How can uncertainty be represented as a chance when we write in Heisenberg's principle? Why do we use Delta for uncertainty and mathematically use it the same way as if it was a change in position or momentum?


closed as unclear what you're asking by Emilio Pisanty, Thomas Fritsch, Jon Custer, John Rennie, Kyle Kanos Aug 27 at 11:56

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    $\begingroup$ How can uncertainty be represented as a chance... should that "chance" be "change"? $\endgroup$ – Kyle Kanos Aug 26 at 11:57
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    $\begingroup$ "Why do we use Delta for uncertainty and mathematically use it the same way as if it was a change in position or momentum?" ─ we don't. What gave you that impression? $\endgroup$ – Emilio Pisanty Aug 26 at 12:02

When analyzing measurements of a continuous quantity like position or momentum, the only complete description of the measurement comes from specifying the full probability density function $p(x)$ for measuring a particle at any position $x$. It's called the probability density because it is used as follows: the probability $P(x_1\leq x\leq x_2)$ of measuring an object to be within the interval $[x_1,x_2]$ is given by

$$P(x_1\leq x\leq x_2)=\int_{x_1}^{x_2}p(x)\;dx$$

In this way, it behaves much like any other density function, like the mass density - to find the mass contained in a given volume, you integrate the mass density over that volume.

This probability density function contains quite a bit of information, just like any other PDF in statistics. But, just like in statistics, in many cases we only need a small portion of the information contained in the full distribution, ideally expressed as a single number, to compare distributions that may be otherwise nontrivial to compare. In this case we want a number that describes roughly how wide a particular distribution is; the wider the distribution, the less certain we are about a measurement's value. This number is what we call the "uncertainty" in a particular measurement, and it is defined as follows, for example for a position distribution:

$$\Delta x=\sqrt{\langle x^2\rangle-\langle x\rangle ^2}$$


$$\langle Q\rangle = \int Qp(x)\; dx$$

for some quantity $Q$. As you can see, the uncertainty $\Delta x$ "throws away" a lot of the information about the distribution by averaging over it, but it turns out to still be useful as a general description of how wide a probability distribution is.

The notation $\Delta x$ also makes sense: it's a measure of roughly how much you can expect the value of $x$ to change between different measurements.

  • $\begingroup$ I am not sure about your final statement. Are you talking about repeatedly measuring the same system? If so, then how can you guarantee that $\Delta x$ is the same before the first measurement and then after subsequent measurements. If you are talking about measuring different systems that start in the same state, then $\Delta x$ would just be what you describe in your answer: a standard deviation of the measurement. I am not sure you could say you would expect one measurement to be $\Delta x$ away from the next one. $\endgroup$ – Aaron Stevens Aug 26 at 13:44
  • $\begingroup$ @AaronStevens I meant to basically say "a standard deviation of the measurement" without saying the words "standard deviation". And I don't necessarily mean adjacent measurements, either; the meaning is closer to "the range over which the measured value of $x$ is expected to differ/vary/change when you measure many copies of the same system." $\endgroup$ – probably_someone Aug 26 at 14:29

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