Ch 12: What are generators in classical mechanics? | Maths of Quantum Mechanics
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- Опубліковано 30 січ 2023
- 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:
quantumsensechannel@gmail.com
Thanks!
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This is the first piece of content that gave me the impression that I understand Noether's theroem. Keep up the great work.
Literally, and I've looked tirelessly for content about derivations of it and the place I find it is one that doesn't even mention it lmao
@@penguinjuice7543 I don't know how useful this will be to you at this point, but I highly recommend seeing the videos made by "physics with Eliot" on the subject. I remember they were also very helpful (not only for noether's theorem, but for advanced physics in general: field theory, Lagrangian mechanics, ...).
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.
I am making notes for this series
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
Hello! Thank you for watching.
There are a lot of great quantum mechanics books, but unfortunately, I don’t know of any that approach the subject in the same way I am in this series. My best recommendation would be to learn QM from a variety of sources: textbooks, lectures, online PDFs, videos etc.
The textbooks that I have studied from (in approx order of difficulty) are Griffiths, Shankar, Sakurai, Nielsen and Chuang, and Ballentine. Each one has a different approach, and has taught me something different about the subject. These are just starting points, however - there are tons of other textbooks that you should also be open to!
-QuantumSense
One book thats not on the list of our beloved QuantumSense is Nourdine Zetteli's textbook, I think it has a thorough and clear walkthrough of the origins, the mathematics, and the posulates of quantum theory (at the level of senior undergraduate~) in the first three chapters. What I like is his emphasis on important results and the connections between them. Definetly check it out
@@UnknownBeast41 thanks for this valuable suggestion, i'll definitely check it out
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!
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.
Best explanation of the Lagrangian I've come across. You've got yourself a new subscriber. Thank you for sharing this!
Another brilliant and illuminating exposition. Thanks for all your effort. Your work will stand as a monument to introducing QM.
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.
Super excited for the next episode!!!
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.
This is by far the best video I have come across in building intuition on lagrangians. Good work.
Your video series is like a visual novel, it shows how physical quantities interact with each other.
Thank you. I enjoy your channel very much.
00:00 Introduction - Generators to help derive Schrodinger equation and Lagrangian formalism
00:52 : Reminder of Classical mechanics and introducing Lagrangian mechanics
06:07 Physics patterns in the change of the Lagrangian
11:25 : Different quantities as generators of changes in the Lagrangian wrt other quantities
12:41 : Relevance in quantum mechanics
that's a great phrase to drop into conversations, "I'm just gonna state without derivation"
it is reallly a fantastic vedio, keep up your work!
So Amazing!!!
Excellent!
We would love a intuitive video on lagrangian mechanics!:D
this is just great.
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.
Incredible😍
AMAZING!!!!!
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)
That's correct. As far as the physics are concerned, there is no distinction between a state of constant nonzero momentum, and a state of no momentum. You can transform between both reference frames almost trivially. An actual change in momentum, though, corresponds to a nonzero net force in the Newtonian framework.
Hello, thank you for watching!
And you bring up an interesting point in the first part of your comment. I’ve thought about that too, and I think it comes down to the following: we are only considering changes to the *state* of our particle. So yes, a constant momentum changes the position of our particle, but this still corresponds to the same physical state.
And this should make a lot of sense. Why? Because we can change the constant momentum of our particle by moving into a different inertial reference frame. We would expect that the “state” of our particle is basically unchanged when we do a Galilean boost into a different reference frame, and this is essentially what the equation is saying. So why does a constant momentum have basically no physical effect on the lagrangian? *Because physics should look the same in every inertial reference frame* . Only when momentum changes do we get more unique spatial changes to the state of our particle, because now forces are involved.
And I considered bringing in conservation laws, but I felt this was something most advanced classical physics classes already cover. But I haven’t really seen anyone cover generators like this, so I figured it was more unique and useful for people to see.
