Ch 13: Where does the Schrödinger equation come from? | Maths of Quantum Mechanics

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Hello!
This is the thirteenth chapter in my series "Maths of Quantum Mechanics." In this episode, we'll finally understand where the Schrödinger equation comes from. We'll first derive that time evolution must be unitary, then use that to get to the form of the equation. We'll then use the fact that energy is the generator of time evolution to take it all the way home (after fixing the units).
If you have any questions or comments, shoot me an email at:
quantumsensechannel@gmail.com
Thanks!
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Опубликовано:

27 сен 2024

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Комментарии : 172

@quantumsensechannel Год назад

Hi everyone! A quick note: In rewatching the video, I don’t think I give enough credit to what Planck’s constant is doing in the Schrodinger equation. It’s true, it really does settle the units, but this is way more important than you might realize. The Schrodinger equation has energy and time in it, but how big/small are these time energies? Mega Joule seconds? Pico Joule seconds? On its own, we don’t know the energy-time scales that show up in the Schrodinger equation. So Planck’s constant does exactly this. It tells us that the energy-time scale is ~10^-34 Joule seconds, ie really really really small!!! So Planck’s constant sets the scale of Schrodinger’s equation (and hence quantum physics) to be on incredibly small energy and time scales!!! So the hbar being there is the reason we don’t regularly see our dog in a superposition of two states: you need to be in tiny energy and time scales to see the effects of quantum mechanics. Hopefully this gave Planck’s constant the respect it deserves in the equation! Units are important because they set the scales of physics! -QuantumSense

@devangshuroy5170 Год назад

Where is the 14th vedio? Am I not allowed to watch it?!!

@davidblauyoutube Год назад

Schrodinger's... dog?

@Alan-zf2tt Год назад

Hmmmmm ... could this imply that there are no real continuums but the least contiguums exist to the order of 10^(-34) ? I hope so - just to see what effect it has on the square root of 2 🙂

@voidisyinyangvoidisyinyang885 Год назад

@@Alan-zf2tt yes study noncommutativity of Alain Connes.

@Alan-zf2tt Год назад

@@voidisyinyangvoidisyinyang885 Downloaded the pdf - will study it. I must admit to being drawn to algebraic/differential geometries. They seem so close to reality and realism. Plus, any measure in physical space more-or-less assumes that empty space is well defined otherwise measures do not work?

@thatdude_93 Год назад

I think you left out the definig characterisitc of one parameter unitary groups U(t), namely that U(t+s)=U(t)U(s), which also makes a lot of sense intuitively. With this the proof that each U(t) is indeed unitary can be shortened quite a bit, also it automatically implies the existence of the inverse U(t)^-1=U(-t). On another note, due to Stones theorem, each unitary group can be written as U(t)=exp(itA) for some unique self-adjoint operator A. This is what i find fascinating. Each unitary group U(t) determines an observable A, and each observable A determines a unitary group via U(t)=exp(itA). It's similar to Noethers theorem in classical mechanics, that each symmetry of a system corresponds to some conserved quantity

@cogwheel42 Год назад

I saw "U(t+s)=U(t)U(s)" and thought "that sure looks like something exponents/logs would do. I wonder how they're related?" And then you delivered

@Woollzable Год назад

@@cogwheel42 You can watch a similar discussion here by Prof Leonard Susskind: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-8mi0PoPvLvs.html I copied the link so that it starts at 29:38. You'll see the derivation which includes writing the time evolution operator explicitly as an exponential. Though he doesn't do it in a mathematically rigorous way, it's still nice.

@Juxtaposed1Nmotion Год назад

The best comments are always in the comments

@DKonigsbach 8 месяцев назад

The best series on quantum mechanics I've seen. If mathematics and physics textbooks were written using your approach, more students would understand and enjoy those subjects. Thank you!!! Please continue your work. With luck, you could become an inspiration for others in how to teach.

@NPCtje 6 месяцев назад

Bro this shit blew my mind. I was easily able to use schrodinger equation but the connection to classical mechanics is enlightened my perception of the formula.

@schmetterling4477 3 месяца назад

That connection is, unfortunately, false. Try to connect superconductivity to classical mechanics and see where that gets you. ;-)

@ReyhanMehta Год назад

Hey Brandon, your Maths of Quantum Mechanics series is amazing! Keep up the great work, and thanks for creating such beautiful, quality content.

@rachidbz01 Год назад

Thank you for making physics as fun and comprehensive as it should be, Huge respect for you Dr Brandon and I hope you continue doing more to humanity!

