The Heisenberg uncertainty principle || Deep dive

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📚 The Heisenberg uncertainty principle is one of those ideas from quantum mechanics that have made it all the way to general audiences. Unfortunately, the principle is very often misunderstood and misrepresented. In this video we explain what the uncertainty principle really tells us about quantum systems. For the mathematical proof of the principle, check out the companion video below.
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⏮️ BACKGROUND
Measurements || Concepts: • Measurements in quantu...
Measurements || Maths: • Measurements in quantu...
Expectation values: • Expectation values in ...
⏭️ WHAT NEXT?
The Heisenberg uncertainty principle || Proof: • The Heisenberg uncerta...
Time-energy uncertainty principle: • The time-energy uncert...
Minimum uncertainty states: • Minimum uncertainty st...
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Director and writer: BM
Producer and designer: MC

[@syedrumman3920]

Your videos should be made standard for all undergrad quantum classes!!

[@ProfessorMdoesScience]

Thanks for your support! :)

[@N1colastoledo]

How lucky we are to have been born in these times where we can see works of art like this video series.

[@ProfessorMdoesScience]

Glad you like the videos! :)

[@narfwhals7843]

The name "uncertainty principle" is unfortunate, indeed. In German it is called "Unschärfe relation" "schärfe" meaning "sharpness". And it is a relation about the sharpness of the distribution peaks. Though, it is worth noting that in his original work Heisenberg did talk about the act of measurement itself and that measuring the position precisely would destroy information about the momentum and vice versa. So we would be more "uncertain" about the other outcome. Though he called it more something along the lines of "imprecise". I think Feynman also introduces it like this in the lectures. Another misconception about the principle I often see(or is this my misconception?) is "The better you know one variable, the less you know about the other". This is only true when their product is already in the range of ihbar. You can constrain position _and_ momentum as much as you want as long as the combination remains larger than this. And since this is a greater or equal to, a broad distribution in one does not enforce a narrow one in the other. You are also free to be completely ignorant of both.

[@ProfessorMdoesScience]

Thanks for your comments! :)

[@TheWingEmpire]

I really needed this generalized explanation. Thank You very much

[@ProfessorMdoesScience]

Glad you found this helpful!

[@kabeerkumar4334]

Outstanding videos for QM!! Unparalleled teaching style!! Really incredible! 💯🎉 Thanks a lot

[@ProfessorMdoesScience]

Thanks for your kind words :)

[@zeio-nara]

Your videos are very useful, thank you a lot. Will certainly recommend to watch it anyone if somebody asks about helpful materials on quantum mechanics) It finally dawned on me that an observable is described as a set of eigenstates and can be interpreted just as a mathematical operation thus we can stack multiple operators to use different basises by transforming our quantum state hence composing required observables.

[@ProfessorMdoesScience]

Thanks for watching, and glad you find the videos useful! :)

[@BruinChang]

Indeed. When I studied time frequency analysis, I saw Gabor uncertainty principle, deriving from H. uncertainty principle!

[@ProfessorMdoesScience]

These ideas are intimately related with Fourier analysis :)

[@garvitmakkar]

lots of love from India 🇮🇳

[@ProfessorMdoesScience]

Glad you like the videos!

[@syedrumman3920]

I hope you can do videos on statistical mechanics too someday! Best of luck!!

[@ProfessorMdoesScience]

We certainly plan to expand to other topics including statistical mechanics once we are done with quantum!

[@mortavor227]

Great video, thank you!

[@ProfessorMdoesScience]

Glad you like it!

[@anibalbezerra]

Again a very nice video. Thanks!

[@ProfessorMdoesScience]

Thanks for your support! :)

[@fabio_air4230]

Thanks!

[@ProfessorMdoesScience]

Wow, thanks!

[@SoloBSD]

What about using the word Undetermined???

[@ProfessorMdoesScience]

Thanks for the suggestion! Unfortunately I think it would be very difficult to change the language that most people have been using for decades...

