Ch 1: Why linear algebra? | Maths of Quantum Mechanics

[@alexanderkruszewski7306]

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.

[@narfwhals7843]

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?

[@jamesbentonticer4706]

@@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'.

[@narfwhals7843]

@@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.

[@Alan-zf2tt]

@@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

[@kashyaptandel5212]

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.

[@Arnaz87]

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.

[@logosecho8530]

It's truly wonderful. "The algorithm" is usually a ruthless beast that wants nothing but your engagement but I keep finding gems of education.

[@miro.s]

SVD is a powerfull tool of projections :)

[@userant]

Decades ago I took QM but failed to appreciate it 'fully' and have always wanted to return to it. Now retired and with KZbin at hand I'm giving it another try. Thank you Quantum Sense for making it sense and ,more importantly, enjoyable. Congratulations, this is brilliant !

[@amoghk.m.6769]

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!

[@schmetterling4477]

Except that he didn't explain to you the actual reason for the linearity and unitarity. ;-)

[@xxFairestxx]

And guess what? Quantum mechanics is only one of the accepted theories. There are others that are incredibly relevant. The idea of unifying the 4 natural forces to finally achieve General Unity is interesting to me. Weak and Strong nuclear power provided by Quantum Mechanics, plus Gravity from General Relativity....and lastly, Electromagnetism. The missing link is Electromagnetism and while far stronger than the other 3 forces, lacks universal attraction required by charge. I hope in my lifetime we can see that. LQC is one of the ways I think it could be achieved by rather than seeking the singularity of a black whole, finding it at Planck scale is an interesting theory at least...

[@ajadamd]

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

[@ruben-en4jz]

I going to watch all of your videos. it seems to me that you are the ''3Blue1Brown'' of physiscs :) please keep making these videos.

[@kashyaptandel5212]

ikr!

[@frankzenter8844]

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?

[@angelmendez-rivera351]

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.

[@wrox2757]

@@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.

[@angelmendez-rivera351]

@@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.

[@angelmendez-rivera351]

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.

[@5ty717]

@@angelmendez-rivera351genius!

[@yungmaz13]

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

[@SoundVoltage]

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!

[@khalidhabib9195]

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.

[@alezzi_mm]

There ARE classical properties of physical systems that are discrete. Examples, modes of standing waves, frequencies coupled oscillating objects like molecules.

[@douglasstrother6584]

Indeed!

[@lucamattioni7690]

Cheers for the new beginning! This videos seem very intuitive indeed! Congrants: I'm very impatient for the next episodes

[@jamiecolclough1852]

Fantastic video, was looking everywhere for a 3blue1brown for Physics and you're the best I've seen.

[@jesussanchezherrero5659]

Same! Only for Quantum Physics so far unfortunately

[@quantum4everyone]

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.

[@z.a.s.5615]

Currently studying Quantum Mechanics at the senior undergraduate level and I am thankful to have stumbled upon this channel!

[@sp3148]

Brilliant！ Exactly the type of super clear lesson I was looking for. Great job!

[@umairahmad92]

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

[@jakublizon6375]

This is why I love KZbin 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.

[@bernardomarques4306]

This is awesome!! I'm majoring in physics and I'm super glad youtube recommended me this series so early on. Keep it up!

[@Arthur-so2cd]

em qual faculdade vc estuda

[@bernardomarques4306]

@@Arthur-so2cd Faculdade de ciências da Universidade de Lisboa

[@virushk]

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!

[@stevenschilizzi4104]

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!

[@frizzarazz]

A very fundamental question, often overlooked by other channels. Well done!

[@ToriKo_]

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

[@quantumsensechannel]

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

[@ToriKo_]

@@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!

[@samuelvaldezgil]

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.

[@meofamily4]

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.

[@alejrandom6592]

It's amazing that 3b1b's essence of LA is so well made that it sets up a standard for LA intuition

[@hubertorhant8884]

Awesome, clear, intuitive, comprehensive. Mapping the physical ontology and the mathematical paradigm attached. Can't wait to build a knowledge of quantum mechanics. 👍

[@tomgargan8339]

Eyyy, let’s go!

[@איילתדמור]

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

[@postpunkjustin]

Fantastic video. I love the approach and the visuals are great. Keep it up!

[@maverick-rs]

YO I LOVE THE 3B1B STYLEEE WE NEED THIS FOR LIKE EINSTEINS STUFF

[@larianton1008]

Lets go! super exited for this!

[@kanhaiyalalrajput3215]

This is the best series to watch after 3blue1brown's linear algebra series❤

[@GH-li3wj]

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.

[@lifeinabubble9091]

The way you explain this topic hints at a relationship with neural networks. Neat!

[@rpdotg7408]

This is such a good video, like its concise, clear, AND factual. Great jumping off point!

[@ayhamhalalsheh221]

finally , we have been waiting for to long

[@hafsabatool8895]

I can not believe this is for free ... thanks so much for this unique approach to teaching quantum mechanics

[@schmetterling4477]

You get what you pay for: a false explanation. ;-)

[@kingspart9828]

​@@schmetterling4477 wym?