@jamesu8033 Год назад

As someone who’s just done an intro to quantum course at uni this series is really brilliant. We didn’t really approach quantum mechanics from the same kind of mathematical perspective of this series. We just started from the schrodinger equation and worked through increasingly complex systems up to hydrogen and helium and the periodic table. But we never really went into the underlying linear algebra with which you do quantum mechanics - we didn’t even learn Dirac notation.. It’s just kinda annoying that all this maths was sort of hidden from view so that we could just trudge through all the usual quantum systems that every undergrad has to learn.

@jaunesmith0947 Год назад

I feel like this series is going to save me a LOT of headache later in my uni course, much like what 3B1B did for me in the past.

@QuarkyMathChannel Год назад

Your series has been helping me so much in my undergrad QM course. Thank you for all the time you put into making these videos, they are both aesthetically pleasing and easy to understand. Would you consider expanding this series to discuss more topics in QM such as expectation values, transmission probabilities, and different potential profiles (ie. Delta Potential, SHO, QMW, etc.)? I hope that your graduate studies are going well so far, and keep up the excellent work!

@BapiKAR Год назад

You rightly reignited my passion to learn QM, which was lost in the last decade or so. Thank you very much from the bottom of my heart.

@ItsJushJosh Год назад

I have really been struggling while learning linear algebra and feeling like I have any appreciation for how all these different concepts I'm learning work together to do something meaningful. This video series has not only been incredibly helpful to me in crystalizing many linear algebra concepts through a fascinating application, but has provided me with fresh motivation to move forward. Really fantastic work. Best of luck in your doctorate work and I can't wait to see more!

@cheesecake370 Год назад

This is the best video series on RU-vid! Thank you!

@krupt5995 Год назад

Beautiful video once again

@franciscopereira2993 Год назад

For someone who is just now starting a course on modern physics (basic concepts of quantum mechanics) I am loving this course although I wont need it until the next semester when I actually take Quantum Mechanics I. Beautifull job!

@146fallon Год назад

You are a legend. Please continue the series.

@mastermatt1958 Год назад

Thanks for the amazing content! :)))) It gave me so much understanding! I am currently taking a Theoretical Quantum Physics Class at TUM and this series really gave me more of an intuitive approach to quantum physics. Really great work, big Thanks and greetings from Austria! :) PS: Looking forward to the next parts of the series :)

@ViliamRockai Год назад

Is this your first channel or do you have more content elsewhere? Your voice is a bit familiar :) This series is fantastic, I want to see everything you've ever made!

@b1ackwollf 5 месяцев назад

this is what i needed which griffiths failed to provide! thank you! please do some more of quantums please. We need a lot of intuitions and perspective in this

@aafeer2227 10 месяцев назад

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.

@namitkamani4732 Год назад

Waiting for your Next video

@corbinh3052 Год назад

fantastic series! looking forward to whatever you release next

@syedkazim786 Год назад

Thank you so much for such an amazing series. Anxiously waiting for next episodes. Kindly upload them soon! Thanks again

@Self-Duality Год назад

This makes quantum sense!

@quantumsensechannel Год назад

Thank you for watching, and I’m glad it made sense! As a side note, I wonder how many people caught on to the idea that “quantum sense” is supposed to be a play off of “common sense”. Either way, I hope this series is making quantum mechanics feel like common sense! -QuantumSense

@Self-Duality Год назад

@@quantumsensechannel It’s a lovely play on the phrase “common sense”! I hope your presentations reach many!

@Miguel_Noether Год назад

@@quantumsensechannel I didn't get that until this comment and your answer 😂 I think it is language dependent (mexican here)

@markev Год назад

Great series! - many thanks for this. One question: If I understand correctly, in the classical case, it is the rate of change of energy which is proportional to the change in the Lagrangian over time. Here in the quantum case, it seems like it is energy itself (not its rate of change) that is proportional to the change in the quantum state over time. Why the difference?

@quantumsensechannel Год назад

Hello, thank you for watching. In truth, I don’t really have a good answer to this besides “that’s what the math led us to.” I’ve tried to find a better reason, but the best I have is that I think it’s an artifact from when we transitioned to thinking about physical observables as operators. Operators are transformations in our space, they “do” something to our quantum state by acting on it. But in classical mechanics, physical quantities are functions, and functions don’t really act on anything. So the way they “act” is by changing through time, hence we get a derivative with time in classical mechanics. Again, this is just me speaking off the top of my brain, and it’s just a loose heuristic. But who knows, maybe there is a deeper reason! Definitely worth thinking about. -QuantumSense

@pepaxxxsvinka3379 Год назад

Is there going to be a continuation of the series? it is incredible!