[@tomgraupner171]

I have a knot in my brain, Professor. Please help me out. I prepare a quantum system: let's say for 1000 times with an electron and that shall have "spin up". Now I measure 500 times with sigma-z (getting 500 times "spin up") and 500 times with sigma-x getting a 50-50 distribution. The standard deviation for sigma-z is "0", right?! Thus the "left side" of Heisenbergs Uncertainty Principle multiplies to "0" as well - independent of sigma-x. And the sigmas do not commute. But "0" is not larger than that. Does that mean that it is impossible to prepare a system in this eigenstate, before I can measure it? At 14:30 in your video you are close to this situation, but you say "very close to 1" and not "prob is 1".

[@ProfessorMdoesScience]

The uncertainty principle states: DeltaA*DeltaB>=0.5*||. If A=sigma_z and B=sigma_x, then note that [A,B]=ihbar*sigma_y. This means that the right hand side of the uncertainty principle is proportional to the expectation value of sigma_y in the spin up state (along the z direction), which is actually zero. Therefore, the right hand side of the uncertainty principle in this case is equal to zero, and your discussion is consistent with the uncertainty principle. I hope this helps!

[@tomgraupner171]

@@ProfessorMdoesScience HAH! You cut the (Gordian) knot in my head. Yep, that makes sense. Thank you for the quick reply and fruitful explanation.

[@nastyavicodin6229]

Best videos!

[@ProfessorMdoesScience]

Glad you like them!

[@jerosq1186]

In minute 16:07, how do you know the expectation value of the commutator i • h bar? Don't you need a state |psi> to calculate the expectation value as such: ? I'm having trouble with that part of the principle. I've seen some examples where they calculate the 'uncertainty' for SpinZ and SpinX and I don't understand how they calculate the expectation value without a particular state.

[@jerosq1186]

Nevermind, just answered myself. Since i x hbar is a constant, and the expected value of a constant is the same constant, then you get i x hbar again. Taking the absolute value yields h bar.

[@ProfessorMdoesScience]

Thanks for watching! You are absolutely correct on both accounts: we do in general take the expectation value with respect to a specific state, but in this instance we are taking the expectation value of a constant, so it is the same for any state.

[@lowerlowerhk]

I can't find the video where you prove the cononical commutation relation between position and momentum operator. I have skimmed through the entire postsulation series of your videos and the video here is the first time the relation is mentioned. Would appreciate if you can give me a link to the video.

[@ProfessorMdoesScience]

We don't have such a video actually. This is a tricky point, as the canonical commutation relation is typically taken as an axiom on which to build the rest of the theory, and this is what we've done for example in the videos on translation operators and the relation between position and momentum representations (wave functions) of states. But these concepts are equivalent, so one could instead take the translation operator as an axiom and then derive the canonical commutation relation from it, or take the relation between position and momentum representations as the axiom and derive the other two. But one of these concepts must be taken as an axiom to develop the theory. We will try to create a video making this more concrete, but I hope this helps as a starting point!

[@jupironnie1]

Hi. Greeting of 2023. I am stuck with this Principle over the Xmas break. The Uncertainty Principle is derived either via a) diffraction of a single slit e.g. kzbin.info/www/bejne/f6jKYWuDi7uqotE from 22 minutes onwards or via b) an elegant mathematical expansion of Cauchy Schwartz inequality. In both cases, I do not see how the simultaneous time measurement interpretation is introduced ? I do not see the time variable introduced in the derivations too. Is there any physics involved behind how Plancks constant as a lower limit is explained i.e the product of two standard deviations gives a physical constant with an origin in black body radiation ?. Moreover, the standard deviation values might not be an actual eigenvalue as it is a statistical measure. Of course, the constant comes out nicely in the derivations. Thx

[@ProfessorMdoesScience]

Could you please expand on what you mean by simultaneous time measurement? In the video we don't really discuss time at all. We do have another video describing what is typically called the "time-energy uncertainty principle", although its nature is somewhat different to that of other uncertainty principles. You can check it out here: kzbin.info/www/bejne/l6avi2WNhLCcp6c

[@jupironnie1]

