Ch 1: Why linear algebra? | Maths of Quantum Mechanics
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- Опубликовано: 2 янв 2023
- Hello!
This is the first chapter in my series "Maths of Quantum Mechanics." In this episode, we'll go over why we should use linear algebra as the starting point for all quantum mechanics.
If you have any questions or comments, shoot me an email at:
quantumsensechannel@gmail.com
Thanks!
3Blue1Brown's "Essence of Linear Algebra":
Link: • Vectors | Chapter 1, E...
Animations:
All animations created by me within Python, using Manim. To learn more about Manim and to support the community, visit here:
Link: www.manim.community/
Music:
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♪ Astral 7 by Patricia Taxxon
Link : patriciataxxon.bandcamp.com/a...
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As a high school physics teacher, I love that content like this is being made which I can share with those students who are hungry for these higher level concepts but are still early on in their mathematical understanding (pre-calculus). Having these clear conceptual bridges with strong essential questions guiding each chapter is both pedagogically sound and a great example of a scientific thinking when walking through the unknown.
I'm curious. Can you often grab your students attention by telling them how weird and physics is going to get when they get to quantum mechanics or is there some recommendation against that in high school?
@@narfwhals7843 I'm not a physics teacher but when I was my final EM course just before QM, my professor said 'get ready to forget everything you've learned in classical mechanics because all that intuition soon goes out the window'.
@@jamesbentonticer4706 And did that make you think "whoa that's awesome I can't wait to get mindblown!" or did you think "wow so all this was just a waste of time so far?"
I'm curious how useful this "undoing" of what we learn first really is.
@@narfwhals7843 That is science. Try reading through any science or technology book written before 1950 or even 1923. You will surely see societal values dominant over science values (or maybe not?). That is why science is neither Pure nor Applied. It is both
definetely me! whenever our sir Introduce new concpets and equations like Heisenberg uncertainty principle, Bohr’s quantization of angular momentum etc I always ask HOW did the physicist even came up with it why is planck’s constant there (seriously it shows up eeverywhere!) he’d just say it’s derived but it’s complicated for you to understand, so I lead my curiousity to youtube.
Proposal for two more chapters:
- What is the tensor product of Hilbert spaces and why does it matter?
- What is the logic of a Hilbert space and why does it matter?
The logic of a Hilbert space is that (a) the concept of an inner product makes sense in a Hilbert space, and this is important, because the inner product is a generalization of the intuitive notion encoded in the dot product. To put it even more simply: Hilbert spaces are spaces where the concept of "projection" makes sense. This is not true for arbitrary vector spaces; (b) Hilbert spaces are spaces with a completeness property. This means that the concept of limits in such spaces is well-defined, and so you can differentiate, integrate, find series expansions, evaluate limits of sequences, and more. These are all absolutely necessary for doing physics. Fundamentally speaking, physics is an applied form of the theory of differential equations, so if the vector space you are working with does not allow differentiation, then you cannot do physics with said space.
Hilbert spaces are exactly those vector spaces satisfying both (a) and (b). This means that if one of the two properties is not satisfied, then the space in question is not a Hilbert space.
@@angelmendez-rivera351 vector spaces that do not allow differentiation is a new thing. Could you please explain it in more detail or provide me some resources. Thanks.
@@wrox2757 There is nothing new about it: most vector spaces you will ever encounter lack a concept of differentiation. For a very elementary example, consider the pairs of rational numbers (q, r). These pairs form a vector space with their usual operations, but you cannot do calculus with them. This is because rational numbers, despite being dense and Archimedean, lack the completeness property that the real numbers have. This is actually what defines the real numbers: they are the completion of the rational numbers. You need the completeness property to do calculus, because you need limits to do calculus, and limits are only well-defined when you have the completeness property.
As for the tensor product of Hilbert spaces, it is basically the most general vector spaces of pairs of vectors, where one vector belongs to one vector space, and the other from the other space, intuitively. It matters, because tensor products are the only mathematically coherent way of studying multi-particle systems. Individual particles are characterized by Hilbert spaces with certain conditions, and operators acting on those spaces. So, systems of multiple particles are characterized by tensor products of those individual systems. This is especially important when talking about quantum entanglement, but we are getting ahead of ourselves. These topics will, if the series continues at the pacing I predict, be covered much later down the line.
