Position and Momentum Operators in Quantum Mechanics

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We've learned a bit about quantum mechanics from a strictly conceptual and qualitative standpoint. But now it's time to dig a little deeper. Quantum mechanics is mathematics, so if we want to understand it on a fundamental level, we have to dig into the math. Unfortunately, the math is rather complex, so a solid understanding of linear algebra and differential equations is assumed here. But luckily, I have covered all of these topics in my mathematics playlist, so if you are committed to understanding quantum mechanics but don't have the math background, just head over there first to get the math down, and then this stuff will make sense! In this video, we start out by learning about operators, what they are and how to use them, the position and momentum operators, and how to apply them to the wavefunction. Don't be scared, it's fun!
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1 авг 2024

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Комментарии : 208

@adamdrewko233 4 года назад

Wow, I'm just so glad you're doing more than just a conceptual course on quantum mechanics and as always your video's crystal clear and high quality. Thank you so much for what you do for all of us learners and I'm waiting for more.

@Yatukih_001 4 года назад

I have at least 5 books about quantum mechanics so I know what he´s talking about. It works in most computer game simulations of a universe so you can utilize it to create your own computer game.

@voidisyinyangvoidisyinyang885 Год назад

please tell me how to live my life.

@petersvideofile Год назад

@@voidisyinyangvoidisyinyang885 hayyyy drew :) I finally watched it. What was the important part you were directing me too ? It was something related to the amplitude being the square of two terms containing the imaginary components, right?

@voidisyinyangvoidisyinyang885 Год назад

@@petersvideofile sounds good.

@adahaholu9691 4 года назад

Yay, another video I won't understand but watch anyway.

@josesaldivar655 4 года назад

ada haholu Learn the "recipes" by heart and learn physics in a lab . true math is only the accounting system of physics.

@Jehannum2000 4 года назад

@@josesaldivar655 Nonsense.

@josesaldivar655 4 года назад

Jehannum Mindlessignorant

@gautamsah5075 5 месяцев назад

Agreed

@nono9555 2 года назад

Now I appreciate the 3 semesters of electrical engineering that taught me the mathematical language needed to be able to patially follow this at two in the morning instead of going to sleep so that I could be productive in the low skill job I have to attend at 8 in order to pay my bills. Your videos have the perfect pace and are super clear and on point. Well done.

@xonikkiecal 3 года назад

I am so happy you posted this. As a physics major there’s only so much conceptual videos will do for me. I can read the textbook multiple times but math needs to be applied (duh). Once you get to higher mathematics and higher level physics, professors don’t bother with showing examples anymore so we have to just figure it out like a physicist should do. So think god for these advanced mathematics videos for modern physics!!

@ivoryas1696 Месяц назад

@xonikkiecal Not even a physics major (ME), but... *_right?_*

@MegaXRadioMan 3 года назад

Amazing work! In my years working in physics, I've never encountered such a clear explanation of quantum mechanics with this level of mathematical rigor. This is an excellent stepping stone for a curious general audience.

@davidhigham1570 2 года назад

I learned more in one hour of taking notes, pausing, and doing the math at my pace with this video than I have in 8 weeks of struggling to keep up in my university course. I may just pass my midterm at the end of this week. Thank you!

@CandidDate 3 года назад

The way he clearly and concisely massages the formulas really stimulates my math senses.

@KK-rg3nj 3 года назад

Ikr

@watchdatclip496 3 года назад

Thank you so much Professor Dave! I was searching for RU-vid videos that explained the mathematical side of quantum mechanics and your videos are honestly the only one I found, and the best too! The way you explain these are so articulate and perfect for anyone to grab the idea. Thank you again so much! Subscribed!

@cguy96 4 года назад

Btw, at about 12:30, where you discuss the partial time derivative of Psi, your explanation was perfect, but I have found that some confusion can be cleared up by first rewriting e^i(kx-wt) as e^ikx * e^-iwt. Then, it is quite easy to see that e^ikx is a constant, call it C, so you are taking the partial time derivative of C e^-iwt, which of course is -iw Ce^-iwt, which is just -iw Psi.

@LunaticBiker11 3 года назад

Never delete these videos, i'll be back next year

@KK-rg3nj 3 года назад

I'm so glad I found this video. This is such an amazingly exhaustive work explaining operators. My physics class is running ahead of my math class so I have to watch youtube to bridge the gap. Thank you soo much for this video!!!!

