The Maths of Quantum Mechanics in 13 Minutes | Functional Operators and Fourier Transforms
ฝัง
- เผยแพร่เมื่อ 15 ก.ค. 2024
- Even physics graduates sometimes misunderstand fundamental quantum mechanics.
Understanding how linear algebra in function space is used to do physics on waves, is paramount to knowing quantum mechanics intuitively.
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Timestamps:
0:00 - Intro
0:55 - Vector Space Review
2:12 - Dot/Inner Product
3:20 - Function Space
5:33 - Fourier Transforms
7:18 - Example QM Calculation
9:17 - Deriving the Momentum Operator
10:30 - Hermitian Operators
12:35 - Ending Words
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Music: Mark Tyner - Close To You
This kind of content is so rare and precious on youtube: stuff about math and quantum mechanics that goes beyond the 101 introduction of concepts. It's also well explained and fun! Thank you.
TYSM!!!!!!!
Agree, we get a lot of talk about interpretations and the symbology, but I've not seen such a sequential attempt to get dirty with the mathematics. The proportion of the population who can deal with partial derivatives, eigenvectors, and basis vectors is probably close to "i".
Shorter videos are better. The concepts aren't that complex if you know calculus and linear algebra, it's just a lot to take in all at once. Breaking it up into smaller pieces makes it easier to get.
5:00 No, integrating a Dirac delta function with itself does not yield 1, it yields infinity (or undefined). They can't be normalized. IMO an excellent choice for a basis is the eigenstates of the harmonic oscillator.
Great video, functional analysis is a tricky subject! If you want to show quantum mechanics to someone for the first time, I still think the best way to do it is using only finite dimensional systems: spin 1/2, two level atoms, etc.
I totally agree that knowing those will allow functional analysis to click like a tick, but the reason I made this video, is that everyone else is pretty good at explaining finite systems.
I’d stick to making videos that no one else have made first, before I start fleshing out the details.
I tried self studying a bit of QM. Had a really good grasp of basic linear algebra as well but I never understood why all the operators were so weird looking. 14 mins of pure bliss and understood everything. Please keep making more videos!
Thank you so much!
I absolutely plan to, and just like everyone else on this platform says, if you share my videos with others who really need it, I would be more than thankful.
it‘s so well explained, that I thought I understood something, but then realised, that I did not 😂
I appreciate the shorter videos that get to the point. If I have trouble then I can rewatch a section on my own.
Very clear explanation, thank you!
it shocks me how underrated this channel is! Incredibly good content, I'll be so excited to learn more from your content!
came here from your SoME2 video. your bear animation is adorable and so inviting!
This video is excellent like all your other ones! Please keep doing them, you deserve to have a lot more subscribers :)
I had more trouble understanding this than I had with the previous video, but still nice video!
Lagrangian mechanics needs 1 novel thing to understand. QM needs 3.
While this one is as tough as well done steak, rewatching it after a while might be more than helpful. It took me 4 years myself to get, so its no wonder you can’t in 13 minutes :3c
@@mindmaster107 yeah, also I already took linear algebra 1 where I learned all the linear algebra related ideas that were relevant to the previous video, but I didn't take linear algebra 2 yet, and that includes all of the linear algebra related ideas in this video, like eigenvalues and eigenvectors.
This is very high-quality content! Thank you very much for making it both highly educational and fun to watch!
highly appreciated work
Excellent video, thanks :)
I'm so thankful for coming across your videos, you're one of the best physics youtubers! Keep doing what you enjoy.
Thank you so much!
I just commented on your othe r video, but I gotta say this one is super nice too! Keep up the good work :)
Also, yes I thin ten to fifteen minutes is the perfect length for such a video. I enjoy longer content too but with such a high topic it's quite often too demanding over a longer time, so maths is perfect in pieces of about fifteen minutes I think
Cool video! 😊
Dear god this is amazing. As the other guy said, TH-cam videos don’t commonly go really beyond 13yo stuff, but I really like this trend (and you) :D keep making them man, love it.
Thanks for the kind words!
I discovered your channel through the #SoME2 video, I've watched also some of your other videos and I really like them!
Thank you!
Great video
Really nice video ty
Now its all crystal clear.than
I love long videos!!! Please please make more content!
Stellar!
What a brilliant video, I've just finished my alevels but did not take physics or maths but have been self teaching them both for the past 5ish months just out of interest. You have a great ability to explain complex concepts simply and i am able to somewhat grasp the concepts (even with no background in maths or physics). Keep up the great work!
Thank you so much for the praise!
Amazing
long explanatory videos would be great,your special relativity video was awesome.
Waiting for general relativity.
I really like your videos and your pedagogy, keep it up! If I remember correctly from my lectures it seems to me that the Dirac function is not really a function but rather a distribution. It is therefore risky to choose Dirac distributions as the basis of our functional space. It would seem to me more convenient (in the framework of quantum mechanics) to choose as a basis the Hermite functions which are a basis of the set of integrable square functions L2[R].
