The Generalized Uncertainty Principle | Proof/Derivation

[@FacultyofKhan]

Alright everyone; if you've been following along this playlist and made it this far, congratulations! You have officially completed the prelude to Quantum Mechanics series, where I teach the mathematical formalism behind QM. Hop on over to my Quantum Mechanics playlist to get started on learning the actual Physics behind QM! kzbin.info/www/bejne/oYbQZaRmiq6jn9E

[@RodriguesNatercia]

Came here for a quick refresh on Dirac notation, ended up staying for the whole playlist this is very useful! Fast pace, but short videos full of important and relevant information presented in a very logical sequence. Thank you so much!

[@jacobvandijk6525]

Who does not like a quickie sometimes? ;-)

[@akarshchaturvedi2803]

Why on earth this channel isn't famous yet?

[@FacultyofKhan]

I consider >4900 subs to be relatively famous :P, but if you want to contribute to my fame, I encourage you to share this with your friends!

[@akarshchaturvedi2803]

Haha, still you deserve way better bro.

[@jacobvandijk6525]

@@akarshchaturvedi2803 Think about the IQ-distribution of KZbin-viewers ;-) Of those who qualify to understand this stuff only a fraction is interested in Physics. That should explain a lot.

[@georgefarah9214]

I genuinely think this channel offer a very valuable content, it offers a lot of well explained videos for an undergrad physics student or anyhow who is just interested in learning physics or math on a higher level . Not too many channels offer that, keep it up man !

[@FacultyofKhan]

Thank you!

[@kipropcollins4220]

I found my gem in a corner of the internet... thanks so much!!!

[@AhmedAli-jy2zv]

I love this, thank you kind sir

[@dustbringer1821]

This playlist was a wonderfully compact refresher. Excellent presentation, thank you.

[@izzyizzy4205]

Was so confused when I first saw this proof 2 years ago...finally it makes a bit more sense hahah. Looking forward to the next part of the series!

[@FacultyofKhan]

Glad you liked it!

[@izzyizzy4205]

Faculty of Khan I’m curious by the way...would you ever consider doing a lecture series on QFT? I’m self-teaching it right now and would enjoy a resource like yours to help! No pressure lol

[@FacultyofKhan]

I'm considering it: others have requested it so it's on my to-do list, so I may get to it eventually!

[@izzyizzy4205]

Faculty of Khan Awesome, thanks!

[@clintonstevens8901]

Thank you for this *amazingly concise* and *incredibly helpful* KZbin series.

[@cerendemir9977]

I studied this from two textbooks and they definitely did not explain it this well. Thank you so much for this!

[@account1307]

This was the most amazing video ever hahaha this is very accessible to undergrads thank you dear teacher:3

[@FacultyofKhan]

Glad you liked it!

[@noism5429]

this channel is so underrated! good work!!

[@brightlin777]

At 3:58, the commutator symbols are confusing, because they should be parentheses if I'm not mistaken (also, at 4:06)

[@douglasstrother6584]

"Quantum Mechanics: Foundations and Applications" by Arno Bohm This is a graduate-level text (or advanced undergrad text), so don't start here; however, the writing style is very clear and crisp. The author approaches Quantum Theory as an algebraic structure built to explain measured phenomena, like Werner Heisenberg. It treats states as vectors and observables as operators on those vectors.

[@schmetterling4477]

Yes, and if you learn the formalism without studying physical phenomenology in parallel (i.e. at least atomic and molecular physics, better still some nuclear and high energy physics), then you will understand absolutely nothing about physics and you will eternally talk total bullshit about the Copenhagen interpretation.

[@RodriguesNatercia]

Looking forward to the rest of the series, by the way, do you have a day of the week you generally upload or do you simply do it when you have time? Will keep and eye out anyway!

[@FacultyofKhan]

Thank you for the support! I don't have a particular day when I upload during the semester; usually, I aim for about 2 videos in the middle of the month. During the winter break, I should have more time which I'll use towards making more videos.

[@MrGaugeBoson]

Excellent video, made my exercises very doable.

[@TifaniRizky]

If you don't yet I'd suggest making a series about linear algebra and mathematical physics. It's very useful, thank you.

[@humblehmathgeo]

Brilliant video series!! Thank you sooo much!!

[@scraps7624]

This is the first video I've seen of you, this channel is amazing. Do you have a video on the derivation using Fourier transforms?

[@unknow9339]

beautiful demonstration.

[@hisashihonda9300]

beautiful demonstration

[@preetamkumar5445]

very Informative video. looking for next playlist

[@samirelzein1978]

Keep it up! Amazing job!

