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So I came across Heisenberg's uncertainty principle in my 11th grade book, and I'm not convinced. It says $\Delta x \Delta p \gt \frac{h}{4\pi}$. Okay, but how does velocity even matter to determine the path?

Now suppose we have, for the case of my argument, 100 exactly identical atoms of hydrogen. Now we find the positions of the lone electron in 100 different cases, and we can do it with 100% accuracy. Now, if we join the dots, won't we get the path of the electron? Why not? Please explain in simple words if you can.

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TheLostGuardian0
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    You're trying to apply your macroscopic intuition to a microscopic event. In your proposal, why do you think you know the path between two measurements? In fact, you have no idea what the momentum at each of those points is at all. And more importantly, you cannot determine position with 100% accuracy. That's the point. – Zhe Oct 27 '17 at 16:34
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    You don't need to be convinced. Nature works this way regardless of whether you're convinced or not. – Zhe Oct 27 '17 at 16:34
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    Colleagues, please refrain from close-voting and downvoting. Neither of us was born with the full knowledge of the subject. The question is completely legitimate and shows some genuine (if mistaken) work of the mind; if this is not welcome here, then I don't know what is. – Ivan Neretin Oct 27 '17 at 17:39
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    I was just responding to your phrasing. It seems very arrogant to say that you're "not convinced" of 100 years of physics built by arguably some of the smartest humans who have ever walked the earth and survived every single attempt to prove it wrong. – Zhe Oct 27 '17 at 18:58
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    My bad, I wasn't convinced as I thought from experience that probably 2 years from now in college they'll tell me that it was something dumbed down so that we could understand, like dual nature for example. – TheLostGuardian0 Oct 27 '17 at 19:02
  • No worries. Hopefully, some of the comments and answers have helped you accept more readily that this is a fundamental property of matter. – Zhe Oct 27 '17 at 21:13
  • Possible duplicate of https://chemistry.stackexchange.com/questions/14848/heisenbergs-uncertainty-principle – Mithoron Oct 27 '17 at 21:35
  • You will be closer to being convinced after you learn some Fourier analysis. – Rodrigo de Azevedo Nov 01 '17 at 04:10

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Some problems with your method: you mention taking 100 copies of the same atom of hydrogen and finding the positions at 100 different times. (i) You actually can't make a copy of a quantum system without knowing its quantum state (the no-cloning theorem), (ii) your measurements of position will disturb the "trajectory" of the electron, so your 100 cases are distinct and cannot represent the path of a single electron, and this is especially true the more accurate you make your measurements.

You should realize, however, that the bound placed by the Heisenberg uncertainty principle is really quite small, and on a macroscopic scale we can essentially determine the trajectory of an electron. A bubble chamber or cloud chamber is essentially a major revision of your proposed method: we fill a vessel with supersaturated vapor, wait for charged ions to pass through the vapor and cause it to condense, and observe the resulting tracks. This describes the ion's motion with a resolution on the order of microns.

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    No-cloning theorem is an overkill. Let's put it another way: just how are you going to make sure that these 100 atoms are 100% identical? – Ivan Neretin Oct 27 '17 at 17:44
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    Okay, so let's say they aren't 100%identical. Now by using our electron microscope we know where the electron "has been" but we can't predict its future because of the energy of the photon. Now even if the hydrogen atoms aren't 100% identical wouldn't my method directly(if the 100pictures are overlapped) give us a picture that looks exactly like the probability density graph for 1s? – TheLostGuardian0 Oct 27 '17 at 17:59
  • @TheLostGuardian0, do note that you'll need to tag Ivan (as I've tagged you here) for him to get a notification about your comment. With regards to your question, you wouldn't get exact agreement, but it will be a better and better approximation as you increase the number of measurements. – a-cyclohexane-molecule Oct 27 '17 at 18:05
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    @a-cyclohexane-molecule I appreciate all your help, but I'm still not clear as to why we need momentum? I mean isn't position enough if we have large samples? Momentum is only required if we have 1 sample right? – TheLostGuardian0 Oct 27 '17 at 18:10
  • @TheLostGuardian0, I'm not sure what you mean by needing momentum. You don't need to measure (or care about) momentum at all if you're only interested in position; the Heisenberg principle just says that you can't measure both simultaneously accurately. – a-cyclohexane-molecule Oct 27 '17 at 18:14
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    @a-cyclohexane-molecule my bad I should have worded my question better. If we didn't need momentum to find the trajectory, then why go for quantum mechanical model instead of sticking to Bohr-Sommerfield model? (We are able to determine the path of the electrons by using a microscope so this is similar to Bohr's model) – TheLostGuardian0 Oct 27 '17 at 18:22
  • There is no microscope that would determine the path of the electrons. Also, Bohr-Sommerfield model fails when it comes to 2 or more electrons. So it goes. – Ivan Neretin Oct 27 '17 at 20:00
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    Here's a good question: if you have something at point A and something at point B at a later time, what's the path between these two points? Turns out the more certainty you have about those two positions, the less you can tell about which path was taken. – Zhe Oct 27 '17 at 21:23
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    @Zhe But what if we have millions of such points? Wouldn't that solve the problem, as the path would be revealed by the various possible places it was in? – TheLostGuardian0 Oct 28 '17 at 14:15
  • @TheLostGuardian0, how will you obtain these points? Each measurement will affect the future path of the electron, so your points will not be an accurate reflection of the electron's original trajectory. – a-cyclohexane-molecule Oct 28 '17 at 17:37
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    It doesn't matter how close these points are. You are trying to use your macroscopic intuition of interpolation. That doesn't work at the quantum level. You cannot tell what path was taken between these two points. – Zhe Oct 28 '17 at 21:27