Ch 2: What are kets and wavefunctions? | Maths of Quantum Mechanics

Hello!
This is the second chapter in my series "Maths of Quantum Mechanics." In this episode, we'll go over how particles are represented by vectors (aka kets) and how wavefunctions relate to the linear algebraic framework.
If you have any questions or comments, shoot me an email at:
[e-mail]
Thanks!
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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♪ blinded by Patricia Taxxon
Link : patriciataxxon.bandcamp.com/a...
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[@pizzarickk333]

This series is exactly what I've always been dreaming about. We finally have the 3b1b of quantum mechanics.

[@kashyaptandel5212]

ikr!

[@andrewferris8169]

Man, these are awesome. I passed QM 1 but these videos would have made it so much clearer. I think a video of this style and quality on Local Gauge Symmetries and Forces would be awesome.

[@padrickbeggs7071]

That would be pretty sweet

[@ToriKo_]

Great suggestion, I would love to learn why it’s natural and useful to describe the ‘symmetries’ of particles etc

[@mikevaldez7684]

Andrew the Fairy, you didn't mention your grade so we can safely assume a D- or C- at best. 🤣🙋 Later dweeb

[@mikevaldez7684]

@@ToriKo_ figure it out dolt

[@ToriKo_]

@@mikevaldez7684 were you having a bad day or do you always make comments like this?

[@jamesbentonticer4706]

I really hope this series does well. A way to conceptualize quantum mechanics could revolutionize how its taught.

[@pacificll8762]

I agrée

[@Juxtaposed1Nmotion]

A lot of smart people are saying that QM needs to be re-formalized

[@it6647]

0:00-Recap 0:54-Formal definition of a vector space 2:12-Benefits of vector spaces 2:55-Quantum state as a vector 5:28-Continuous physical quantities (position) 9:35-Wavefunctions as coefficients of ket vectors for continuous list of kets

[@joshdeconcentrated2674]

These videos are SO incredibly helpful. understanding the concepts better is always a good thing and especially for futureproofing. super underrated channel!

[@brunofagherazzi9903]

What an amazing content you're building right here. I've been waiting for this kind of videos for years. Thank you SO much!

[@roelofvuurboom5939]

Great explanation. Explanation of why linear algebra in QM is so simple and intuitive. Really cool.

[@alexba88ify]

Loving this series! Thanks so much for doing this!

[@Miguel_Noether]

I already have QM notions but the way you are presenting this is so good👌

[@Self-Duality]

Beautifully explained!

[@sanderdude3901]

MAN this series is amazing! Thank you for putting in the time and effort to make this!

[@e.s.r5809]

Amazing video, really clearly explained! Thanks! This is fantastic prereading for my next semester. :)

[@michaeledwardharris]

Wow, this was excellent! Really great presentation style. I'm looking forward to watching the remaining videos. Thanks!

[@blacksmith1634]

I'v been waiting for something like this a while. The mathematics behind quantum physics always seem to be like understandable math in a language I don't know.

[@davidlearnforus]

This series is brilliant! Thank you so much for all great work!

[@WestOfEarth]

Really enjoying this series! Thank you so much.

[@yenbinh2239]

As a student who is intensively learning Quantum Mechanics, this video is great!!!! Thanks a lot

[@niranjanca3534]

THE BEST KZbin channel which made me understand the quantum things all the best brother you really should have bright future.....❤

[@Marevks]

This is what i needed in my life right now. Wow. So incredibly well explained… i needed to dig deep to find this channel thanks god i did

[@nablarnermk8844]

THANKYOU FOR EXISTING, I HAVE MY QUANTUM MECHANICS EXAM IN 2 MONTHS AND A YEAR AGO THIS WASN'T AROUND YET TO HELP!!!! KEEP UP THE EXCITING AND GOOD WORK

[@vincentlin7240]

Thank you so much for saving my quantum mechanics midterm and my life. Best quantum lecture ever.

[@ES-qe1nh]

Amazing playlist. Not overly reductive or too in depth.

[@johnhamilton7762]

Love the series. Great work.

[@physicsbutawesome]

These videoas really have a nice flow and are interesting to watch.

[@nandagopalgopakumar5626]

Another Amazing channel! Thank you!