@mosalethoba5267 Год назад

Please allow save button some of us are normally busy so wed like to save and watch when we free from work please .and when we not on the wifi . Your series is the best out there .

@banderfargoyl Год назад

Very well presented. Thank you.

@alexpapas99 Год назад

Loves these videos! When are we getting the rest? ❤

@mastershooter64 Год назад

more quantum when? relativistic quantum when?

@ziyingwang2113 6 месяцев назад

the best derivation video!

@riccardoguareschi193 4 месяца назад

Well done. 15’ of great physics

@creativenametxt2960 Год назад

What's the intuition behind time evolution being the same as multiplyng by a linear operator and not some arbitrarily chosen function? To rephrase, why do we expect the time evolution function to be linear?

@cademosley4886 Год назад

Instead of Joule seconds, which is kind of hard to picture, I more intuitively understand h-bar as Joules per Hertz (energy per frequency or energy / (1/time)), that is, the amount of energy the system gets for each cycle around the complex plane that that "i" is contributing to the state in the time evolution. I think it's something like that. But I think we need to wait for a future video to see more explicitly how a frequency gets into the picture.

@quantumsensechannel Год назад

Hello, thank you for watching. And this is an interpretation of hbar that I’ve never considered, but I think is very intuitive! I’d have to think a bit more to make it more concrete, but it makes for good intuition (especially once the time evolution phase factor is introduced, as you say). I’ll remember this interpretation moving forward! -QuantumSense

@voidisyinyangvoidisyinyang885 Год назад

study noncommutativity. Penrose points out the origin of the de Broglie-Einstein principle is noncommutative nonlocality. Same with Basil J. Hiley and Alain Connes. negative frequency should not be discounted. Quantum algebra is more foundational as a process than geometry. Math Professor Lou Kauffman explains this also. thanks

@baptiste5216 Год назад

This was amazing thank you

@picobarco4407 Год назад

ALL I wanted to say is, "I am Amazed as to how you were able to show HOW the imaginary number 'i' got into the Schrödinger equation. The closest explanation was via Partial Differential Equations theory, where if you put complex/imaginary 'i' into the Diffusion Equation, the solutions would yield Sinusoidal type functions, just like the Wave Equation of classical Partial Differential Equations. But I see from your approach, you came at it from Linear Algebra from idea of Hermitian Operators. I can't believe I did not see this way of thinking about it, it's due to the fact that for some reason I forgot that the Hamilitonian is a Hermitian or Anti-Hermitian operator. SO it makes sense that we need to include this basic property, that we learn from Linear Algebra. This complex/imaginary number, was something that seemed to kind of come out of no place, it was a strange mystery to me. Thanks for clearing this up, and clearing it up so "Naturally", it is really really great what you have shown here! Thanks so much!

@pippenpoodle 8 месяцев назад

7:12 what exactly is that symbol used to reference the future terms in the taylor expansion

@jesussanchezherrero5659 Год назад

I love your videos. Please show us more!

@pizzarickk333 Год назад

12:45 I find it not convincing and arbitrary to introduce constants out of nowhere inorder to make our equations work. If our deductive math steps were correct, why haven't we arrived at an equation that has units that match? isn't this a sign that there's something wrong with our steps?

@quantumsensechannel Год назад

Hello, thank you for watching! And ah, I think you are thinking too much like a mathematician! In terms of the math, yes, it is somewhat arbitrary and “illegal” feeling. But as a physics equation, this “constant” insertion should actually be comforting. First, math really doesn’t care about units, so we can have 100% correct math and still have incorrect dimensions. As far as pure math is concerned, a mathematical object is just some abstract object. But it goes deeper than that. Inserting a constant actually does something incredibly important: it sets the scales of quantum mechanics. We have energy and time in the schrodinger equation, but how large/small are these time-energies? Joules seconds? Mega joule seconds? Pico joule seconds? On its own, math doesn’t know the overall scale of quantum mechanics, this is our job to figure out as physicists, and is the reason experimental physicists still have a job. So by inserting plancks constant, we’re saying that the scale of Schrodingers equation is ~10^-34 Joule seconds, which is really really really small. So inserting that constant actually sets the scale of quantum mechanics to be really really small, which is what we know to be true. Math doesn’t know that on the outset, math couldn’t care less about what the scale of quantum mechanics is. But we care, and hence that constant is actually incredibly important (hopefully now intuitive) as physicists. Let me know if this didn’t clear it up! PS: This is also somewhat why in advanced theoretical physics, we work in “natural units”, where we set hbar = c = 1. We work in nature’s physics scales, and just let the math run free without these annoying constants. But they are still there! We’re just assuming that we are working in whatever scale is set by those constants. -QuantumSense

@pavelrozhkov3239 Год назад

H operator we got here, without reduced plank's constant, correspond not to energy, but to frequency. It's de Broglie relation that connects energy and frequensy by a constant, and it was verified experimentaly.

