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If we have some quantum system (ex. a free particle under a uniform force), how can we calculate the evolution of its (position) uncertainty? In other words, how could we find $(\Delta x)^2(t)$?

I've seen some suggestions to treat $(\Delta x)^2$ as an operator and insert it into the Heisenberg equation, while others use the definition $(\Delta)^2=\langle x^2 \rangle -(\langle x \rangle )^2$. However, I'm not sure if both are correct or if there's a better option.

Charlie
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What's wrong with computing \begin{align} \langle x(t)\rangle & = \int\,dx\,\Psi(x,t)^*\,x\,\Psi(x,t)\, ,\\ \langle x^2(t)\rangle & = \int\,dx\,\Psi(x,t)^*\,x^2\,\Psi(x,t)\, ,\\ \end{align} and then $(\Delta x(t))^2=\langle x^2(t)\rangle - \langle x(t)\rangle^2$?

ZeroTheHero
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