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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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  • Didn't I just see this closed with a comment about how someone had seen something identical closed yesterday? – JMac Aug 27 '18 at 18:27
  • Not exactly, there was a bit of confusion. That comment was about my former question which wasn't properly formulated. I actually was modifying the previous question but since it had changed considerably I thought it was better to delete it and start a new one. – Charlie Aug 27 '18 at 18:36
  • For future reference, it's usually not better to delete it and start a new one. Having questions get closed/deleted both bring you closer to potential question bans where you're unable to ask new questions for an amount of time. – JMac Aug 27 '18 at 18:38
  • Thank you for telling me, I wasn't aware of that situation. I'll be more careful next time. – Charlie Aug 27 '18 at 18:40
  • This is just a more general version of https://physics.stackexchange.com/questions/424951/degenerate-parametric-amplifier-quadratures – ZeroTheHero Aug 27 '18 at 18:48
  • From the way I reformulated the question, you're right (they actually belong to two different exercises I'm working on, so it wasn't intentional). Is it possible to fuse questions? – Charlie Aug 27 '18 at 19:02

1 Answers1

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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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  • I wanted to make sure if that was a correct way of getting the expression, as someone told I could also apply the Heisenber equation. But if it's correct I'll continue under that route. Thank you. – Charlie Aug 27 '18 at 19:01
  • @Charlie sure you can use H's equations of motion but why start with the complicated stuff first? – ZeroTheHero Aug 27 '18 at 19:17
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    @Charlie although not the same problem this https://physics.stackexchange.com/a/353893/36194 might give you a template. – ZeroTheHero Aug 27 '18 at 19:31
  • Thank you. I actually tried with the definitions but I still can't seem to arrive at the solution stated in the problem. Particularly, I used the Heisenberg equation since it was suggested by the problem. There's a very similar problem in the number 5 of this link: https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/assignments/MIT8_05F13_ps7.pdf – Charlie Aug 27 '18 at 22:09