@quantumsensechannel Год назад

@@pavelrozhkov3239 hello, thanks for watching. And in that case, we are still assuming that we have H/hbar, and hence we are still doing some constant introduction. And we want H to really be the energy operator, so in that case, we do indeed need that extra constant. As we have framed quantum mechanics, we do not have a “frequency” operator, frequency is only introduced through dividing the energy eigenvalues by hbar, as you say. -QuantumSense

@pizzarickk333 Год назад

@@quantumsensechannel Thank you for replying! I understand that it's has important consequences for setting the scale with which we work, but I'm stil wrapping my head around why it is a reasonable thing to do.

@quantumsensechannel Год назад

@@pizzarickk333 hello, I completely understand why it feels weird. The trouble comes from the fact that we suddenly introduced dimensioned quantities in an undimensioned theory, and so we are sort of retroactively fixing it. If we wanted to introduce hbar “rigorously”, we would define a “dimensionless” time parameter t/tau, and dimensionless energy operator H/eta. So this way we’d keep with dimensionless QM the entire time. Putting this in the schrodinger equation would give us the product tau*eta in the equation, which we would then call a new constant that has to be experimentally measured. This is a bit more rigorous, and we never introduced dimensioned quantities at all, so we stay consistent. But this is every bit as valid as our approach, and they’re both worth understanding. -QuantumSense

@senaveli 4 месяца назад

at 8:48, when we multiply the two berackets, we got one identity operator from the multiplication of the two identitiy operators in both the brackets. don't we get another one from the multiplication of the dagger \dot U with \dot U? so shouldn't there be two identity operators on the right side of the equation at 9:01?

@quantum4everyone Год назад

Another great video. I think you have to be very careful with your proof of unitarity in the way you chose to prove it. While it is easy to see that the operator you are working with is Hermitian, one actually has to show it is self adjoint in order to use the spectral theorem as you do. It may be better to start from probability conservation as a requirement for time evolution from the start and the fact that unitary operators preserve inner products, and hence preserve probability. I personally prefer the semi group property to show the exponential behavior of the evolution operator (for constant Hamiltonians) because it does not require using the differential equation. Sure that does not then derive the Schroedinger equation, but one does not really need the Schroedinger equation if you want to work in its “integrated” form. Then using the Trotter product formula, you obtain a clean “derivation” of the time-ordered product.

@ashutoshtechnology Год назад

While checking hermiticity of time evolution operator you take adj.(U*.U)* as U*.U but it should be U.U* until they commute. So do time evolution and unitary operators commute ?

@observer137 Год назад

Most of the arguments can be applied to a real vector instead of a quantum state, until the final part where the operator matrix becomes anti-symmetric instead of anti-hermitian. Anti-hermitian operator can be written as i*H which H is hermitian and hermitian matrix has real eigenvalues. But anti-symmetric matrix cannot be written as something*symmetric matrix with real eigenvalues. Is this the origin of the i from reality?

@nicolasPi_ 9 месяцев назад

Can we derive the equation of understanding? I started at chapter 1 with the definite state 'understand'. But after some time watching more chapters, I evolved into a superposition of 'understand' and 'not understand'.

@Qrudi234 Год назад

Could i ask; Why are there so many complex numbers in quantum mechanics?

@beta3physiaacademy-925 9 месяцев назад

What book do you use my friend?

@KevinZomberTV Год назад

Love this videos as a bachelor physicist

@cogwheel42 Год назад

My reward is that I get to watch this 3 more times and then maybe understand a bit of it

@quantumsensechannel Год назад

Hello! Thank you for watching! And don’t be afraid to ask questions! I want to make the derivation understandable, so let me know if I can help you with anything that didn’t quite make sense. -QuantumSense

@cogwheel42 Год назад

@@quantumsensechannel My comment was more about my own information absorbing process. First pass is usually pretty superficial. I can follow along and get the gist. Second pass I see some of the things I missed or misinterpreted the first time, and fill in details. Third pass is when it really starts to click. That said, there were a couple moments earlier in the series where an explanation was followed with "I hope you can see how that leads to X". I couldn't really at the time, but maybe just need another pass. If I notice them again, I'll point them out.