@Yatukih_001 4 года назад

A truly enjoyable video to watch. Professor Dave feels at home when describing the concept of anything highly complex with regard to math and physics . He may find it a bit funny, but math fascinates me as well. Every day I read chapters from an over 1000 page thick Princeton companion to Mathematics, with the intent to just calm the mind and be more logical during the rest of the day. Momentum operators in quantum mechanics are indeed welcome and further research is required to decipher their remaining mysteries. Thanks for this!!

@matiassantacruz5487 Год назад

These videos are a godsend. I am a second year student in university, and even though our professor is pretty good, the flow of lectures and of most textbooks introduce far too many concepts at once do not explain the rationale for creating them. Thanks so much Professor Dave!

@rholin0997 3 года назад

I absolutely love how you introduce this material, and I agree, there really is no way to understand this material without the math.

@voidisyinyangvoidisyinyang885 Год назад

I wish he went into the de Broglie-Einstein Relation as Penrose does - realizing it is noncommutative nonlocality with mass originating from frequency.

@aleksanderkardyka6013 4 года назад

It just boggles my mind how scientifically sophisticated this man is! Kudos to you, Mr Farina! ^^

@thatomokoena4673 4 года назад

I'm no physics student but I love your videos and try to learn what I can. Thank you for your work Prof

@OwenMcKinley 4 года назад

Haven't even watched, and I already know this is going to rock. Thank you, Professor Dave.

@tassosmanganaris9150 2 года назад

Woah. I really enjoy your drumming covers. What a surprising connection between channels

@user-qx8zg4de6p 3 года назад

I really do love this math stuff. I'm telling you there are no videos talking about math of this depth

@MiltonRosso 3 года назад

I am so glad you're doing this, I have been wanting to learn about the math of quantum mechanics for a long time.

@josesaldivar655 4 года назад

Very cool , crystal clear. You know, while starting my grad studies, I wanted to film videos of each class as O had recorded audios on mini tape. I was dreaming on a computer based system. The internet days were at using menus or **** and a year later Netscape appeared. Now it is so good to have these accurate well pronounced lessons on demand , thanks to professors like Dave. Many thanks , and It should be in each And every lesson.

@mccawpa 4 года назад

Didn't understand most of this (understood a little) but had my mind blown by the calculations needed at this quantum level.

@rafaelcrispin5519 3 года назад

You sir. Are national treasure. Your explanations where clear and well thought out and I deeply appreciate what you are doing. Thank you.

@BigNes445 11 месяцев назад

You’re literally the biggest life savor, Professor Dave. I hope everything good that can happen to a person happens to you man

@lolygagger5991 3 года назад

it was really cool seeing something i learned in pre-calc like multiplying the conjugate of complex numbers being used in advanced math courses like quantum mechanics. thanks!

@cliffordwilliams9597 3 года назад

Thank you so much for uploading this!!! I just finished calculus and am going into O-Chem and biochem; this will be a great way to keep up the maths skills !!!

@subrajeetmaharana 4 года назад

Probably best video on internet to date. Thanks professor.

@Sailesh004 2 года назад

Information overloaded... Everything is explained very clearly... Thank you so much...

@geraldlarson5498 3 года назад

This video has helped tremendously with homework, thank you!

@juliogodel 3 года назад

Thanks for going an extra mile on math explaining this! Well done!

@johnc4900 4 года назад

Prof Dave really do be dropping a better explanation of this then my university professor. Wish I had this video during modern last semester RIP. Anyhow thanks Professor Dave!

@sathirapromodya6498 9 месяцев назад

The commutator provides information about the uncertainty or lack of simultaneous measurability of the observables represented by A and B. If the commutator [A, B] is nonzero, it implies that A and B do not commute, and you cannot measure both observables with arbitrary precision simultaneously. This principle is known as the Heisenberg Uncertainty Principle, which is a foundational concept in quantum mechanics.

@Gregegeg 3 месяца назад

Maybe he explains this in another video, but how does A and B being communative effect observability? This is related to me not understanding why you are able to take the momentum of a position, and the position of a momentum. Let me know if you have any insight on this, any help is appreciated.

@parthverma6652 2 года назад

I actually finished the textbooks first to get a very vague idea of what quantum mechanics was... I was still figuring out the true sense and trying to fit it in my mind. But after this Lesson I feel like I have understood the basics clearly.!! Thanks.