@@angelmendez-rivera351genius!
I going to watch all of your videos.
it seems to me that you are the ''3Blue1Brown'' of physiscs :)
please keep making these videos.
ikr!
I love that youtube is learning to recommend good and interesting videos super early now.
This is a great concept with beautiful style, and I'm excited to see the full series! Please keep it up, I hope you get the audience you deserve.
It's truly wonderful. "The algorithm" is usually a ruthless beast that wants nothing but your engagement but I keep finding gems of education.
SVD is a powerfull tool of projections :)
I'm really glad to see your series truly start. Your preview videos were pretty influential in how my own recent videos have been produced. Here's to a long, successful run!
as a physics undergrad student, this is one of the best videos i’ve seen on qm and really helps to give an appreciation of the subject
Cheers for the new beginning! This videos seem very intuitive indeed! Congrants: I'm very impatient for the next episodes
Decades ago I took QM but failed to appreciate it 'fully' and have always wanted to return to it. Now retired and with RUclips at hand I'm giving it another try. Thank you Quantum Sense for making it sense and ,more importantly, enjoyable. Congratulations, this is brilliant !
Fantastic video, was looking everywhere for a 3blue1brown for Physics and you're the best I've seen.
Awesome, clear, intuitive, comprehensive. Mapping the physical ontology and the mathematical paradigm attached. Can't wait to build a knowledge of quantum mechanics. 👍
It has been a pleasure to find such series, and I wanted to thank you for spending the time and effort into making this fascinating model more understandable. From the perspective of a second year physic's bachelor student I find it extraordinarily useful. It is through passionate and genuine people like you that this world becomes a better place. Thanks again.
A brilliant idea this series! Thanks for putting the effort. It’s quite illuminating for non-specialists like myself but who nevertheless are not satisfied with the usual “hand waving” approach using analogies. Getting an intuition for the math is probably the best way to “understand” QM fir what it is: a rather strange beast from the point of view of our everyday experience. Thanks again!
A very fundamental question, often overlooked by other channels. Well done!
Currently studying Quantum Mechanics at the senior undergraduate level and I am thankful to have stumbled upon this channel!
It's amazing that 3b1b's essence of LA is so well made that it sets up a standard for LA intuition
This video and those promised to come from the same teacher are extremely important and unique in that they go into the mathematics of quantum physics, something that is not touched on in most of the videos currently available online.
Please, continue to make these videos as you proposed and promised ❤ Thank you so much.
Brilliant！ Exactly the type of super clear lesson I was looking for. Great job!
i shall remain indebted to u all my life,, i have been waiting for such a course for a decade now.
Thank u very much sir.
Fantastic video. I love the approach and the visuals are great. Keep it up!
Great video, perfect timing. I had a random urge today to explore this topic. Will be g through the whole series ☺️.
0:42 There is this youtuber I recommend called Parth G who doesn’t just explain the maths of a few concepts of quantum mechanics but also other concepts in physics like Maxwell’s equations
Damn, I'm very happy to discover your channel. Love the work thank you for all this effort. A very happy new year !
Wonderful! Love this series already. Thanks for sharing!
OMG. This is the series I have been waiting for. Thank you sir for spreading the knowledge ❤
Wow, Just finishing ch 1 I can already tell this is going to be a great series. You have a gift for teaching this complicated subject no one else can!
Lets go! super exited for this!
Great stuff, excited for the rest
I was looking for a series like this for a long time! I took a course in quantum mechanics that is not really for physcists so I missed on most of the math behind it
Before you got into the argument, I was skeptical of the "good start" you spoke of. Yet, within a few steps, your presentation looked just like a vector. Well done.
Very good episode !! Now it really makes more sense how the quantum physics works !!
Thanks a lot and keep it up 😊
Bro, you're a life saver please do more content!! I wish I found this before my midterm.
I like how you can explain such an opaque subject with clarity and insight. Thank you.