Quantum mechanics is quite a naughty mathematical theory
Even if dirac functions are unsafe (I’ve had to take derivatives of the dirac function before lol), it’s the basis used by the position operator, so has to be explored just to teach QM.
Can you cover the Hamiltonian and also how to determine shapes of orbitals ?? My teacher told that the Schrodinger wave function is very complex and you just need to have faith in what's written in the textbook.
Once you understand electrons are waves, then you can figure it out yourself from these two pieces of info combined.
1. Around a electrical/gravitational -1/r potential, waves naturally form “spherical harmonics”
2. Electrons can only go into 1 harmonic at a time.
Spherical harmonics are just a list of possible waves that can exist spherically, which is used in earthquake prediction and looking at the Sun. Usually, waves are a huge combination of these waves, but specially to quantum mechanics, electrons only occupy a single harmonic at a time.
The rest should be accessible on wikipedia or in your textbook. Good luck with your studies!
Well, nonhermitian quantum mechanics is possible, it's math is just not neat for quite a few problems. But loss and gain problems are apparently ok to be treated by it mathematically, at least some. I'm sure someone will eventually find that we just need to use adequate math for the adequate problems eventually.
Regardless, yes, I had quantum mechanics right after linear algebra unlike my peers, and to me it seemed a lot more intuitive than to them exactly because of that, but quantum mechanics does have other properties that are less intuitive - nonhermitian approach also uses a few of linear algebra concepts, it's just usually less neat. And a few unsolved (or maybe unsolvable) problems as in measurements, nature of time (which isn't a problem for nonrelativistic physics), and nonlocality.
There's a lot of other nuance things that I could comment, but let's keep this short.
Non-hermitian would simply not relate to observables (the raising and lowering operators come to mind). Other formulations of QM are mainly for theoretical reasons, and at least wasn’t relevant to the fundamentals.
The problems with QM however, are a topic others have done infinitely better than I. I may make a video on it in the future, but I would rather make videos on topics untouched by others first.
You are very good 👍🏻
Other interesting topics that are not so well covered elsewhere.
Entropy and information theory
Quantum information
Symmetries and conservation of momentum
Ooooooo yes, I'll make videos on them soon
My Lagrangian video introduces symmetries for the first time, but yes I plan to make a deeper video.
Translations and rotations are BOTH SPATIAL
Why does it go 1, 2, 4, 8, WRT complex, quaternion and octonions? I have learned so far that you lose certain properties, like quaternions produce different results and lose the a•b=b•a property and as far as i understand, octonions go a step further with a+b no longer equal to b+a. Why is this the case or are there other number systems? Ive heard that octonions find home in quantum physics? Are they useful to try understand, in general? Primarily as graphics devloper and "weekend maths enthusiast"... just would really like to understand this particular aspect! TY
Numbers are useful because of how they add and multiply together. Their algebra defines them.
Quarternions and the such follow different algebras, and hence are extremely counter-intuitive. Despite this, they find huge use in quantum field theory and General Relativity.
I plan to make videos on those one day
@@mindmaster107 thanks for responding. Looking forward to it!!
Can you please explain Fourier transformation
Fourier transform:
A coordinate transformation for vectors, is when the basis vectors change, and the vector is rewritten by those vectors.
A function-space transformation for functions, is when instead of the basis function being the dirac function, it is a function of x.
For Fourier Transforms, the basis function is e^iwx.
This is the explanation I chose for the video, but 3blue1brown made a video going over a different way to imagine the fourier transform. th-cam.com/video/spUNpyF58BY/w-d-xo.html
It is a huge topic, so different explanations might click better for you.
Is this weird that I'm a fan of quantum mechanics more than the classical mechanics ? Bruh I ended up hating classical actually. My physics teacher 🫠 well kinda brainwashed my intuition and fundamentals. Successfully wasted my 11th grade! Yay! 😭
Classical mechanics is about as challenging as quantum mechanics at its very core.
I think the mainstream problem people have with classical mechanics is it is taught very robotically, despite how interesting it can be. I understand part of that is this robotic method helps weaker students, but it leaves nothing to the imagination to stronger students.
@@mindmaster107 Any lecture/video recommendations for classical mechanics ? I'm tired of searching for books and resources. Tried mit open courseware/morin/hcv/problem books/ school books/endless loop of online teachers/ and now it's all a messy clutter. Now I even doubt the key assumptions. Help is what I need. And I'm preparing for an entrance exam.
Havent understood a shit, but that's very interesting
Hey this may be random, but I swear I recognise your username from SOME discord server… I’m totally clueless on which though, would you happen to have any idea?
Might be osu related, but I will leave the adventure through the forest of memories up to you
hoe can you multiply two 1,2, etc, column VECTORS, You CANNOT! NO CAN DO need to get the transport of the first one.