[@weinsim3856]

Thank you so much! youre explaining it in a very clear and understandable way, which i think is going to help me a lot for uni

[@MaruriPorzio]

I follow Benson, this episodes fills all math I need to satsfatorely understand QM. TKS

[@NovaWarrior77]

MY GUY YOU LITERALLY KILLING IT

[@Mac-zl4po]

My man

[@MaruriPorzio]

Excellent, suits perfectly to what I need to better understand QM. Thanks & congrats

[@neil6477]

Fantastic! It was many years ago that I took a course on quantum mechanics (late 1970s) and found that little was explained about where the mathematics came about. Rather an equation was written on the board, followed by some words spoken by the lecturer - most of which I didn't follow. I passed the course by doing the usual student trick of practising sufficient past papers in the hope that my own exam would be similar - it was! However, despite being a physics student I was totally put off the subject of QM and didn't take any more classes (much to my regret). Now, in my 70s and long since retired I find these videos both educational and, more importantly, thoroughly enjoyable. Thank you so much for your work and I hope to learn a lot more in the coming weeks. 😀👍 (I am wondering whether we shall see actual worked examples which use the maths - but I guess I shall find out later?)

[@GreenFlyter]

That is brilliant work! Thank you

[@zeropotential6830]

this is just a blessing. thank you so much

[@speedspeed121]

I just graduated in June. This video gave me a better intuition than two quarters of QM

[@atanumaulik7093]

Amazing! Keep up the good work.

[@waltertoki1]

This is a very nice introductory approach to learn Quantum Mechanics. However a traditional approach of Planck’s constant, the Bohr model, de Broglie particle wave duality and finally Schroedinger’s wave equation with eigenvalue solution’s is more complete and easier to digest. Finally matrices can be introduced with unitary and hermitian operators and eventually the description of the electron spinors.

[@family-accountemail9111]

Thanks for this series! It's very valuable to me. I have only watch a few so far but this approach of explaining the maths and why it is suitable is right for me.

[@family-accountemail9111]

If I was teaching a course on am I would use this and ask students to watch this

[@keroshehab1543]

Wow ,your on fire broo ♥️♥️

[@allanolave2701]

Thank you so much! I love your explanation.

[@ashheralikhan6043]

Its brilliant. Go on . Keep it up

[@kholitakhawla3622]

Please keep creating series like this

[@sergiolucas38]

Excellent video, man, thanks :)

[@vianadon]

You are awesome! Thanks for everything!

[@EriiikaGuerra]

This is incredible! Why is QM making so much sense now?

[@ohidulislam5545]

Hey man! Great job! Would love to see long videos like 20 or 30 minutes

[@ToriKo_]

Cool video. Even though I can see that I’m not grasping everything, it’s so appealing how it seems like you’re making it a priority to get us on board with the packaging these ideas come with, helping us to see that actually this is a super natural way of working with these physical phenomena, and helping us feel like we actually *want* these notations. As small as it was, I got so much joy out of saying “position” out loud as a guess for a continuous quantity, and having that confirmed by you! One thing I don’t understand is how, 11:13, if we have use a ket to represent *all* the possible information about our particle, then why do have different outcome kets that represent only partial information about our particle, like energy or angular momentum. 11:27. How can we label one |psi> ‘energy’ and another |psi> as ‘angular momentum’, when our ket is supposed to represent *all* the possible information of our particle. Which should cover all information about our particle, like energy, angular momentum, spin, mass etc?

[@narfwhals7843]

Are you familiar with linear algebra? This is a change of basis. When we write |E1> we have chosen to represent our state in the "Energy basis" and when we write |p1> we chose the "momentum basis". These are both valid choices to _represent_ our general state vector |psi> and there are many more. In any basis |psi> will be a superposition of basis vectors. |psi>=c1|E1>+c2|E2>+c3... or |psi>=C1|p1>+C2|p2>+C3... Where the c's and C's are the coefficients for that particular basis. Any basis that spans the entire space will contain the full information, but some(like spin) only span a subspace.

[@ToriKo_]

@@narfwhals7843 wow okay that’s super interesting. I’m not really familiar with linear algebra, but I’ve seen quite a few videos explaining basis vectors. Your explanation makes sense to me but I imagine there are a bunch of subtleties and inner workings to the explanation that I’m failing to grasp. Thanks for ur time and explanation

[@stanislavtsybyshev7453]

Exactly the question that popped into my mind after watching - thanks for asking this!

