Ch 12: What are generators in classical mechanics? | Maths of Quantum Mechanics

Hello!

This is the twelfth chapter in my series "Maths of Quantum Mechanics." In this episode, we'll take a detour into classical physics to learn about generators in the Lagrangian framework. We'll see that each physical quantity generates a change in the state of our particle, which will be key in deriving the Schrodinger equation next episode.

If you have any questions or comments, shoot me an email at:
Thanks!

Animations:
All animations created by me within Python, using Manim. To learn more about Manim and to support the community, visit here:

Music:
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♪ Imagination by Rosentwig
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[mohammedbelgoumri]

This is the first piece of content that gave me the impression that I understand Noether's theroem. Keep up the great work.

[pizzarickk]

The best thing about this series is that it develops the theory and shows the motivations for each concept instead of just presenting it.
can you provide us textbooks that explain quantum mechanics this way? that'd be much appreciated

[aafeer]

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.

[lexinwonderland]

waaaait, this is the last video out?! WAIT THATS 8 HOURS AGO?! shit, king, i'm keeping up with this series real time, keep it up it's been fantastic and i love the insight you provide to your students. your commentary on the commutator was especially unique and insightful!

[DarkNight]

What an amazing way of explaining concepts. This guy understands how human brain and thinking works and that's why every concept makes sense to everyone. Extremely few people posses such ability and intuition. A lot of respect for you.

[ernestoherreralegorreta]

Best explanation of the Lagrangian I've come across. You've got yourself a new subscriber. Thank you for sharing this!

[stevenschilizzi]

Another brilliant and illuminating exposition. Thanks for all your effort. Your work will stand as a monument to introducing QM.

[peterhunt]

These are some of the best videos I have seen. You keep things simple! The presentation is really crisp and clear, so one can actually see the equations clearly. Nice work. Thank you.

[MichaelGowland]

This is wonderfully clear and accessible.

I do find that one of the problems with math teaching is that maths is easiest for people who enjoy playing the game of manipulating symbols according to a set of rules without actually worrying about semantics, indeed in the case of abstract algebra without having any semantic content. But for those of us of a more philosophical disposition, the questions of what does it all mean and why do we do it like this are central.

I am minded of an event in the second year of my undergraduate course in symbolic logic, where the lecturer had been developing a symbolic language and a set theoretical model built out of the empty set, when one of the students put his hand up and said, "I don't understand what you are doing!"

The lecturer tried several times repeating the derivation of the WWF he was demonstrating, finally in exasperation he said, "Look its just addition".

At that point you could see a look of relief and understanding sweep across the faces of half the class. The lecturer had never even thought it relevant to explain he was developing a set theoretical model of Peano arithmetic, at the start if the class he simply laid out the symbols of set theory and the language he was going to develop, and explained the rules we were going to use for manipulating them - so even those of us who could follow the game of symbol manipulation he was playing for the most part had no idea why he was doing it.

These videos very much remedy that issue where the math of QM is concerned.

[me-kihs]

Your video series is like a visual novel, it shows how physical quantities interact with each other.

[ronoronyi]

This is by far the best video I have come across in building intuition on lagrangians. Good work.

[wadelamble]

Your videos are among the best of the best, and I love your particular angle of trying to derive as much as possible from intuition and reasonable guesses. Thank you.

I have a question (perhaps an argument, actually) with this video.

I don't see how del-L/del-A = dB/dt says "B generates A." From a purely math definition, I think "generator" means, more or less, g(t) = exp(at), where a is a certain group element, t is some parameter, and g is any other group element. Or, in differential form, dg(t)/dt = ag(t). We should then have, choosing the "momentum generates position" example, that del-L/del-x = dp/dt <=> dx(t)/dt = px(t), which I don't see. Also, if L is our "state, " time translation would mean, I would think, any change to L over t. But del-L/del-t by definition only includes explicit time dependence of L on t. But L surely changes when there is no explicit dependence on time. In other words, equating "del-L/del-t = -dE/dt" to "energy generates time" would imply that there is no change to our state under constant energy, but that is clearly not the case. I sense this comes back to analogizing L to the |psi> as being the "full state, " even though I do get that knowing the functional form and value of the Lagrangian provides a full description of the state.

[MikeMagTech]

Thank you. I enjoy your channel very much.

[voidisyinyangvoidisyinyang]

that's a great phrase to drop into conversations, "I'm just gonna state without derivation"

[rpgtalkout]

We would love a intuitive video on lagrangian mechanics!:D

[strawberry_cake]

5:32 pls do so someday... I've seen proves which work on paper but with those I have no idea what and why we're manipulating the things we're manipulating to get the equation.

[narfwhals]

It doesn't seem clear to me why a change in position is generated by a _change_ in momentum rather than a constant one.
Does this mean that, if the lagrangian does not change with position, a constant momentum has no physical effect?

I would have actually expected a short aside on conservation laws (If partialL/partialt=0 then dE/dt=0 meaning Energy is conserved)

[xiaolian]

it is reallly a fantastic vedio, keep up your work!

[jimmylander]

Me, a computer scientist: "Ah yes. So this is the one that makes me cry. Finally."

[azizurrehman]

It is very series for understanding quantum mechanic