Thanks for the video. I look forward to the rest of the series. I do have a few comments, and I suspect you know these, but I am of the philosophy that we should teach in a way that we do not have to unteach. The first comment is that the experiment you describe is a quite fictitious one. The closest one could get to the experiment you propose would be to shine light of an energy higher than the ionization potential and then measure the kinetic energy of the emitted electrons and from that infer the bound-state energy. But, of course, each measurement must have a spread to it as the experiment is repeated, because the light shone in will be from a wavepacket so its energy is not well-defined, and unless you have actively excited the atoms, they will all be in the ground state so it won’t show the behavior you desire to use for the example. Second, I believe it is very important to distinguish between abstract vectors and operators and the representation of them as coordinates of vectors and matrices. But here, they are described as the same, or at least could be easily confused as being the same. Finally, the example you give, of an atom, is a classic example that has both discrete bound states and continuum states. But, your presentations sounds like it has only bound states.
Great job! As a student, i always wanted this specific approach to QM. Thanks a lot!!
this seems so great, i’m in ib hl physics and my teacher who is a mechanical engineer doesn’t like nuclear and quantum physics but i find quantum to be so fascinating so this is a great way to learn more. I’m only in calculus ab so we’ll see how this goes and i might have to do some background research to understand the more complex math
I absolutely love videos like this, great quality!
This is awesome!! I'm majoring in physics and I'm super glad youtube recommended me this series so early on. Keep it up!
em qual faculdade vc estuda
@@Arthur-so2cd Faculdade de ciências da Universidade de Lisboa
Pleased keep continue to teach the concepts of quantum mechanics in easy way
I watched your video firsts time and I loved it.
Usually i don't comment on any video but your fantastic video and your style force me to comment
Cool. Looking forward to the series. RUclips recommendation finally worked.
This is amazing. I love the use of an actual experiment to start developing a mathematical representation
great video love to follow full series
Thanks so much! Really enjoyed this and you’re right, it fills a hole in the material that’s available!
This series is awesome! Keep going and thanks
Amazing work! Keep up with it!
Thank you very much for all of this. Looking forward to it
Simply brilliant, a magnificent video!
Wonderful, clear, refined, concise style. Illuminating in so many ways. Thank you!
Hope you can do a series on quantum computing.
hey how r u !! i just want to thank you for this kind of video.understanding the true meaning of physics and mathematics is the essence of these sciences.i hope you produce a lot of content like this
Now, I can finally watch this after being done watcing the Essence of Linear Algebra by 3Blue1Brown
An introduction to spinors would be amazing too!
finally , we have been waiting for to long
Eyyy, let’s go!
Quantum mechanics is a subject which I learnt by just blindly accepting a lot of things. While everything seemed more and more consistent the deeper I got into it, in the beginning, it was such an ordeal to feel confident in the subject and know whether I understood stuff right. A lot of times, things didn't make sense. And that feeling lingered for a very long time.
After having gone through all fourteen videos in this series, I think your approach to the subject is really helpful and refreshing!
Except that he didn't explain to you the actual reason for the linearity and unitarity. ;-)
wow, it does give me more intuitive image on the connection between math and quantum, thanks
This is why I love RUclips as of late. The educational videos are just so damn good. Especially with newer creators who understand what previous lessons were missing.
You
Highly Entropic Mind
3b1b (he's big, but for good reason)
I still like PBS spacetime but it's not mathy enough for me.
Great initiative. Very niche approach .thank you so much
Thank you so much for your awesomely clear and deep explanations!
Yu r serving curiosity ❤️ nd thts grt
Hello. Wahoo such a good way to explain the maths of quantum mechanics. I eagerly await follow-up. many thanks from france
Excellent video!
Interesting question, realy awesome video, few remarks, Linear algebra is already there in classical mechanic, Rotation, translation are linear operations , Newton law is linear relation between vectors. Eigen vectors etc are computed to determine the proper modes of a classical mechanical system. I am not sure the Linear Maths are bound to the Quantum world. The peculiar nature of Quantum world is elsewhere, the discrete nature is definitly the signature of the quantum world.
This is really beautifully done! Your urge to explain is highly appreciated. Any plans for a similar QFT series?
This seems really interesting! I always wanted to dive deeper in the world of Quantum Mechanics but I never really found a good opportunity to. But now I did.
