rangle,\;|11\rangle,\;\hbox{and}\;|S\rangle \equiv{\textstyle{1\over\sqrt2}}\bigl(|01\rangle+|10\rangle\bigr).[/tex]

    Then the most general state of the two particles is

    [tex]a|00\rangle+b|11\rangle+c|S\rangle[/tex]

    with [itex]|a|^2+|b|^2+|c|^2=1[/itex].

    What can you deduce about [itex]a[/itex], [itex]b[/itex], and [itex]c[/itex] from the requirement that each particle be equally likely to be in [itex]|0\rangle[/itex] or [itex]|1\rangle[/itex]?
     
    Last edited: Dec 6, 2008
  4. Dec 7, 2008 #3
    Well, I guess if a=b=c, then they'd each have to be 1/sqrt3... unless we also have to take into account the 1/sqrt2 factor in front of |S>?
     
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