Ch 14: Where does the momentum operator come from? | Maths of Quantum Mechanics
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- Опубліковано 6 лип 2023
- Hello!
This is the fourteenth chapter in my series "Maths of Quantum Mechanics." In this episode, we'll take a look at where the momentum operator really comes from, and we'll discuss how all observables follow a pattern that not many textbooks show. We'll then finish up by showing what the Schrodinger equation looks like in the position, momentum, and energy bases.
If you have any questions or comments, shoot me an email at:
quantumsensechannel@gmail.com
Thanks!
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Music:
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♪ Overworld by Patricia Taxxon
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Hey everyone! One quick important note:
At 11:29, I was a bit too loose with my language. It is actually possible to construct a wavefunction that is normalizable, but does NOT vanish at infinity. For an example, see this stack exchange thread: physics.stackexchange.com/questions/382324/why-are-wave-functions-required-to-vanish-at-infinity .
The condition I am actually applying here is that our wavefunction be *localized* in space, which by definition means the wavefunction vanishes at infinity. I got a little loose with my language, and I apologize (leave it to the physicist to be a bit sloppy with the math!).
This argument for getting rid of the boundary term is used all over physics, and is just a statement on our intuition that particles are always somewhere “here”, not doing something crazy at infinity.
-QuantumSense
Dear quantum sense. I can't thank you enough. It's been a great learning journey. Please come out with a new series as soon as possible.
This is THE quantum mechanics playlist for dumbos 101. Please make more videos. 🙏
Let's goooo the series is back!!
The clarity of this series is very impressive. The author relies on physical reasoning with a minimal number of steps in his presentation. It provides an excellent starting point for one who wishes to learn quantum mechanics.
As a chemistry graduate who is trying to understand QM a little more I really can't thank you enough for this brilliant series of videos. This was amazing, I have never come across such a well-explained course in quantum mechanics. It's a real wonder! Thank you for every single episode!
If possible please make videos on QUANTUM FIELD THEORY, we love your way of explanation and clarity over the subject..... Thank you
Thanks for continuing further with this series of brilliant videos.Waiting eagerly for path integrals and other stuff.
I love how the pattern just goes in there, thanks for doing this.
change in energy drives time evolution… and change in momentum drives spacial transformation… that’s astonishing, never thought about this approach in understanding the quantum states being described by this equation!
6:42 Here you should notice that what you have is not the momentum operator but in fact the "wavenumber (k) operator" k_hat which is in units of inverse meters.
k_hat = - i d/dx or in 3-dimensions k_hat = - i (del operator)
Then using deBroglie momentum, you can reach conclusion that momentum operator,
p_hat = h_bar * k_hat
= - i * (h_bar) * d/dx
The wait was worthwhile, your videos are awesome !
Glad to see you back! Really enjoyed your other videos. I think I need to recap and then rewatch this one!
Great to see you again. This series is just too good.
Finally! Have been waiting for so long!
So glad to see you back! 😊
YAY Finally.
It's a treat when you upload videos. Thanks for this one. And feel free to take your time on the next. 🖤
Will you do videos on the Klein-Gordon equation and the generalisation of the Schrodinger equations to special relativity ?
These videos are thoroughly appreciated. 👏👏👏 Can’t wait to watch the next series!
Binged this "series"... just left me in awe. you're awesome! Q.M. is awesome! thanks mate!
As with previous videos, simply brillant! I was looking forward to this chapter and I am far from disappointed. Thank you!!!
my friend, we are eager to complete the series in quantum. waiiiiiiiiting for you
Brilliant is an understatement. Please do more intuitive series like this in all areas of physics. This is very very helpful. Thank you so much for the job well done!
Amazing series, thank you so much for these videos.
These were beautifully-made videos on the topic. I've learned a great deal from them and hope you consider making more in the future.
Your videos are seriously top notch. The physical sense you add to the maths we know is amazing, and much less common than the actual maths behind.
Would you be interested in continuing this series afterwards, for instance by adding this sort physical insight into QFT? Or other physics subjects
Welcome back!
This helped me understand why we can call the momentum operator the generator of translation. The division by dx made the penny drop for me.
Around 5:55, calling the unknown hermitian operator H seems unfortunate, because it looks like you are using the Hamiltonian. Was this on purpose to draw the parallel to the SE?
What happens to this pattern when we introduce a spacetime translation? Does the 4-momentum pop out, or do we run into issues with relativity?
Wow this series really helped crystallize QM for me. Great job.