@cogwheel42 Год назад

And as a counterexample, I was also surprised to find myself anticipating some of the conclusions in this video before you worked through them. (like when moving the operators to the middle of the ket, how it must equal zero)

@yenbinh2239 Год назад

This series is great, which book should I read to further understand those concepts ?

@Dekoherence-ii8pw 9 месяцев назад

GRIFFITHS

@phildurre9492 Год назад

Perfect series! I am left with one question, if i understand this correctly, conserving probabilities makes only sense in an unchanged quantumstate? Ass soon as something observable happens like emission of a photon, the waveform collapses, probabilities change and the whole thing can not be applied? So quantum theory is a good theory as long as nothing happens, we can predict energy levels and probabilities, but ass soon as me measure something, or anything interacts with the system it’s completely useless? Thats very unsatisfying.😰

@quantumsensechannel Год назад

Hello! Thank you for watching! And you bring up an interesting point, and you have just derived the fact that *measurement in quantum mechanics is not a unitary process* . Exactly as you say, collapsing the wavefunction modifies all probabilities to coincide with being in a particular eigenstate, so measurement can’t be unitary. That being said, the whole framework of QM still applies. You just start applying the Schrodinger equation after you’ve made the measurement, using the eigenstate as your initial state. Is it weird that we have to do this? Absolutely, and much debate still exists in physics about the nature of collapse. But the framework and theory itself works just fine. So don’t let that one fact deter you from seeing QM as a wholly satisfying theory! -QuantumSense

@starocket8710 Год назад

IIIIIIII nnnneeeeeeeeddddddd mmoooooorrreeeeee of your videos pleaseeeeeeee!!! I hate how this subject is taught in my class

@briansteven6318 Год назад

Gracias por la serie.

@Alan-zf2tt Год назад

What amazes me is how pure math and pure/applied physics sort of coalesce into something that closely models reality. But I guess that is in the nature of mathematics plus! a couple of questions if I may. 1theimportanceofspaceandpunctuation or rather 1 - the importance of space and punctuation (if you see what I mean)? EDIT: or more strictly speaking fsctioneampndunctppceottaa1heiornu the importance of ordering space and punctuation if you see what I mean 🙂 and 2 - does the value and units attributed to Plank's constant imply the event space is not a continuity but a very dense quotient space? (sorta helps with a leap from micro-physics to macro-physics and allows for things clumping together ~ a continuity is a continuity - quotient in time and also quotient in energy) okay 3 (sheesh) 3 - space seems too well behaved to be a no-thing so maybe relatively matter free space is a relatively non-thing space therefore not a no thing space? (difference between an almost empty set and a null set)

@kansuerdem2799 Год назад

Deep Respect !

@danielwallace1968 Год назад

This is a great video!

@subhashishmukherjee405 Год назад

Loving it!

@beta3physiaacademy-925 9 месяцев назад

I have a question, why did we bother ourselves with proving that Udager is U^-1 isn't it enough to say if UUdager=I then Udager= U^-1

@yuminti3368 4 месяца назад

By evolving backwards in time, i believe you meant we can back track to find where is was.not we go back in time. So i see! Breaking time symmetry means we can not back track Which state it was in!

@schmetterling4477 3 месяца назад

Which state what is in? A single quantum system is in no state whatsoever. The ensemble is in a state or, more generally, in a mix of states.

@kisho2679 4 месяца назад

How can be verified that probability remains invariant (conserved) over time evolution?

@schmetterling4477 3 месяца назад

That's an axiom of the theory. In an actual experiment you will inevitably see losses and we account for those with concepts like "quantum efficiency".

@michaelschnell5633 Год назад

Just Great !

@Inceyb Год назад

HI I think it would be great if you could make a video on bound and scattering states and the infinite potential well.

@Dekoherence-ii8pw 9 месяцев назад

Brant Carlson

@aramsarkisyan8061 Год назад

can someone help mi out with calculation at 9:00 ? Why we only got 4 arguments left. Does smth cancel out?