[@angelmendez-rivera351]

@@ToriKo_ Well, the entire point of Chapter 1 in the series was precisely the point that you *need* linear algebra to have a solid grasp on these subjects, because ultimately, quantum mechanics is just one particular way of doing linear algebra. In fact, the video explicitly tells you that you need to have at least some minimal education in linear algebra, even if not formal. The video recommended 3b1b's linear algebra series on YT, which I agree with. Having the basics down is absolutely fundamental if you want to have a solid grasp of the intuition behind the mathematics of quantum mechanics.

[@AndreKowalczyk]

So far it's going great! Thank you. Still not clear how a continuous x can be represented by a ket vector (which is a list of discrete values). I hope this will become clear later.

[@pandiest7764]

as a starting physics major, i enjoy watching videos of all of the higher divisions of physics whilst i'm still in classical physics. it's fun to see what i will be learning later on in my education. thank you!

[@moslynmoslyn679]

Same here bro

[@bibek2599]

Very nice explanation

[@curtpiazza1688]

WOW! Great stuff! 😊

[@mariocesarsousa]

Excellent bro✍️✍️✍️ Thanks for sharing. 💚💚💚💚👽👽👽👽

[@admiretsikayi8238]

Good work.

[@TheFireBrozTFB]

Keep it up!! Love the content

[@Mouse-qm8wn]

Super Nice videos, thank you so much 😊

[@DarkNight0411]

Beautiful!

[@kwintenderijck3110]

This is amazing

[@jorgesaxon3781]

I find it fascinating and also a bit terrifying how looking at quantum mechanics through the lens of computer science trivalizes it massively (arrays, functions, mappings etc)

[@faenzarfaenzar2636]

Amazing serie !!

[@Masrawy_79]

More than excellent 👍👍

[@jinishgaming3240]

Excellent buddy

[@reefu]

Finally!!! Let’s go!

[@florisv559]

Well done. I do have a gripe though with how you describe a function as something that is necessarily continuous. But the sequence 1, 1/2, 1/3, ... is also a function, from the natural numbers to the rationals, because it links each natural number to at most one rational number.

[@blusham4629]

Love the series

[@san99539]

Why you are so good!

[@aramsarkisyan8061]

This is extremelu useful

[@jdbrinton]

Thank you thank you thank you!

[@PETERTRITSCH]

Awesome !

[@siamsama2581]

Very good

[@andreaq6529]

These videos are awesome, instantly subscribed. I also have a question: why is energy considered as a vector?

[@quantumsensechannel]

Hello! Thank you for watching. I think there may be some confusion into what we mean by “vector”. Energy itself is a scalar quantity. However, in the quantum mechanical framework, our particle can be in a state representing a certain energy measurement outcome. This state is represented by a vector, called a ket. The terminology is weird, but the vectors we’re talking about in quantum mechanics are a bit different than the vectors in classical mechanics. So energy is still a scalar quantity when measured. -QuantumSense

[@andreaq6529]

@@quantumsensechannel Thank you!

[@angelmendez-rivera351]

Energy is not a vector, but there are vectors associated with a particular energy. These are called the "eigenstates" for that energy.

[@opd-cp3ee]

Please create a Patreon page, if you haven't done so already! I'd definitely support you there :) Also, for videos in the future you might want to reduce the breaths in the audio (via editing or with a different mic or angle?) Sorry!! I feel a little bad for nitpicking, because I really love the way you explain and am extremely grateful for the time and energy you put into these videos. I've even thought about starting a series myself, because this really was missing on KZbin. (although I don't think I'd reach the ease at which you explain, not to speak of the animation!) Thanks thanks thanks! maxi

[@manstuckinabox3679]

even if it wasn't continous (with plank's constant coming in mind) the absurdly large amount of possibilities AND the fact that by definition dx is kind of an approximation, I think integral is quite the best way with dealing with the super-position.

[@quanrumride1027]

damn...such a nice class..

[@davidhuo6902]

just love it

[@johannbrrr8065]

When we go from a discrete sum to an integral do we have to change the meaning of the coefficients from probability to probability density?

[@davidgruzman5750]

Thank you a lot for very clear explanation! I am a bit confused by picture on timem point 11:21 . In one hande - Phi is said to be vector containing all information about the particle. Than i see on the picture that it is equal to linear combinartion of energies and, in the same time - of angular momentums. Please tell me what i miss here..