This was an amazingly well-organized and inspiring series. I really enjoyed watching this, and while I’ve tried to take the time to get into understanding these things before, I have failed to stay focused because of the poor presentation and uninspiring way it was communicated. I am literally blown away by how simple you made all of this. It was all it needed to be: straight to the point, capturing and addressing what a newcomer might question along the way and making each step exciting, giving the student a both sense of progress and achievement.
I really hope that you continue these videos - you are extraordinarily good at it! Thank you so much for this!
Great Work!
I love videos you made!
These videos are really good. I hope you continue making them for a long time.
Excellent work, at my age I will have to watch the entire series at least one more!
Words cannot convey my thrill when I finally finished the series! Can't thank you enough for helping me realize my long-held dream of getting an intuition of the Quantum world! Also, I am wondering whether you could put on a new series on Largrangian and Hamiltonian in classical physics? I get the idea that you derive Schrodinger equation from classical mechanic concepts, but I'm not so familiar with those. Thanks again!
Very good series, this is the first time I am able to make a little sense of the structure of the maths of the quantum world. Thanks a lot...❤
Life is colorful again. Love these Chapters.
Hello Brandon , really captivated by the way you presented and explained the complex mathematics behind quantum mechanics in such an elegant and intuitive manner.
If you ever see this message at any point in the future , can you suggest the best books to get the most complete and holistic understanding of quantum mechanics with all its theoretical and mathematical rigor.
Also ,In the future , if you can do a similar video series on the mathematics of general relativity it will be immensely appreciated because most of the channels don't really touch the mathematics part of it.
Once again , thank you for this amazing series and all the best for your PhD.
loved the series very much....
I'm hyped for this more than any movies!
I'm enjoying this series thanks
Really great work.
I appreciate 🙏
Thanks for this kind of series..
Bro, your videos are so good. Thanks a lot.
Amazing video series, thank you so much. I always thought quantum mechanics was beyond my comprehension but you made me gain a mathematical intuition of it. :) Btw, I think a video delving into the basics of quantum computing based on this would be very interesting. I believe you could explain it in a nice and concise way.
I´ve complete the hole course, and i´m grateful of how u explain math mixing with physics and geometry. I don´t think so I can´t appreciate enought ur effort. thank u a lot. i bet i´ll pass the subjet with no problem. if it so, i´ll tell u.
That's beautiful man. What a moment to be alive ♡
If you make a PDF I will buy it, and probably I will not be the only one. This really associates the formal math and the intuition in a striking way. Well designed, brilliant accomplishment.
Great lectures!! I really wish I found these a little earlier
He's back!🎉❤
🎉🎉🎉🎉🎉 Thank you for updating
The best channel!
This is an excellent series: among other things it goes back to classical physics introducing the way 19th century physicists reinterpreted Newtonian physics in a system approach, laying the foundations for Schrodinger's interpretation of the wave function evolving over time.
the return of the king ❤
Welcome back 🙏
YOU'RE BACKKKKKKKK!!!!
Great video
Hello! Thanks for the amazing content, any idea when you're continuing this series?
thanks, thanks thanks for you quantum sense, i hope you come out with a new series❤❤❤❤
Welcome back 🎉
Finally you came back! Be waiting for so long 😭😭
Thank you so much!
Amazing!
Excellent...
I'd really like to see you deriving the Dirac equation too!
What is |x>? A position eigenstate or the quantum state in the position basis? In equations at 9:35 , what are the ket vectors? Are they the same quantum state represented in different basis or are they specific eigenstates for the corresponding observable?
Edit: the answer comes later in the video, it is the basis used to represent the state.
Thanks for your amazing work on distilling intuitive understanding of the equations.
omg i was getting scared that u wernt gonna continue with the series!!
Fantastic! :)
Please, please more such videos.....whenever you can.
Thanks for stressing the basis-dependence of those different forms of the Schrodinger equation and how they're all really the same information. The position basis is often given undeserved privilege over the others.
Welcome back :)
I would love to see a next episode on time in QM. Why is there a time-energy uncertainty? Why nonetheless is there said to be no time observable operator? What is the derivative of action with respect to energy? Help me understand!
Regarding your question why there is no time operator: Well spotted: time is treated as a parameter in non-relativistic Quantum Mechanics. The differential equations evolve with respect to time. However in Relativistic setting time appears on the same footing as space and in that sense time also becomes an operator. See Quantum Field Theory by Paul Dirac. He resolves (as far as my understanding goes) Quantum Mechanics with Special Relativity, but despite his genius, falls short on the grand unification with General Relativity and the effects of Gravity as the curvature of SpaceTime.
Thank you a lot
More please!