@krupt5995 Год назад

There are infinite arguments actually. The big O just shows that the rest of them are a coefficient times a dt² or dt³ e.t.c. So when you calculate that expression and you multiply for instance U(dagger)(0)dt with U(0)dt this becomes a term of order dt²

@wdobni Год назад

i don't see how you can state that time evolution is reversible (if i am quoting you correctly) ... how can you state that we can move forwards and backwards through time when nobody and nothing has ever moved backwards in time

@quantumsensechannel Год назад

Hello, thank you for watching. Note that time reversal is a statement on the equations of motion of our system. Note that this is possible in classical mechanics as well. Write down Newton’s second law, and replace t with -t (so you are now moving backwards in time). The laws of motion would still be valid! This isn’t saying that a particle can be sent backwards in time, it’s just saying that it’s possible to mathematically trace back your steps to get back exactly where you were. Let me know if this didn’t clear it up. -QuantumSense

@terry1357913579 Год назад

Although it should be obvious but I doubt that if I really correct. In the whole series of the video, you talk about physical observable and could be call an operator. one of the example would be the Ê. So, what exactly is the operator Ê? What is the physical meaning of Ê. This is one of the things that I still cannot get it through your video. Does it means that if you measure the Energy of a quantum state, you apply some Ê on |ψ> and you should get the result ( probability ) of that particular Engery State Ê?

@clmasse Год назад

It is iℏ∂/∂t. The energy is associated with the evolution in time, like in classical mechanics. The Hamiltonian is the energy as a function of the coordinates and the conjugate momenta, so that naturally H |ψ> = iℏ∂/∂t |ψ>

@krupt5995 Год назад

I guess this was meant to be answered in the next video but I don't see it coming..

@lantonovbg 8 месяцев назад

just one word - rotation

@krupt5995 Год назад

When is the next video coming out?

@san99539 Год назад

Just wow

@coolman8488 Год назад

Hey there, I'm making a 5 minute on Shor's algorithm for college and I'd like to use manim to do so. I'm brand new to the library and I'm trying to find some resources on using bra-ket notation and I haven't found anything. Could you help me?

@quantumsensechannel Год назад

Hello! Thank you for watching. In using Manim for this series, I didn’t do much fancy for bra-ket notation. I wrote kets as | \psi angle and bras as \langle psi | . There do exist some packages, but I didn’t find them necessary. In terms of moving things around in bra ket notation, I just used the tools within Manim by shifting and fading things in/out. Hopefully this answered your question! -QuantumSense

@i7887 Год назад

For me I need the steps between seen.

@stefanopogany3767 4 месяца назад

What would happen if the time evolution operator were NOT unitary? 0:04

@schmetterling4477 3 месяца назад

Then we would lose (or pick up) quantum systems along the way. Experimental physicists account for that with terms like "quantum efficiency". There is no simple theory for these losses, so it's pointless to drag that into the theory itself, but it's being extensively discussed and dealt with by experimentalists.

@Miguel_Noether Год назад

There is a video from Science Asylum showing that this equation is not a wave equation, actually it's a diffusion equation, what is diffusing? The probability! check that video is very helpful

@clmasse Год назад

Yes, but this video is wrong. Many proficient physicists have pointed out that the imaginary unit changes everything, but he keep on with his rubbish. There is no diffusion equation for the probability density, there is an equation of continuity together with the probability current, which is of first order both in space and time. He censored me for having corrected him, he is not honest. Perhaps Quantum Sense will confirm.

@eliaderenda2041 Год назад

Chapter 14?

@MSloCvideos Год назад

Wonderful series. Will you expand it to N particle systems and relativistic QM?

@geoffreyfaust3443 3 месяца назад

Thank you for this series. In reference to the i in the equation, you refer to the i as needed to conserve probability. What do you think about what my browser's AI says, when I asked? "Does the i in the schrodinger equation conserve probability The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time-evolution of a quantum system. The equation is given by: iℏ(∂ψ/∂t) = Hψ where ψ is the wave function of the system, H is the Hamiltonian operator, i is the imaginary unit, ℏ is the reduced Planck constant, and t is time. The question of whether the i in the Schrödinger equation conserves probability is a subtle one. The answer is that the i does not directly conserve probability. However, the Schrödinger equation as a whole does conserve probability. Local Conservation of Probability The Schrödinger equation conserves probability in a local sense, meaning that the probability of finding the system in a particular state remains constant over time. This is because the probability density |ψ(x,t)|² is a conserved quantity, meaning that it does not change over time. Non-Local Conservation of Probability However, the i in the Schrödinger equation does not directly conserve probability. In fact, the imaginary unit i is responsible for the non-local behavior of the wave function, which can lead to non-conservation of probability in certain situations. For example, in the presence of a potential barrier, the wave function can tunnel through the barrier, leading to a non-local transfer of probability. This is known as quantum tunneling. Conclusion In conclusion, the i in the Schrödinger equation does not directly conserve probability. However, the Schrödinger equation as a whole conserves probability in a local sense, meaning that the probability of finding the system in a particular state remains constant over time. The non-local behavior of the wave function, facilitated by the imaginary unit i, can lead to non-conservation of probability in certain situations." THANKS AGAIN FOR THE SERIES OF VIDEOS! Geoffrey Faust

