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So $\Delta x$ and $\Delta p$ are uncertainties in quantum mechanics. What exactly does the term "uncertainties" mean?

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It means that if you were to try to take a measurement the "uncertainty" would characterize how large the standard deviation would be (for that measurement).

A large uncertainty in x means that it's position is uncertain, so a large standard deviation if you were to make a measurement. (That is, if you were to repeat the same scenario over and over and collect data - you could end up with with an estimate of the probability distribution and the standard deviation is the uncertainty)

This is often explained that if you tried to make a measurement of something position you have to "hit it" with something and distort its momentum. This isn't exactly a correct interpetation as it's really the wave-particle duality (particles are waves of probability) that is what is causing this.

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  • $\begingroup$ Thanks very much $\endgroup$ – Kong Kuyjeu Jan 10 at 8:59
  • $\begingroup$ You're welcome. Also (since you're new I'm mentioning this) you should upvote and accept the answer if this is satisfactory. $\endgroup$ – Steven Sagona Jan 10 at 9:01
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If you look at any macroscopic object such as books, houses etc. (pretty much everything your eyes can see), they have an exact position and momentum. They same is not true for electrons, photons and even atoms and molecules (for more information, see Why doesn't the nucleus have “nucleus-probability cloud”?).

So uncertainty in position and momentum simply means that very small particles do not have a well-defined position and momentum.

Let's take an electron as an example and assume it is bound to an atomic nucleus. In the quantum mechanic model of the atom, electrons are represented by probability clouds (atomic orbitals) instead of small particles with one specific position:

enter image description here
(Source)

These probability clouds are mathematically described by a wavefunction and represent the probability of "finding" the electron in the shown position when measuring the same (which "forces" the electron to take one specific position).

Hope this helps.

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