-QuantumSense
I think part of the reason for your confusion comes from the line around 7:09 "if we interpret the Lagrangian as the state of the particle, then changes in momentum seem to be indimately connected to changes in the position of our state". This is - in my probably uninformed opinion - a slight mischaracterization. I wouldn't go as far as to call it a mistake; Quantum Sense seems to know his stuff, and the point of the video was mostly to give a basic motivation, so he probably made the conscious decision to simplify it this way. If you are interested, I try to give a short clarification:
The Lagrangian itself is tyically not identified with the state of a system (neither in classical nor quantum mechanics). Instead it is something that gives you information about how the state will change into other states. In classical mechanics a "state" is usually just a point in phase space, i.e. a given position and a given momentum. This is because by just knowing the position and the momentum at one point in time you can typically calculate this trajectory of the particle for future times. Due to uncertainty that can't be the case in quantum physics but there the state is just given by the wave function or more generally the state vector.
The change of the position is indeed generated by the momentum, not just the time derivative of the momentum. We will see that the spacial derivative of the wave function will correspond to p, not dp/dt (this got also mentioned in the video). A constant nonzero momentum does in fact change the position of our state. The fact that we can "transform away" that change by going into another inertial frame of reference - while being true - is no reason to dismiss that in the first frame of reference the position still changes. This is not very surprising, because going into another frame of reference "redefines" what you mean by "position", so it is only this "new position" in the new frame that doesn't change.
Another way to see from the Lagrangian picture that momenta generate coordinate changes and energy changes in time, is to just consider the action along an infinitesimal time interval dt, which is Ldt. Inserting L=pv-E (compare with what is given in the video to convince yourself that this is true) gives pv dt-E dt. Because v=dx/dt we have v dt=dx, so we arrive at p dx-E dt. So p is fundamentally connected with the change in position dx and E with the change in time dt (even in a seemingly relativistic way due to the relative sign, although we were talking about classical mechanics). Of course this could be completely arbitrary and only when we exponentiate this action we see how it actually "translates in time and space". If Quantum Sense plans a video about the path integral we will undoubtedly see where that takes us.
Of course the initial statement of the video " changes in momentum seem to be indimately connected to changes in the position of our state" is technically true. If momentum generates changes in position, a change in momentum will also be somehow be connected to changes in position. But it is the momentum itself that generates translations, not its time derivative.
Please don't let that take away from your enjoyment of the video. The series is great and the video definitely points us in the right direction. Also take my comment with a grain or two of salt, I might be wrong at some places. Still I hope this could help a bit.
Just adding more to the answers already given, The new key concept that nobody has mentioned is that x and p are called conjugate variables. You can look for Hamilton-Jacobi equations for more information.
But the thing I would like to add is that in quantum mechanics, the conmmutation relation between the operators [x,p] is intimately related to the Poisson brackets in classical mechanics, of the conjugates variables {x,p} . There are a lot of ways you can "quantize classical mechanics" to get quantum mechanics. This is just one way. Another way is the one that this channel is going to show us that is the path integral formulation. There are more ways, and every single one gives different insights.
Perhaps we ought to interpret “x” then not as position, but as inertial state.
A change in the momentum generates a change in inertial state, and a change in inertial state generates a change in momentum.
Me, a computer scientist: "Ah yes. So this is the one that makes me cry. Finally."
thanks, my teacher.
I'll be honest, after reading a bit on Wikipedia, I realized that I have much less of an intuition for classical mechanics than I imagined. Like, why is the kinetic energy that formula with the squared velocity, and not the same as the momentum (which intuitively describes "kinetic energy" quite well in my opinion)?
It is very series for understanding quantum mechanic
Very nice
Love it
Bravo
Can someone try to intuitively explain what “thing 1 is the generator of thing 2” means. I get what the equation says, but I can’t seem to get any intuition of it.
The word "generator" is going to be more obvious in the quantum part. In quantum mechanics, we need an unitary operator U to act on a state in order to change it, if the state changes its position it's because U has the particular for of U(p) = Exp [i * f(p)] with, f(p) some function of the operator p. So, we say that p is the generator of changes in position. The same with time, U has the particular form of U(t) = Exp[i * H], with H the Hamiltonian (or energy) being the generator of changes in time of your state. And so on so forth with the other conjugates variables.