@narfwhals7843 Год назад

Very cool video and a great payoff for the work done so far. It's really interesting to see different ways to derive the Schrödinger Equation. In this video (I hope its okay to link another channel) it is derived from E=KE+PE and the assumption we need is that we want to find a wave solution for the de Broglie relations. Imaginary numbers then pop up because we need to get a wave solution from a first order time derivative. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-2WPA1L9uJqo.html A question: at 1:50 you say that U(0) should be the identity, but our _physical_ intuition only tells us about the physics. So U(0) could in principle introduce a complex phase that doesn't change the physics of the state, right?

@cleon_teunissen Год назад

In an earlier comment I had linked to that same video. Here is my take: in order to satisfy the de Broglie relation the time derivative must be first order. In order to accomodate that a property is introduced that acts as a phase. This phase property could have been mathematically implemented in a number of ways; using complex number notation happens to be a particularly efficient notation. My point is: it is about accomodating a phase property, it's not about the mathematical concept of complex numbers, that is just how it happens to be notated. Further reading: Article by the historian of physics Ricardo Karam, 2020: 'Schrödinger's original struggles with a complex wave function' Comparison: In electronics engineering: the oscillations in LC-circuits are modeled using exponential notation, because that makes going through the mathematical operations more efficient. So in the calculations there is a real current and an imaginary current. Karam describes that Schrödingers initial expectation for the new wave mechanics was an analogy of that; that he was using complex number notation only for making the operations proceed more efficiently, without attributing any physical significance. Over time Schrödinger recognized that the complex number notation was representing something significant.

@quantumsensechannel Год назад

Hello! Thank you for watching. And yes, I’ve seen that video and also found it incredible. I think it’s a statement on how many ways you can understand quantum mechanics. They took the de Broglie relation and wave dynamics as a starting point, and that is just another way of viewing QM. It reminds me of videos I’ve seen that discuss the uncertainty principle in terms of wave packets, while I discussed it in terms of commutators. Same principle, but completely different (and valid!) understandings. What you’re seeing is the correspondence between wave mechanics and matrix mechanics, but they’re both quantum mechanics. And in relation to your point about a complex phase, I would be iffy about that. Let’s say we had a linear combination |a> + |b>. Well we said that U(0) time evolved by zero seconds, so it leaves a state as it was. So there’s no consequence to applying it to the b state: |a> + U(0) |b>. If U(0) were a complex phase, then we’ve now introduced a relative phase between the states, which IS measurable! So if U(0) were a complex phase, then it would contradict our point that it “does nothing”. So now we would have the idea that a particle sitting there for 0seconds changes something, which doesn’t make sense. So it wouldn’t be an issue for a global phase, but for a relative phase, it would be a bit of a problem. -QuantumSense

@wjshih3786 7 месяцев назад

❤👍

@Jhaashish.004 Год назад

Sorry starting 12 are good but i don't understand this video it seems just mathmatical stuff only to beginner like me

@larianton1008 5 месяцев назад

This is the point where I drop

@Abc19853 Год назад

I didn't understand anything 1. Bcz it has 0 animations 2. You didn't explained about all the terms Plz make such videos again with following points

@Re_Soraa Год назад

Absolutely amazing. I've never thought of proving time evolution operator have to be unitary, that's so pretty and neat derivation, love it!

@wondererasl Год назад

At 2:05, "we want to use the intuition that we expect that time evolution is reversible." Why I do not have that intuition? And, our intuitions are shaky most of time.

@Duskull666 Год назад

Thank you so very much for this amazing video series! My Quantum Mechanics I professor literally butchered the foundation of quantum for me with his dry "shut up and calculate approach". He barley gave us any context, motivation or intuition for any of the derivation he thought us, so when i finished that course i could say i somewhat _knew_ the basics of quantum mechanics but i certainly didn't understand any of it.. but this video series really helped me fill in the gaps! Please keep up the good work! you are really helping physics students everywhere understand this wonderful and confusing subject!

@michaelgonzalez9058 Год назад

Shrodinger equate Hamiltonium

@gideonwiersma2794 Год назад

I currently following quantum physics 2 and this helped my a lot in understanding the basics! I am looking forward to the rest of the series.