[@nicolasPi_]

11:12 shall we say that the quantum state contains all the information about the particle at an instant t? Does the quantum state change over time or is its time evolution self-contained?

[@serenowsky1284]

When you say physical properties, does this include all innate properties that a particle would have by definition? For example, would a quantum state hold the property of a -1 charge in an electron, or would that be unnecessary?

[@shreenathwalvekar1009]

Keep it up

[@drewnoren8416]

At 4:24 you say that we can describe the same quantum state with a linear combination of energies, and with a linear combination of momentums. Does this mean that this combination of energies is equal to the combination of momentums (representing an energy state with momentums), or are these two linear combinations measuring completely different quantities? If they are unrelated, then how can we tell the difference between them if we use the same symbol to represent the quantum states?

[@quantumsensechannel]

Hello, thank you for watching! This is a good clarifying question. You are correct that those two linear combinations describe the same quantum state. So in that quantum state, you are in a superposition of possible angular momenta AND superposition of possible energies. I would be careful in saying “an energy state with momenta”, since we are not in an energy state, we are in a superposition of energy states. And although I showed those two, the particle could also simultaneously be in a superposition for position outcomes, or any other physical quantity. In a later episode, we formalize this a bit by showing that these “outcome states” are the eigenstates of the corresponding observable, which form a basis. So these different linear combinations are just ways to write our quantum state in different bases. So how do we distinguish between the energy and angular momenta linear combinations? You don’t! They exist at the same time, under the same quantum state. They just show up when expanding our quantum state in that respective linear combination. In order to break the superposition, you have to make a measurement, which changes your quantum state (and we’ll also discuss this more in a later episode). Let me know if this doesn’t clear it up! -QuantumSense

[@kennethhou912]

could the ket of some particle be thought of as the weighted (by probability) summation of all possible positions?

[@angelmendez-rivera351]

You are close, but not quite there. A superposition is indeed just a weighted summation of possible "elementary" states, as you suggest, but those states often have nothing to do with position. What these states are ultimately depends on what exactly the system is.

[@samsonling3142]

when will the square of wavefunction kick in to be probability density function of position? Is that we do an inner product?

[@ayhamhalalsheh221]

that was adorable

[@user-ui5lc3kp7g]

Please make a same for General Relativity

[@exploring197]

Please explain about hermitian conjugate? Physical significance of wavefunction being hermitian.

[@quantumsensechannel]

Hello, thank you for watching. I have an episode released on hermitian operators, where we define what they are. Also, in general the wavefunction is not hermitian (since it can be complex). -QuantumSense

[@yuminti3368]

I still find it hard to twist my mind around vector space is just describing patern because in my mind I see vector as arrows. It would be great if you could show me an example of vector space made by a different set of object! Very please!

[@bharath__100]

4:47 - is it like, we can use any operator to find a quantum state? Like energy operator or momentum operator?

[@drdca8263]

In some systems, some operators will have for each possible value you might measure for it, a 1D space of vectors, and in this case this works as a nice basis for the vector space. In many systems, this will be true for energy. However, not all operators will, by themselves, pick out a good basis.

[@enderw88]

Does anyone know of a textbook that takes this approach?

[@kennethhou912]

is the fact that the linear combination of outcome kets equaling the quantum state an axiom or a consequence?

[@quantumsensechannel]

Hello! Thank you for watching, this is a great question. In truth, it is an axiom of the quantum framework. We haven't derived this fact, since we have nothing to derive it from! But given what we showed in the first episode, hopefully it makes some intuitive sense why we would have such an axiom in our quantum theory. -QuantumSense

[@kennethhou912]

@@quantumsensechannel thanks so much for the response! it does make sense why it would be an axiom of the system rather than a consequence of how addition and vectors are defined. i can’t wait to continue exploring your series!

[@SSNewberry]

The vector space requirement are the axioms for vectors.

[@leventegyorgydeak1300]

10:13 - There is something I havent understood for a long time here. psi is in position representation, right? Here you just turn the position wave function into a "continuous vector". However psi can also be expressed in terms of momentum, then it would be |psi> = integral(c(p)*|p>) right? but that means that |psi> = integral(psi(x)*|x>) = integral(c(p)*|p>) which I am pretty sure is not true. Do those psi-s then represent a different hilbert space element, and it is just poor notation that we use the same letters for them? Can someone please explain?