Great job on showing how the variation in the certain physical variable in the Lagragian corresponds to the time-evolution of the other conjugate variable. However, I think it would be more enlightening to do this in the context of Hamiltonians as opposed to Lagrangians, as the “standard” quantum mechanics is based on the Hamiltonian formulation. I think the Lagrangian formalism is better-suited towards the path-integral method.
For example, I would introduce Hamilton’s equation and show how the Poisson bracket with the Hamiltonian gives rise to the time-evolution of that conjugate variable.
However, I don’t really understand how the time-evolution for the wavefunction corresponds to the energy operator. I understand that the Lagrangian and the wave function corresponds to a state, as they are both mathematical object that encode the physical information of the particle. Then assuming that they have a similar correspondence, shouldn't the time-evolution of the wave function correspond to the time variation in energy of the state.
Sorry for the naive question, my expertise lies in general relativity(if you want to learn more about advanced topics in GR please visit my channel), and quantum mechanics is my kryptonite. So I want to learn more, but when learning quantum in University, I always took the operators for granted and never understood the underlying reason/rationale as to why these operators gave rise to variations in a particular variable.
Quantum mechanics is my most favorite branch in mathematics ,,, me Ahmed El.Qafass bachelor of Science Mathematics department... Egypt
do you recommend any series explaining analytical mechanics to understand lagrangian frame work better?
Genius
i'm gonna go out on a limb here and guess that the fact that the position and momentum relationships shown at 13:49 are inverses of one another will have something to do with those variables being linked by the uncertainty principle?
I wouldn’t have thought to think of the Lagrangian as being like a state. I would generally think of it as a description of the physical system that something might be a state of. Like, a Hamiltonian isn’t a state?
could we regard the position of the particle as a function of time x(t) as the ''central objetc'' in classical physics?
it contains all the information about the particle:
-it contains the position of the particle at each time.
-it contains the velocity of the particle at each time ( you just have to derivate x(t) ).
-it contains the energy of the particle at each time ( you justa have to calculate E(t)=m*v(t)^2/m+V(x(t)) )
and so on
Hello, thank you for watching!
And not quite. What if we had a time varying potential V(x,t)? In that case, energy is not purely a function of just position, but now an explicit function of time! So here the position alone isn’t enough to capture the overall behavior of our system.
So we do somewhat need a larger abstract object to capture this behavior. You might then ask, “ok, then why not use the energy as that central object?” And with that, you would have discovered Hamiltonian mechanics. But the Lagrangian has no preference to any physical observable (unlike the Hamiltonian which is based entirely on energy), so that’s why I like to think of it more as a “state function”.
Again, these are all just heuristics to try and understand the patterns we are seeing: there is no “proof” that the Lagrangian is a central object. It is merely an interpretation to help us learn.
-QuantumSense
When you talk about the principle of least action, you mention that it should just principle of stationary point. Then it leads to another question: is this stationary point unique?(I know it should be unique, but is there a proof?)
4:20 Common mistake - you mixed up "saddle" with "inflection point". A saddle is a maximum (for z) in the x direction and a minimum (for z) in the y direction. (Or the other way round). Whereas an inflection point is what we have here.
5:35 You seem to be assuming that d/dx (1/2)mẋ^2 = 0. Why is this the case? ẋ is dependent on x, so I thought that the chain rule is needed here, right?
11:38 What does it mean to "change the position" there? How does one change the position without referring to time?
(I have a bunch of similar confusions that make me unable to understand a large portion of the video.)
Hello, thank you for watching!
This is the case because we are taking a partial derivative with respect to position, which treats velocity as an independent variable. So with that derivative, I’m asking “what is the change in my Lagrangian when I keep time and velocity the same, and only move the position of my particle.” Likewise on the right, we have a partial derivative with respect to velocity, which instead asks “what is the change in my Lagrangian when I keep time and position the same, and only change the velocity of my particle.” We don’t need the chain rule because we are asking what happens when we assume either position or velocity is constant (even though they are related!). If we had a total derivative d/dx instead of a partial derivative partial/partial x, then indeed things would get complicated.
It can be weird to get used to, so I totally understand the confusion. Let me know if this doesn’t clear it up.
-QuantumSense
@@quantumsensechannel Oh. So the lagrangian takes a single position, a single instantaneous velocity, and a single point in time, and spits out a number. That makes a lot more sense.