@rafaelwendel1400 Год назад

6 days without a video, I'm dying on the inside

@mangiailpane8534 Год назад

Great lecture series, I am teaching a Molecular modelling grad level course this semester and I wish I could give students an indepth QM and CM intro before jumping into DFT and MD. Keep up the good work.

@liuwei3102 2 месяца назад

perfect videos. removing a lot of my confusions in QM. I recommend a Springer book to you, which says exactly I what you covered here: Quantum Mechanics Axiomatic Approach and Understanding Through Mathematics, written by Tapan Kumar Das.

@nerdphysics6402 4 месяца назад

OMG worth every second. And probably then most beautiful seconds of my life. The Beauty of quantum physics can't be explained better than this.

@silentbubble Год назад

What about time dependent Hamiltonian? It seems that this derivation is not valid for time dependent Hamiltonian.

@olix182 Год назад

Hi will be chapter 14 of your amazing series?

@tondog54 Год назад

Bro when you multiplied U dot by i my brain almost exploded. BOOM

@ryannunes2862 20 дней назад

Nice finish lol. Digging the series

@alex1lane Год назад

how do you make these videos?

@Higgsboson20 Год назад

where's episode 14 :')

@jaylenchiang1478 Год назад

Absolutely mind-blowing. If I may ask, did you figure out this elegantly understandable path of quantum mechanics all by yourself or was there a book (books) that mentioned these brilliant ideas?

@quantumsensechannel Год назад

Hello! Thank you for watching. In truth, everything in this series came from a combination of textbooks, lectures, and self derivation. For example: the proof that observables eigenstates must be orthogonal was inspired by a similar argument in Nielsen and Chuang’s quantum computing book. The derivation of Born’s rule was something I figured out myself after messing with the math for a while. And the derivation in this episode is largely inspired by a lecture given by Prof. Patrick Hayden, in which he did most of this derivation in the first lecture of my graduate QM course (you can imagine how amazed I was! The very first lecture and he shows us this!). So over the course of ~5 years, I’ve learned bits and pieces about QM from all over the place. And this series is just me putting it all together to share to the world. So I can give you some textbook recommendations, but my biggest recommendation is to learn from everyone and everything. Each source teaches you something different. After some time, all these viewpoints will give you an incredibly wide and thorough understanding of QM. Hopefully that somewhat answered your question! -QuantumSense

@jaylenchiang1478 Год назад

@@quantumsensechannel Wow, thanks a lot, not only for the reply but also the effort into these videos. Extensive learning on one topic greatly widens one's comprehension. I totally agree with you. I am a chemistry freshman now, and am really interested in quantum mechanics, so I would very much appreciate it if you could recommend some literature or sources!

@BlackHole-qw9qg Год назад

Will you do a "Maths of Quantum Field Theory" ?

@voidisyinyangvoidisyinyang885 Год назад

study Professor Jean Bricmont on Bell's Inequality first. He proves it debunks QFT.

@kartikrao7439 6 месяцев назад

Can someone recommend me a book, which dives even deeper into this?

@schmetterling4477 3 месяца назад

On how to derive quantum mechanics from first principles? It's not in any books I know, but search for papers on how to derive it from Kolmogorov's axioms. There are, at least, a couple out there that I have seen over the course of my life. They are not very interesting. In essence, Kolmogorov is a partition of unity. One can partition 1 in different ways: p + (1-p) =1 gives you probability theory. sin^2(phi) + cos^2(phi) = 1 gives you quantum mechanics. Combine the two and you get the density matrix. Why does Kolmogorov apply? Because quantum mechanics is a theory of ensembles of independent experiments. If we would start teaching quantum mechanics with that sentence, then 100% of all students would be 200% more comfortable with it. That's the difference ten unspoken words can make in your life. Don't make the same mistake with your girlfriend or wife. Never leave things unspoken that need to be said out loud. ;-)

@funbangla1333 Год назад

Thank you so much for making these videos. Please keep making them. These videos helped me intuitively "realize" the connection between quantum world and the macroscopic world in a very gullible manner.

@gvsgaius 3 месяца назад

Thank you so much for this series. I remember failing my Quantum Mechanics II class in university more than 10 years ago, and subsequently never really studying the field in any detail. This not only brings back a lot of old memory but also finally things are put into place in my brain. Really helpful and it feels very good to finally start to really understand the meaning of all these bras, kets and operators. Thanks!

@karthikasokan5741 Год назад

Waiting for the next set of videos!! Really couldn't wait to be honest. Thank you for these wonderful explanations!

@ihmejakki2731 Год назад

I was really puzzled about a lot of things in my qm classes last autumn. This series is a fantastic refresher and has helped me to finally understand many concepts!