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Boson statistics and the uncertainty principle

  1. Jul 24, 2013 #1
    This question is regarding the boson statistics and it’s relation to the uncertainty principle. Consider we have a vacuum state and we apply a field operator on it to create a particle at position x, we end up with state like

    \begin{array}{l}
    \left| \psi \right\rangle = {\psi ^\dag }(x)\left| 0 \right\rangle \\
    \left| \psi \right\rangle = \sum\limits_p^{} {{a^\dag }(p){e^{ - ipx}}} \left| 0 \right\rangle \\
    \left| \psi \right\rangle = \sum\limits_p^{} {{e^{ - ipx}}} \left| p \right\rangle
    \end{array}

    So there is an equal probability for all momentum. And the result is completely consistent with the uncertainty principle.
    Now consider a situation where we have created 99 bosons with momentum k. Using the statistics of creation of creation operators, when we now add an additional particle, we have


    \begin{array}{l}
    \left| \psi \right\rangle = {\psi ^\dag }(x)\left| {0,0,99,0,...} \right\rangle \\
    \left| \psi \right\rangle = \sum\limits_p^{} {{a^\dag }(p){e^{ - ipx}}} \left| {0,0,99,0,...} \right\rangle \\
    \left| \psi \right\rangle = {e^{ - i{p_1}x}}\left| {1,0,99,0,...} \right\rangle + {e^{ - i{p_2}x}}\left| {0,1,99,0,...} \right\rangle + {e^{ - i{p_k}x}}\sqrt {100} \left| {0,0,100,0,...} \right\rangle + {e^{ - i{p_{k + 1}}x}}\left| {0,0,99,1,...} \right\rangle
    \end{array}/

    We see that that when we create the now particle, it is more likely to have momentum k than any other momentum by a factor of 100. In the limit that we have a thousand billion particles in a certain momentum and then create a particle at a certain position, with almost 100% we can determine the momentum that it will have. Does this not violate the uncertainty principle?
     
  2. jcsd
  3. Jul 24, 2013 #2
    just to clarify

    /dag is the dagger symbol (hermitian conjugate).
     
  4. Jul 24, 2013 #3

    stevendaryl

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    Staff Emeritus
    Science Advisor


    What does your notation [itex]\left| {0,0,99,1,...} \right\rangle[/itex] mean?
     
  5. Jul 24, 2013 #4

    Cthugha

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    Science Advisor

    In order to have an effect of boson statistics, you need indistinguishable particles. So you do not get n individual particles, but one n-particle state. So in adding a particle, you go from the n-1 to the n-particle state. It is the position and momentum of the whole state of indistinguishable particles you need to consider. If this whole state has a pretty sharp momentum, it will be rather delocalized. If you (let us just assume that is possible somehow) create a particle that is localized better than the n-1 particle state, the added particle and the other particles cannot be completely indistinguishable, so the bosonic enhancement factor will not be n, but smaller.
     
  6. Jul 29, 2013 #5
    Thank you Cthuga. That makes sense
     
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