That said, I'd probably study the prerequisites first before watching this series since I feel like it would be a waste if I just watched this without understanding anything.
Elegant explanation, thanks a ton!
Finally, you did it!
You ought to have more subscribers. You have wonderful content
Amazing video!
Awesome, been wondering about this for a while now. I just bought the 3rd edition QM by David Griffiths. Really looking forward to your series.
Once you finish griffths, do shankar, it's very good!
@@mastershooter64 I will. Thank you for the recommendation.
plz never stop making these videos!
Hi man, good job, please as soon as possible put next chapters in youtube, i need your knowledge. Thanks lots of millions millions ... ❤❤❤❤❤❤
Can't wait for the rest!
I simply loved your video, sir.
I’m very ignorant, so the leaps I’m making (and the words I am using) are very likely wrong.
But this is the first time I’ve ever seen how the ‘shape/form/description’ of the wave function is just an outcome of us wanting to describe the type of data we get when we try to measure/interact with quantum (quantized) systems. Instead of it being some arbitrarily abstract object sent to us from the heavens. And we can compared to classical systems where we can represent stuff with a continuous function. Thank you for letting me see something about our world for the first time
The assumptions I’m making here are that the ‘linear combination’ of the set values (8:35) is going to turn into a ‘wave function’
One important thing I am still unclear on however, is if the language-based description of a wave function as ‘all the states happening, only after which we get a singular answer’ is an artefact of the mathematics we use to describe the system or an actual transparent view of the underlying physical phenomena
Hello! Thank you for watching.
You are on the right track! If you watch episode two, you’ll see how the wavefunction comes about.
And your second question is actually incredibly profound. I think what you’re asking is if quantum mechanics is inherently random before we measure things, or if that’s just some artifact of our mathematical model. I may one day make a video on this, but you can actually *prove* that quantum mechanics is inherently random! (If you assume that causality is a fact I.e. nothing travels faster than light) This is done through Bell inequalities, the experimental verification of which was the subject of the 2022 Nobel Prize in Physics.
You call yourself ignorant, yet you are thinking about physics problems that have won the most recent Nobel Prize. So don’t worry about being “wrong”, physics is the art of being wrong, while having the passion to figure out *why* we are wrong.
Hope you enjoy the rest of the series!
-QuantumSense
@@quantumsensechannel thank you so much for your measured, thoughtful response. I really enjoyed episode 2 as well! I would love to see the bell inequality video from you, I’ve watched other yt videos on the topic, but you seem to have a real knack for helping us see why, and agree with, the mathematical decisions that are being implicitly made along the way. ‘Ppreciate ya!
YO I LOVE THE 3B1B STYLEEE WE NEED THIS FOR LIKE EINSTEINS STUFF
This is exactly what I need.
thank you very much for making this series. you explain very well
thank you for your video ! I started a course of quantum mechanics this morning and haven't understand nothing (the teacher did not help us in fact...) but know I know this channel, I am safe for the rest of the semester :)
Excellent! I really love your bold claims at the beginning of the video that you intuitively understand QM. I do not know why some (experts) claim they don't. You just have to throw away some of your old beliefs, and adopt new beliefs, and that's it. Thank you. :)
You have a new subscriber in me. Great content 😊
Excited !
good luck for the series
very well done. i am on second video and watch and try to soak in all the videos. thanks again.
you cant stop now, we want it all baby!
This is very good!
The perfect teacher ever
Love you from Pakistan
Genuinely thank you for this ❤
The best quantum theory series!
i am starting my first semester at university at UCSC, and am looking forward to my physics major. thanks for uploading this playlist for me to look forward to viewing the first time, not knowing much of anything, so i can come back in the future and laugh at how much i didnt know.
That is incredible, thank you so much
Great lecture, thank you!
-- awesome series
I am looking forward to also see videos about the topics covered in chapter 6 to 11 of Griffiths!
Brilliant. To the point.
I see the amount of effort gone into making this high quality video.
Love the mathematical explanation.
thank you ....keep going😃
Good video, man :)
Great explanation!
this is SUPER helpful
Great intro! Thanx! 🎉