@estebancanizales3303 3 года назад

that was so good and clear wow everything clicked when watching this video thank you so much

@justin5455 3 года назад

I totally agree with your ideas. Math is the key to understand QM in depth!

@tahoor4083 3 года назад

super awesome video, cleared all my doubts, loved it

@nikhil2453 Год назад

thank you so much for this video! You have no idea how helpful it is to me!

@nyluong5427 3 года назад

Luckily I find this before my Physical Chemistry Exam tmr. Thank you Prof. Dave!

@MrZzzjjj 3 года назад

this is so good. Thank you Professor Dave

@karatheythemharkins2393 4 года назад

Did my physics degree at UNH ... after the classical stuff my next course was modern physics ... taught by Dr Heisenberg's son ... it was hilarious when he started talking about the HUP as it was obvious he wanted to spend as little time as possible saying his dad was in the textbook and he was not achieving that

@Jehannum2000 4 года назад

Wow. You were taught by Heisenberg Jr?! Cool. It must have been weird to see a Heisenberg talking about the HUP!

@karatheythemharkins2393 4 года назад

Jehannum yes, it was odd, but I was used to Heisenberg’s grandson hanging around my dorm (he was a business major) ... one day he launched a rocket signed by his grandfather, got it caught in some power lines, then just went inside .... meanwhile myself and a chem e were wondering who would be the first to try and retrieve it

@KK-rg3nj 3 года назад

That is so cool!!

@yourmom-nv9ui 2 года назад

You are so good 💕💕💕💕 thank you so so much for these lectures and I will watch all the ads in this video without a skip

@manpreetrana4484 2 года назад

Really amazing work 👏 🙌 thanks alot for this conceptual explanation.

@jalalabdallah7879 3 года назад

very good to Explain , Prof. Dave

@unknownbeing8222 4 года назад

as a 14 year old watching this, i can’t thank you more professor davis.

@lakshmiamruthayechuri4005 4 года назад

Wow. Thats great to hear Young learner

@unknownbeing8222 4 года назад

Lakshmi Amrutha Yechuri always gotta educate yourself no matter what’s your age :)

@erikawimmer7908 3 года назад

@@unknownbeing8222 I am also 14 and I understand it at least a little bit

@leon_noel1687 3 года назад

Hey Dave, I will have Quantum Mechanics in University next Semester and I´m highly intersted, try to understand Operators for months, this video was extremly good for beginners. Thank you.

@Ccaste1967 2 года назад

very very good explanations, Thanks

@Jehannum2000 4 года назад

I've been studying this subject hard for more than a year and am just at the stage where I can follow at the level of this video. But somehow I hadn't realised that the commutator is itself an operator - so I've learned something. And for that I give a grateful 'like'.

@voidisyinyangvoidisyinyang885 Год назад

it's different whether it's Heisenberg or Schroedringer. Quantum PHysics Professor Basil J. Hiley uses noncommutative math - much better. NO need for a wavefunction.

@kimberlypena2898 4 года назад

How I wish I knew you during my college years. But thankyou for your videos.

@yourmom-nv9ui 2 года назад

Wow you don't know how much i needed this video 😢😢😢😢😢

@cguy96 4 года назад

Nice! I was good with QM, it was QC that started my morning drinking.

@nikitavasilev7829 11 месяцев назад

Thank you! Maybe it is more for a physics student rather then for the general public, but that is exactly what i am and the videos are perfect!

@olafurjons Год назад

Extremely helpful

@criskity 4 года назад

I'm trying to imagine a flat earther attempting to understand even the most basic parts of this.

@cguy96 4 года назад

CNVideos oh yeah, I am going to say that most flerfers suffered radical cranial expansion at 0:20

@Yatukih_001 4 года назад

Easy peasy. I just have to use Wikipedia or other dictionaries. But even that is not truly understanding this. If I talk with my friends and we find out how to create computer simulations out of this, then you can say I understood something, even if I believed the world to be a globe.