Nice video! When you derived the generator for the momentum and position in classical mechanics, it seems that they have the same a positive sign. However, when you derive it in quantum mechanics the signs are opposite (i.e. one is positive and the other is negative). Is there a mathematical way of obtaining this sign difference? And how does the sign difference make quantum mechanics consistent?
Once again, apologies for my naive questions. I work on GR and am a QM novice at best, but I wish to learn more QM.
Legend!
I almost thought this man died. BTW looking forward to the path integral video
Can you do a video on the mathematical relationship between entropy and caratheodorys theorem?
Which Platfrom you used for Animation ?How you Animate ? Which Softweres you used ?
hello there I like the way u teach . would u mind make a video and teach lagrangian mechanics and every thing about it I've been searching a lot and couldn't find some useful and ofcourse mathematical about it. thank u so much
I would like to see example problems where these equations are used and solved. I also wouldn’t mind seeing some attention given to special unitary groups, what they mean, and how they apply.
Great. So the Schrodinger equation is a quantized Euler-Lagrange equation that unites elementary QM with Noether's theorem.
How exactly we interpret the potential operation V in other basis? For example if the potential is the electrostatic (~1/r), how to substitute the i*h*d/dp in that?
Hello! Thank you for watching.
And this is a really great question, and there’s a couple ways to go about this. One way is to think about this logically: 1/x_hat as an operator can be interpreted as the inverse of x_hat. So, when moving to the momentum basis, what is the inverse of d/dp? The integral! So for a 1/x^hat potential, you would get an integral for the potential term (Note: in spherical coordinates like 1/r, we need to be a bit more careful about exactly what integral we use - but in any case, if you look at the hydrogen Schrodinger equation in the momentum basis, you’ll see that you do get an integral).
Now what about in general? Well note that for a well behaved V(x), we can usually Taylor expand it, and then insert d/dp for each power of x (although you see that the 1/r case is an exception! No easy Taylor expansion!).
As an example, consider a particle moving in a periodic potential V(x) = cos(kx) = 1/2(e^ikx + e^-ikx). In the momentum basis, the exponentials become e^(+-ik d/dp), and if you Taylor expand this and act on a momentum wavefunction, you’ll see that this is just the momentum translation operator (ie, it shifts our momentum wavefunction by amount +-k!). So you see that in the momentum basis, a periodic potential term in the Schrodinger equation can be interpreted as taking your momentum wavefunction and kicking it forward with momentum k and backwards by momentum k. This is the foundation of how band structures arise in solid state physics, and hopefully you see that working in the momentum basis allowed us to see a really intuitive interpretation of what V(x) is doing (which is why it’s worth remembering that form of the Schrodinger equation!).
A bit of a long answer, but hopefully I answered your question! Let me know if there’s any other questions you have!
-QuantumSense
@@quantumsensechannel Thank you so much!!! I'll ponder over it
we miss you
2:31 "I mean all we're doing is moving stuff over. It'd be weird if the total probably were no longer 1 for some reason"
Queue the Banach Tarski paradox: I mean, all we're doing is rotating stuff. It'd be weird if that makes two spheres out of one :D
Hey @quantumsense can u suggest some books for improvement in mathematical part of quantum mechanics.
First chapter of Shankar is really good imo. Basically goes through all the linear algebra you need for undergrad QM with lots of Bra-ket notation.
Physicists: let’s just Taylor expand it and see what it leads to.
Mathematicians: why the hell is this differentiable?
I binged watch the whole rest of the series like a few months ago and now I don’t remember anything 😭
11:34, that gets completely glossed over, why does normalizability imply that the boundary term vanishes?
I can't help but think that we didn't actually derive the Schrodinger equation last chapter. The last chapter derived the energy operator alone. The Schrodinger equation puts the energy operator, on a wavefunction, equal to the Hamiltonian of that wavefunction.
chapter 15 what are hamiltonian matrices
How would the Momentum based SE Look If WE Had a 1/x Potential?
Multiple on impact
7:40
I fucking love this
YASSS
👏🏻👏🏻👏🏻
10:33
Sometimes I wonder how we could include standardized, proven to work material like this one in our education system.
It's a tough challenge, but there are way too many professors with their own poorly made slides which nobody wants to see. There are of course good professors but sometimes I just wish I had these kinds of videos instead.
On the other hand, I wouldn't want to take away the independence of professors and their methods.
Of course, I can fully take advantage of these videos, but most students just go to class and that's it. It's so much wasted potential.
Also, I know I said "proven to work" without proving it... But hey, this content is clearly engaging (a much important part of education) and in my opinion, as clear as air.
see yaaa. hope you come soon XD
still waiting here