@@ProfessorMdoesScience Heisenberg Principle is often explained in textbooks etc as we are unable to measure the product of momentum and position simultaneously with accuracy higher than lower bound of Planck constant . I assume this means both measurements are made at the same time of a single experiment. This exist a time variable in this interpretation. If one accepts Heisenberg Principle to mean that the variance of an ensemble of position and momentum measurements taken at the same time i.e of many different experiments measuring either position or momentum of the same initial state conducted at the same time, does the just shows a quirk of Nature ? There is no time variable in this interpretation. I find it very odd that the lower limit is Planck constant for a product of position and momentum. I agree the maths of conjugate pairs leads to this result. Maybe the lower limit has it source in one of the assumptions used in the maths e.g. psi is zero at infinity, square integerable / summable functions Note: I hate the maths on this subject. In any case, classical Physics measure velocity as the difference in position, during a small time interval. This implies we cannot measure both velocity and position at the same time. So what is the big deal about Heisenberg principle ? ( maybe related to the collapse to one of the states during measurement and the variance to imply the existence of superposition of various states of a wave state psi ? ) Many thx.

[@ProfessorMdoesScience]

@@jupironnie1 If I understand you correctly, you are considering two observables that don't commute, which means that they cannot share a common set of eigenstates. As after a measurement the state becomes an eigenstate of the observable we are measuring, then if we consider a second non-commuting observable, if we measure this second observable the starting point will not be an eigenstate of this second observable, and therefore the measurement outcome is not fixed. In this context, the best way to think about this is to consider compatible observables, which we do in this (rather mathematical) video: kzbin.info/www/bejne/f5mtp4tqfZyroaM The relation with the Heisenberg uncertainty principle is that, when two observables do not commute, then the lower bound of the product of mean square deviations is larger than zero. From this point of view, the Heisenberg uncertainty principle is telling us something about whether observables commute or don't, and then in turn we can relate this to the discussion at the beginning. Overall, I would say that the Heisenberg uncertainty principle is very often misunderstood and explained very badly. We have tried to be very precise in our discussion to avoid misunderstandings. From this point of view, I may suggest you take a "fresh" approach to the principle from our video without using any of your previous knowledge, and try and see if that makes sense. I hope this helps!

[@jupironnie1]

@@ProfessorMdoesScience A fantastic response !!! You guys do indeed have an oversight. Thanks You for removing 4 weeks of mental gymnastics with your simple but correct explanation. Bravo...once again

[@amaljeevk3950]

❤

[@schmetterling4477]

One should greatly stress that the uncertainty principle has absolutely nothing to do with quantum mechanics. It applies everywhere where linear operators come into play, e.g. in Fourier transformations. Classical waves are just as much subject to these limitations as quantum mechanical wave functions. This is a constant problem in engineering applications involving the time vs. frequency domain representations of audio signals, for instance. It is also being put to use for the spectral filtering of wireless communication signals.

[@ProfessorMdoesScience]

The fact that an analogous feature appears in other contexts does not mean that this has "nothing to do with quantum mechanics". On the contrary, it is a feature of quantum mechanics (as well as other contexts).

[@schmetterling4477]

@@ProfessorMdoesScience If I can replicate it on water waves (and I can), then it's not a quantum mechanical feature. It's much more general than that. There are things that one can not replicate classically (like the quantum mechanical phase space) but the uncertainty principle is not one of these.

[@ProfessorMdoesScience]

I am not sure what you are responding to: I have not claimed this to be an exclusively quantum effect. I have simply pointed out this is an effect that occurs in quantum mechanics.

[@schmetterling4477]

@@ProfessorMdoesScience My point is simply that many students (and laymen) who are hearing about the uncertainty principle in the quantum mechanical context first are getting the false impression that it is a quantum mechanical effect. That is simply not the case and I find it important to point this out as part of the wider education of students and laymen. This does not impact the role of the uncertainty principle at all, but it does take some of the unnecessary woo out.

[@ProfessorMdoesScience]

@@schmetterling4477 Fair enough! I generally agree that some aspects of quantum mechanics are overhyped in non-useful ways...