@cguy96 4 года назад

Yatukih001 you could simulate this, without any understanding. Why do I say this? Because Prof. Dave is just listing all the relevant equations (or deriving them). So you could certainly simulate these equations, using various numerical techniques, but it wouldn’t mean anything. You wouldn’t have a concept of the initial or boundary conditions, and you wouldn’t know how to interpret the results of said simulation (assuming there were any meaningful results, which is highly doubtful). My justification for these claims? 30 plus years of large-scale numerical simulations of the Navier-Stokes equations on basin to global scales, which is qualitatively not that different from QM systems, and benefit from some of the very methods used to study QM systems. This is not something that you can just Wiki to understand (ok, maybe there is somebody out there who could... maybe).

@kevinclydeeguia2543 2 года назад

Im gonna come back in six months since im gonna study first the chemistry, classical physics and mathematics in order for me to understand this section of the playlist

@moeenkhan4803 Год назад

brilliantly explain

@DAMN__________ 4 дня назад

This is such a good video

@shivamep-8757 Год назад

Best i ve found on RU-vid

@rocketpsyence Год назад

Yes this is great. The math is stuff I had in college and isn't hard to brush up on. I think this is JUST right for me and is what I've been looking for. Everything out there seems to be limited to just conceptual explanations OR just throws you in the deep end with no background for where any of the equations come from. But this is just the right balance for someone like me - there's math, but you don't assume we know every single thing about the equations going in. I wouldn't trust myself to sit down and give a lecture on any of it, but for someone who can vaguely conjure up their old diff eq and linear algebra concepts while being bored in a waiting room somewhere this is fine so far.

@geaninaalecu4567 4 года назад

Exceptional!

@zapp485 4 года назад

Underrated !

@fernandoportal5422 Год назад

motivation to keep on with calculus and linear algebra

@speedspeed121 3 года назад

I have to say, this video and the following video cover the first chapter in Griffiths fairly well.

@JohnJoss1 3 месяца назад

I've been reading Jim Baggott's wonderful book: "The Quantum Cookbook" and have got to the chapter on Paul Diracs derivation of the Wave Eqn with spin and Relativity. But I just couldn't find how out the momentum operator was derived anywhere. It'd been driving me nuts all day. You nailed it at around the 17 minute mark. Thank you kindly! Nope! I Still don’t understand this. I get p psi = h-bar k psi but I don’t understand where the h-bar comes from in partial by partial x of psi. I just get ik psi. So where does the h-bar come from in the partial derivative please Prof Dave?

@HWC_CHEMISTRY 2 года назад

Outstanding

@wakeawake2950 4 года назад

Thanks alot!

@HanBurritoz 4 года назад

18:16 Sel-adjoint and having real Eigenvalues are not the same. An operator can have real Eigenvalues, but still not be hermitian. For example the matrix A = (1 1; 0 1) has real Eigenvalues, but is not equal to A^T.

@nadiames8061 3 года назад

Merhaba, who wants that professor Dave makes videos about crystallography?!! Like

@priyadharshinivenkataraman5212 2 года назад

Hello prof dave .. I am studying 6th ... I have just found how to measure both momentum & position with a simple eqeation ..just shoot a photon on the sub atomic particle so find the speed of sub atomic particle + the photon's force on it... Then just subtract sub photons speed with sub atomic particle , then it will be easy to find position or momentum

@Boooommerang 2 года назад

Excellent job, wowww! One question: how much time do you need to prepare a video like this?

@minaemad2393 4 года назад

Marvelous

@QuicksilverSG 8 месяцев назад

Thanks for this nuts and bolts introduction to Hermitian operators, it lays bare the mathematical operations. It may sound hairsplitting, but in this context is it literally the case that "quantum particles are in several places at once"? While the application of the position operator on psi produces multiple solutions, each with a probability density of psi squared, in practice, only one of these solutions describes where the quantum particle is actually observed. Is it that describing a particle as being in several places at once is correct, but only on a statistical rather than physical level?

@user-pk9qo1gd6r 4 года назад

Will you talk about where these two operators come from? Especially the momentum operator which seems so random the way it is.

@alcoll1038 4 года назад

Look up Brant Carlson's lecture on this

@gallapettisingaram5792 2 года назад

Thanks

@yourmom-nv9ui 2 года назад

Guys don't skip ads on his videos if you are not supporting his patreon ..he is actually putting alot of efforts and I would kill to be his student 😓😓😓😓

@vikramnagarjuna3549 4 года назад

I love it...

@gnikkings Год назад

15:16 I would have liked for you to elaborate why you think all quantum objects can be expressed as exponential functions.