[@Alphabetagama434]

Very nice explanation ..

[@tayyabahaider3872]

Thank you so much ..was a great help

[@richardshane456]

1:00:00 If we are in a trajectory spin, you can not forecast with any certainty the future Our trajectory spin system that never allows the observer to exist as a communicative function but as a conjugate system hence the trajectory spin observation as conjugate symmetry

[@avarice9686]

Thank you for this series

[@Dubickimus]

I wonder if you plan on doing manifolds

[@FacultyofKhan]

Perhaps later. I need to introduce some preliminary concepts first. Thank you for the suggestion!

[@romanfanta3990]

Hello, perfect series. I just want to ask which books/sources did you use for this series and if you can give some recommendations on what to read. Most of the series reminds me of Quantum Mechanics: Concepts and applications by Nouredine Zettili.

[@Gamma_Digamma]

Yes... Finally

[@Radheradheparbhu209]

Very helpful sir I am from India

[@valor36az]

Great derivation

[@violinsheetmusicblog]

If the expectation of A is a scalar, how can you subtract it from a Hermitian operator? Are you subtracting it from each element in the Hermitian operator?

[@FacultyofKhan]

You're technically allowed to subtract a real number from an operator: that doesn't invalidate anything or violate any rules. Recall that in quantum mechanics, the position operator x^hat is just x (a real number denoting the position), so a real number itself can serve as an operator - you don't need a specific 'operation' per se to have as your operator. Also bear in mind that even if delta A^hat corresponded to, say, the momentum operator, meaning that delta A^hat = hbar/i (d/dx) - , this doesn't break any rules even though the expression itself is kind of jarring (i.e. you're subtracting a derivative operator and a real number ). This is fine because delta A^hat isn't meant to be a stand-alone expression; it's an operator itself. So in most use cases, you'd have delta A^hat operating on a function (or you'd be taking its expectation value like we do here), which is perfectly fine since you can both differentiate a function and subtract the function multiplied by a scalar value and you wouldn't be breaking any rules. Hope that helps!

[@17LeeCheYu]

Great series! thank you.

[@ostensiblyquerulous]

Could you perhaps explain to me why when you square the expectation value of delta A hat delta B hat, you don't end up with cross terms? Is it because the cross terms would be purely imaginary and thus a norm squared would just be equal to the real parts of the square? This makes sense, but I'm struggling to see how you can take the norm of the expectation value of the commutator of A hat and B hat before the square and still end up with no cross terms. Edit: I realized that none of the above is necessary/makes sense. The expectation value of delta A hat delta B hat is just a complex number with imaginary component from the commutator and real component from the anti-commutator. Thus finding the norm square is just equal to multiplying the value by its complex conjugate, which will get rid of cross terms, and any negative quantities.

[@oivindification]

Hmm, how? How is 1/2 = 0? This is the cross term, right? Or am I missing something?

[@sagargour2024]

Becaue its mod squared.... like (x+iy)^2=x^2+y^2+2xyi; but lx+iyl^2=(x+iy)(x-iy)x^2+y^2 (the cross term terminates with mod squared) since is Imaginary (bcoz [A,B] is antiHermittian) [denoted by ''iy'' in upper example] and is Real (bcoz {A,B} is Hermittian) [denoted by ''x' in upper example].

[@thepleasantcatprincess]

Finally I understood everybit of the equations. Sir could you suggest me some good understandable books on quantum mechanics

[@romanfanta3990]

As introduction I can recommend these books: Quantum Mechanics : Concepts and Applications by Nouredine Zettili Introduction to Quantum Mechanics A Time-Dependent Perspective by David J. Tannor Introduction to Quantum Mechanics by David J. Griffiths For total begginer you can try: Quantum Mechanics: The Theoretical Minimum by Leonard Susskind and Art Friedman

[@thepleasantcatprincess]

@@romanfanta3990 finished all of these already😀....something a little deeper than these?

[@romanfanta3990]

​@@thepleasantcatprincess at this point it depends on what is your interest. Every advanced quantum mechanic books differ in some topics. In general, I would recommend Advanced Quantum Mechanics by J. J. Sakurai (maybe first try to go through his Modern Quantum Mechanics book to be accustomed to his writing/explaining). If you want to dig deep then there are 3 volume book Quantum Mechanics by Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë.

[@brightlin777]

How do you subtract a real number from an operator???