[@schmetterling4477]

@@kingspart9828 As soon as he starts talking about "particles" you know that he is merely repeating the nonsense he has heard on the internet. There are no "particles" in quantum mechanics. Quanta are small amounts of energy. We even teach that in high school (and have for at least 40 years), but absolutely nobody seems to be paying attention.

[@kingspart9828]

@@schmetterling4477 interesting,to be honest i know next to nothing about Quantums but your comments were interesting,how do you recommand someone to learn this stuff while avoiding misinformation,thank you so much.

[@schmetterling4477]

@@kingspart9828 That's the challenge... even many introductory QM textbooks that are correct on the math are presenting the theory without any rational explanation where it comes from (even though we know how to derive it from first principles, which are basically the Kolmogorov axioms). I basically had to piecemeal it together by reading the papers of the founder (especially Heisenberg, not so much Schroedinger) and a few modern ones. That quanta are energy, momentum, angular momentum and charges is best appreciated within the high energy physics community, at least by experimentalists like me who have built machines that measure said properties and nothing else. Even the theory is very clear about this, if you take the time to read the theory and you are not focused on the atrociously bad language that physicists have been using for about a century now to describe quantum mechanics in terms of imaginary objects called "particles". This is nothing new, by the way. Before modern thermodynamicists discovered that heat was form of energy, there had been the phlogiston around... a mythical "Stoff", that was supposed to be exchanged between system whenever heat flow was taking place. Similarly the aether was supposed to be the "elastic medium" that could transport electromagnetic waves. No such medium exists. We know that space has to be empty for relativity to hold. This phenomenon of humans imagining material carriers of properties is not restricted to physics, by the way. In economics theory it's the favorite of the gold standard crowd. They just can't imagine that money does not need a material backing. Money is a function that facilitates exchange of goods and services. It is not some shiny metal that has to be stored in some large vault from where it never moves, no matter how much we are spending of it. I am sure the psychologists have a name for this "material fallacy", but that's not my field. ;-)

[@TekCroach]

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. :)

[@p.m.rangarajan1055]

OMG. This is the series I have been waiting for. Thank you sir for spreading the knowledge ❤

[@vishrutpandya3257]

Great job! As a student, i always wanted this specific approach to QM. Thanks a lot!!

[@jamesbentonticer4706]

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.

[@mastershooter64]

Once you finish griffths, do shankar, it's very good!

[@jamesbentonticer4706]

@@mastershooter64 I will. Thank you for the recommendation.

[@user-wr4yl7tx3w]

Cool. Looking forward to the series. KZbin recommendation finally worked.

[@durandesther1506]

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 :)

[@kaushikroy4041]

This is truly amazing -- I have a very limited grasp of the mathematics assumed in this video, and yet the intuition was crystal clear to me. You have a unique talent for teaching. My one confusion is that to my knowledge, in standard linear algebra, matrices operate on continous spaces. The quantum experiment you simulated suggests that these matrices operate on a kind of lattice space, and therefore demand a particularly special kind of linear algebra. This is unless your telling me that recording probabilities of the points on the lattice now makes it a continuous space, in which case I dont get why normal functions wouldnt work.

[@amritawasthi7030]

Damn, I'm very happy to discover your channel. Love the work thank you for all this effort. A very happy new year !

[@anonymoushawk962]

Bro, you're a life saver please do more content!! I wish I found this before my midterm.

[@it6647]

0:00-Preface 1:05-Topics covered 1:47-Prerequisites 2:53-Introduction 4:26-Classical vs Quantum world-Hydrogen atom 7:13-A preliminary mathematical model 10:29-Speedrunning Historical Developments of Quantum Mechanics and further questions

[@colorx6030]

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.

[@rodomoyo9122]

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

[@pandiest]

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.

[@elyazrhiyassine1915]

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

[@diegopg7186]

Simply brilliant, a magnificent video!

[@physicsbutawesome]

Finally, you did it!

[@khaliliskarous2225]

This is really beautifully done! Your urge to explain is highly appreciated. Any plans for a similar QFT series?

[@ishaqktktech4372]

The perfect teacher ever Love you from Pakistan

[@LostinTime0310]

Now, I can finally watch this after being done watcing the Essence of Linear Algebra by 3Blue1Brown

[@Wallfacer2306]

This is so good for Quantum Club

[@JMC-35]

Hello. Wahoo such a good way to explain the maths of quantum mechanics. I eagerly await follow-up. many thanks from france

[@davidhuo6902]

wow, it does give me more intuitive image on the connection between math and quantum, thanks

[@karimshariff7379]

Just wanted to say that I don't completely understand this: In classical mechanics, position and momentum are independent coordinates in the phase space of a particle. However, in QM there is only a _single_ state vector with a representation in different bases (which is complete for describing a given observable such as position, momentum, spin, etc.). I need to understand why we can go back and forth (via a Fourier transform) between the position and momentum basis, in describing the same state.