@marckhanna861 2 года назад

Hi I am 10 years old and I like your videos I understand them all and can even do sums

@moulibeu Год назад

Sure you do

@harrytruman5700 4 года назад

Hi Professor dave

@MS-cj8uw 3 года назад

Thank you prof ... its beautiful and amazing...... I have question.....What is the meaning of : the potential energy multiplied by psi in the Schrödinger equation ...what is the meaning of potential energy oscillation is there any meaning for that...thank you

@lcy6434 Год назад

As Barrier in prognosis is as difficult as Magellan’s voyage using Celestial Navigation before any civilization’s comprehension.

@rishavraj3099 2 года назад

can you please provide us with these slides or notes

@TJ-hs1qm 3 года назад

Why is Heisenberg's uncertainty principle expressed in terms of Δx and Δp instead of just Δx and Δv? Now, you have to measure mass and velocity instead of just velocity. Basically I'm trying to understand, why we are using the momentum operator instead of the velocity operator ( iow.. assuming m is constant, why deal with it?)

@aymenbk9138 3 года назад

in 16:47 is k the wave-vector or the wave-number?

@lasbutious116 4 года назад

I believe the wavefunction we get after applying the operators on the wavefunction help us find the probability distribution of the operator type Is it ?

@xonikkiecal 3 года назад

Yea so you can find the probability of an electron being between an interval by doing an integral and then you can find which is the expectation value.

@hassaanwaqas8896 2 года назад

I have a question. I fully understood the math until the end, I just don’t understand how the commutator of position and momentum calculated lead to the uncertainty principle

@zohrebashtani2990 2 года назад

perfect

@nicholspikos3136 2 года назад

Could it be true that something as simple as a single quantum particle has many or infinite degrees of freedom and once you have groups of particles clumped together into molecules you get many times less degrees of freedom and so on so that once you have regular sized objects...,....

@dolfanatic314 4 года назад

Damn I wish I knew you years ago. Math always clicked for me. Some kids in high school and I would get a different test. Even kind of taught my friend way back in 1981. Flat Earthers tell me I impossible because no one can live this long. I had all the numbers. And flushed them a few years ago. The symbols I know what they do, but cant get past that. Be well keep up the posting

@alexdred5750 2 года назад

Dude do you really know any good books for quantum or modern physics

@venkataramanadasari7465 4 года назад

Hi dave

@overlordprincekhan 2 года назад

BTW, is that "e^(ikx-Ωit) " derived from fourier series?

@kjl3080 2 года назад

I recognized it as the heat equation

@elivaughan1192 2 года назад

Haha. I'm that guy. I'm the one that saw the prelude of this and said "pssh. I dont need to go back and watch those others. My math strong.". 10 minutes later... uhhh

@hwetherell6250 4 года назад

Just for clarity, since it does not detract from what you're saying: around 7:20, you mean "linear operators", not operators in a general fashion. There are non-linear operators.

@benmcwhirter4566 2 года назад

You problem won't understand but may get something from it anyway. Proceed's to explain operators so. I better understand c++. Thanks

@freshely9309 3 года назад

Oh yes I am very much aboard the quantum train, but unfortunately I don't know exactly where on that train 🤔

@ansariraiyyan1879 Год назад

Sir, Uncertainty is not just position and momentum "You can't not measure two dynamic operators which are canonically conjugate to infinitesimal accuracy" There is uncertainty between energy and time also

@ProfessorDaveExplains Год назад

Yes, I know.

@garysamuel9521 Год назад

i times i is 1 not negative 1, correct?

@ProfessorDaveExplains Год назад

No it is negative one.

@utkarshraj3272 8 месяцев назад

👏

@profphilbell2075 4 года назад

Dr. Dave, we do have a value for mass of an electron or proton. 9.1 x 10^-31 Kg. and 1.7 x 10^-27 Kg respectively. E=p/c would apply to photons. while classic physics can apply to Electrons (E = 1/2mv^2) which are huge in comparison. HUP would certainly apply to photon sized particles. Happy to be corrected.

@ProfessorDaveExplains 4 года назад

I'm not sure what any of that has to do with this tutorial.

@profphilbell2075 4 года назад

@@ProfessorDaveExplains Never mind. A great tutorial as usual. I will add the url to the resources list for my students.

@Jehannum2000 4 года назад

HUP also applies to electrons. The rest mass of an electron may be 'certain' but uncertainty in its momentum and position are still inversely proportional.