[@TJ-hs1qm]

So this is how you calculate the expected value with the wave function representing the probability density ?

[@angeldude101]

Sure, I can take an infinitely long vector written as an integral. It really is just a linear combination of basis vectors, just with a continuous range of such basis vectors.

[@agentprismarine2778]

5:53 isn't the smallest possible length the plank length ? Which should make measures of length discrete?

[@amoghk.m.6769]

The plank length is many many orders smaller than the length scales we are operating at.

[@jaybae8056]

so what does: (-1/2)del squared minus 1/r) |2s》 mean?

[@vatsuu8865]

How do you even un descritize the position at 9:00

[@pefactz9.9m3]

Good❤❤❤❤

[@kennethhou912]

how important is the knowledge that the mapping of a ket to it's probability is continuous to the calculation of the integral?

[@quantumsensechannel]

Hello! The continuity of the coefficient function is actually very important, and in all honesty, I felt kind of bad brushing it off to later in the series. Remember that the coefficient function is the wavefunction, so we're asking how important the continuity of the wavefunction is. If you've ever solved the Schrodinger equation before, you might have seen that continuity is a consequence of solving that equation. More intuitively, we'll show that the momentum operator is proportional to the first derivative of the wavefunction. So if our wavefunction weren't continuous, then the resulting derivative would blow up at a point, which gives us nonsense for the resulting momentum. This is more of a physical interpretation, but I think it gives good intuition regardless. Hopefully this answered some of your question! -QuantumSense

[@kennethhou912]

@@quantumsensechannel not going to lie, this is my first introduction to quantum mechanics. I am simply a math major that decided to learn quantum mechanics out of interest, but it is cool to see that there is a proof for why the coefficient is always continuous. hopefully my questions aren’t too annoying, and thank you for the time you take to answer them!

[@HilbertXVI]

@@kennethhou912 If you're a math major check out Brian Hall's "Quantum Theory for Mathematicians". It's very rigorous and probably much better for a mathematically inclined person than the average QM textbook.

[@PhotonicJerk]

You are saying that KET is nothing but another form of vector notation. Does this mean that It is the same plain old vector that we're used to or is it just an analogy? At 4:12 in the linear combination you have used energies in the KET notation. As far as I know energy is not a vector. I believe I am missing something but I am not sure what.

[@narfwhals7843]

What a vector is is defined earlier in the video. At 1:39. Objects that obey these rules are vectors. If by "plain old vector" you mean arrow, then sort of. Arrows generally are vectors. So you can use the vector addition rules you are used to for an intuition. Energy itself is not a vector. But Energy _states_ are objects in our vector space. The energy of that state is the measurement outcome and just a number, but we can collect the different possibilities of outcomes into a vector. Similar to how a basis vector can basically be represented by a single number because all the other coefficients are 0.

[@angelmendez-rivera351]

This is why actually taking a linear algebra course, as was explicitly recommended in Chapter 1, is important. This video series is not meant to teach you linear algebra. This video is meant for you to already know linear algebra, and from there, to build on top of those linear-algebraic concepts to achieve an understanding of quantum mechanics.

[@Nightmare-iq9tb]

can somebody please tell the font used for psi vector at 3:11

[@quantumsensechannel]

Hello, It’s the default math typeface used by LaTex, which I believe is Latin Modern Math. -QuantumSense

[@Nightmare-iq9tb]

Thanks a lot!

[@kurrennischal235]

An integral over a vector space, summing up infinite vectors to produce another vector. Is there a name for this notion that I can google?

[@angelmendez-rivera351]

Formal sums

[@lolsadboi3895]

whenever I hear you refer to kets as vectors, I keep on wanting to ask "how many dimensions does a ket have? How is 'all the information about a particle' arranged in the vector? Why is it a vector instead of a matrix or whatever has more dimensions than a matrix?" 11:22 throws me off more because looks like you can say |ψ> = |ψ> ∴ c₁|E₁> + c₂|E₂> + c₃|E₃> + c₄|E₄> = a₁|L₁> + a₂|L₂> + a₃|L₃> ∴ energy=angular momentum which,,, I don't think is right... i mean, they're related for sure but they're not equal, right? i'm confused by the notation x.x