[@FacultyofKhan]

Ok nevermind I see it: you're referring to the definitions of delta A^hat and delta B^hat. You're technically allowed to subtract a real number from an operator: that doesn't invalidate anything or violate any rules. Recall that in quantum mechanics, the position operator x^hat is just x (a real number denoting the position), so a real number itself can serve as an operator - you don't need a specific 'operation' per se to have as your operator. Also bear in mind that even if delta A^hat corresponded to, say, the momentum operator, meaning that delta A^hat = hbar/i (d/dx) - , this doesn't break any rules even though the expression itself is kind of jarring (i.e. you're subtracting a derivative operator and a real number ). This is fine because delta A^hat isn't meant to be a stand-alone expression; it's an operator itself. So in most use cases, you'd have delta A^hat operating on a function (or you'd be taking its expectation value like we do here), which is perfectly fine since you can both differentiate a function and subtract the function multiplied by a scalar value and you wouldn't be breaking any rules. Hope that helps!

[@brightlin777]

@@FacultyofKhan Omg, I see. Thank you very much.

[@زينبناظم-ر1و]

Thanks very much

[@gg-bn4gv]

great stuff

[@oivindification]

4:05 How is it obvious that Delta A * Delta B = 1/2 [A,B] + 1/2 {A,B}?

[@__-op4qm]

dAdB = [dA,dB]/2 + {dA,dB}/2 == (dAdB-dBdA+dAdB+dBdA)/2, but before this he showed that the first part has [dA,dB] = [A,B].

[@Radheradheparbhu209]

One doubt in this derivation sir

[@shivangi3030]

thanks

[@schmetterling4477]

It is very cool that you derive a general mathematical theorem about linear operators, but you didn't teach the student any physics here. I would actually suggest that you are misleading the student into believing that you were teaching quantum mechanics here when the exact same formulas apply to classical wave systems and are being used in signal processing applications in audio, video and RF systems, basically everywhere where we are using Fourier transformations and their generalizations.

[@rifeayy1558]

They didn't teach any physics because well, this is a series on mathematical basis of quantum mechanics. Just a brief introduction to linear algebra required to do any QM at all. They do a good job at teaching "physics" in the other series.

[@paulg444]

bra ket notation... why ? why? ... just use vectors and conj. transpose. .. a standard inner product space.

[@schmetterling4477]

That's simply how physicists are used to writing these things. As a mathematician you can ignore the notation, you already know what we mean and that there is absolutely nothing novel here.

[@youngwoongcho]

Is this guy 3B1B

[@FacultyofKhan]

No.

[@youngwoongcho]

Lol ma bad. Well explained tho

[@wdobni]

that was completely unintelligible but its interesting to follow along behind somebody to whom these hieroglyphs seems to convey sensible rational reasonable logical provable methodology ...... it would require 5 years of university study for me to become convinced that anything about this video corresponded to anything in the outside world, and even then i would still wonder if i had seen some revealed truth or simply become brainwashed ..... i tend to see this material as an entirely imaginary edifice inside the human mind rather than an obvious self-revelatory expression of the actual nature of external reality as revealed by the senses and puzzled out by inquisitive geniuses.......in other words quantum mechanics is to the real world as Pablo Picasso's Nude Descending A Staircase is to real women

[@FacultyofKhan]

Sir...this is a Wendy's.

[@richardshane456]

I think the 1st question ought a be what forces are acting on us to help permeate this imagination called mathematics to explain our reality using matrix vectors in hermiston vectors I beg to differ with mathematics not as a functioning language of displacement of a thing called real numbers in your own mathematical construct your own mathematical preponderance and mathematical construct States that mathematics is lower ranking system than imagination. it's a much lower ranking system than reality, therefore, how can your mathematics ever ever be able to explain anything about this reality other than creating subset sub standardized technology that we observe now that occupy in R space, this thing called a reality, created from a subset of imagination called mathematics that's propelled by a hierarchy of imagination and reality? Back to the quantum mechanics observation of reality as far as from our human perspective Here is my observation after observing and comparing to your mathematical observations What your quantum mechanics physics is describing is a kinetic energy system that manifests itself as a trajectory with spin, I'm not talking about electron orbitals of our kinetic energy system, however as a trajectory with spin as a hierarchy of trajectory with spins of all matter mass I even suspect a natural explanation of our Trajectory Spin energy system as a simple entropy scalar back to conjugate symmetry by particle attraction of spiraling mass called spiraling galaxies which then emit leading and trailing singularities perpendicular of the galactic plane creating trajectories spins weaving a fabric of particles creating branes theorized in string theory My thought imagination describes a big Bang with singularities as its beginning or observed from our theorized beginning as a big bang and does not violate any of our known laws of physics, and it's observed all over our universe as leading and trailing singularities

[@Ihsan_khan00]

thanks