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(adsbygoogle = window.adsbygoogle || []).push({});

\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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Boson statistics and the uncertainty principle

**Physics Forums | Science Articles, Homework Help, Discussion**