[@williamworthy4091]

Great stuff, excited for the rest

[@davidpalomino9138]

An introduction to spinors would be amazing too!

[@Faridbuza]

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.

[@tune490]

Subscribed and liked! Love the channel

[@Forever._.curious..]

Yu r serving curiosity ❤️ nd thts grt

[@assburgers3457]

This is amazing. I love the use of an actual experiment to start developing a mathematical representation

[@kamkwok3835]

I like how you can explain such an opaque subject with clarity and insight. Thank you.

[@sythatsokmontrey8879]

This is exactly what I need.

[@RickB500]

Great introduction. Love your graphics. What do you mean by "intuitive"? Some take is a synonyme for understandable. But in historical and philosophical contexts its was more like demonstrable, you can SEE it. Leibniz had some issues with "intuitive cognition". An if you can't realise something inutitively you have to under it by symbols. And math is such a symbolic language, so math is never intuitive in that sense. The famous term "calculate and don't think" follows exactly this tradition. But I very optimistic, that you can solve the problem and combine this views. This was a good start!

[@juniorcyans2988]

This is my second time to watch this series. I watched all of them in the beginning of the semester but I remembered nothing. Now I’m reviewing the materials as I will have my first quantum exam next week.

[@nemooverdrive760]

Great video, perfect timing. I had a random urge today to explore this topic. Will be g through the whole series ☺️.

[@hrperformance]

Thanks so much for making this series!

[@vextronx]

I absolutely love videos like this, great quality!

[@osotrop]

great video love to follow full series

[@Hamza-zz3zc]

Very good episode !! Now it really makes more sense how the quantum physics works !! Thanks a lot and keep it up 😊

[@Luca-jy2rg]

The best quantum theory series!

[@jneal4154]

Wonderful! Love this series already. Thanks for sharing!

[@kashyaptandel5212]

plz never stop making these videos!

[@vaibhavnakrani2983]

Getting serious 3blue1brown vibes from this. Looking forward to attending the entire series! If this is the future of education then the ivy leagues might as well retire!

[@baloshi69]

Thanks a lot, i really love to learn about quantum computing and be able to do quantum programing, but as i lack the understanding of Math behind quantum everything :D. it was really hard, but i hope this series give my the much needed understanding to be able to walk on Quantum Realm.

[@physicsnabo]

thank you ....keep going😃

[@davidwright8432]

Many thanks! Rather than just plunging in to calculations, you motivate the choice of, and reason for, mathematical descriptions. Why doesn't every intro quantum (or other) course do this?

[@simpaticode]

7:13 "A continuous function won't work [to model discrete values]" but that's not strictly true. You could define a convention to indicate discontinuity with a continuous function. For example, adopt the convention that if there is a discontinuity the function "oscillates quickly" within some small bound. This might sound pedantic, but I think it highlights the often ignored fact of math's arbitrariness and that it matters as much how we interpret the output as it does how we explain the input.

[@sergiolucas38]

Good video, man :)

[@r4fa3l59]

Amazing work! Keep up with it!

[@johnstuder847]

Wonderful, clear, refined, concise style. Illuminating in so many ways. Thank you! Hope you can do a series on quantum computing.

[@andrejmajstorovic1437]

you cant stop now, we want it all baby!

[@nutmeggaming11261]

You ought to have more subscribers. You have wonderful content

[@kevalan1042]

If you think you understand quantum mechanics, you don't understand quantum mechanics, or you've watched Quantum Sense

[@miro.s]

Exactly!

[@bilyayakubu1718]

Space 4 u do it's now

[@NoActuallyGo-KCUF-Yourself]

Or a superposition of the two - properly normalized of course.

[@angelmendez-rivera351]

I have always hated the saying "If you think you understand quantum mechanics, then you don't understand quantum mechanics." It is one of the most insidious, anti-intellectual sayings a genius has come up with, and it is far too prone to being misapplied.

[@TekCroach]

This explanation is the most intuitive one I have ever seen. Definitely he understands quantum. Don’t just parrot what other great scientists had proclaimed in the past. If you allow sufficient time to let your brain digest quantum phenomena, quantum mechanics is easy to understand as the classical physics.

[@eliaderenda2041]

This series is awesome! Keep going and thanks

[@Anamikaaaa]

Great initiative. Very niche approach .thank you so much

[@JennySiede]

Agree, Linear Algebra is the fundamental mathematics behind Quantum computing.

[@lucahaines4655]

Amazing video!

[@mofazzalhossain2944]

I simply loved your video, sir.

[@shamantdesai930]

Elegant explanation, thanks a ton!

[@tkansal]

very well done. i am on second video and watch and try to soak in all the videos. thanks again.

[@trigocuantico]

good luck for the series

[@jonathanporter7601]

Thanks so much! Really enjoyed this and you’re right, it fills a hole in the material that’s available!

[@rgudduu]

I see the amount of effort gone into making this high quality video.