(I got confused because I thought, based on the visuals at 3:29, that the lagrangian took at input the entire function for the position as input. Turns out that you were talking about the action, not the lagrangian.)
Remember that derivatives, when you boil them all the way down, really come from f(x) - f(x+h). In other words, they compare the difference in the output of some function between some point and a nearby point. So the "change in position" is just a manner of speaking. What's really happening is we are comparing the output of the Langrangian from two positions which are vanishingly close to each other, but at the same point in time.
@@quantumsensechannel Since you "MOVE the position of my particle", how could you do not change the velocity of the particle ? You meant relative velocity in the same frame ? Thanks!
I don't get why change in position generates change of momentum of L. Like, okay, change and only change in momentum can cause the change of coordinate of our state since constant momentum is vulnerable to the change of frame of reference. But dx/dt does not immune to that change. We can always find frame where dx/dt = 0 and so dL/dp = 0. And beside that, how the coordinate can generate change of momentum, if it's like result causing the reason.
incredible 「令人難以置信」
My teacher once gave me a formula relating light frequency, the distance between slit, distance between slit and screen for the double slit experiment but explain nothing on that formula. Can we derive it from quantum mechanic?
You don’t even need quantum mechanics for this.
All you need is:
- light is a wave with constant frequency (or a combination of separate waves with some frequencies)
- light waves, when combined, are “stronger”/brighter when in-phase, and “weaker”/dimmer when out-of-phase
That’s it. Just make a diagram with two sources, two paths, and the point where they combine.
The brightness of the light on the target (for a given point and given frequency of light) will be related to “how well” the two phases align after traveling different distances.
--•-•--
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kind of sounds like the formula that gives you the intensity of light given the position?
I thought that one could be derived by seeing how the phases of the wave overlap, in terms of school level physics
(the phase at the start of the slits is equal cause those are waves from the same source traveling the same distance to the slits,
then you postulate that hitting a slit is the same as creating a new source with same phase and wavelength, aka dispercing it in all directions,
then for each point you can calculate the distance to each slit,
knowing speed of light and the wavelength you can calculate difference in time and phase and that gives you the result for intensity)
don't know how you would calculate that with quantum physics though, but it should still have to do with 2 paths having different travel time (and therefore phases) and interfering with each other
Googling for "double slit derivation" gives you a bunch of relevant results.
I see! So if position Does not change, There is no momentum. When we are certain There is a momentum, the position has changed so we don't know where it is! Like wise, When we are certain of the position(no change in position,fixed?), momentum has changed so we don't know what it is?
Why is the lagrangian the form this video assumed? I can't relate information from this video or the next episode with the assumed form of lagrangian.
What about a Patreon?
Hello! Thank you for watching.
I appreciate the question, but I don’t currently have plans to set up a Patreon or other monetary support platform. These videos are a passion project, and I am fortunate enough to be well supported through my PhD program (but if funding gets short, maybe that will change, haha!). Thanks again!
-QuantumSense
@@quantumsensechannel Love to hear that. Should you change your mind let us know ahah keep up the amazing work! You’re truly worth experiencing
Ps I study physics of complex systems and find this series illuminating
You should be showing how Newton and Lagrange work together, not using one part to attack another and tear physics apart.
This almost be the case if you are not a physics student. Outside physics it seems that Newtonian mechanics are the best that physics has to offer, but inside physics it is just an incomplete view of the patterns (as discussed in this video) that shows up in nature. You only can see these patterns with Lagrangian, Hamiltonian or Hamilton-Jacobi formulations
In classical mechanics a state is _NOT_ a Lagrangian. A state is just a point of the symplectic manifold with local coordinates (q,p). In classical mechanics a Lagrangian is a physical system (without a state specified). Likewise, in quantum mechanics a Hamiltonian is a physical system (without a state specified).
now I don't feel so bad that my vids are so boring - no offense. haha
ot tip for speaking and narration. Never use the word "what" unless you are asking a quastion. Avoid all those "What we want to do is ..." that has creaped into everyone's public speaking and wastes so many man-years on UA-cam.