Let's Learn Physics: A Surprise to Be Sure, but a Welcome One
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- Опубліковано 11 вер 2021
- In this stream, we will look at how to incorporate more general "canonical transformations" into our Hamiltonian framework. Along the way, we will see some very deep relations between transformations, symmetries, and conservation, as well as some other interesting relations between parameters. Comparing some of these relations with some previous mathematical results, we will find something quite amazing!
This is excellent. It's the first time I've seen plainly how QM emerges from classical mechanics without the terrible and unmotivating cliché of "we just put a hat on x and p and on the one hand it just means x and on the other hand it's -ihbar d/dx". Great stuff!
41:40 - Rather than the Stokes theorem (which is true on any manifold with boundary), it's the Poincaré lemma: every closed differential form (dω=0) is locally exact (ω=dψ loc.) and on simply connected manifolds (such as R^6) it is even globally exact.
For a 1-form ω on a simply connected manifold you can define its potential f (df=ω) by f(x):= line integral of ω along a path γ starting from a point x_0 (chosen at the beginning) to x. f(x) will not depend on γ, only on x (and x_0). Clearly changing x_0 changes f by an additive constant.
So anyway, here's quantum mechanics.
This was so cool! And we haven't even quantized anything! Oh and also here's the uncertainty principle (sort of) just because _it has to be there_ .
This is what we should show people when they say quantum mechanics makes no sense, or that they don't like superposition.
When we make the leap from the Poisson Bracket the the Fourier Transform, does that represent anything "physical"? It isn't the same as just asking "What if they're waves...?", is it?
Also why do we keep ignoring time? Wouldn't it make sense to just make it part of our generalized coordinates? What's so special about time anyway...(he asked innocently)
@Narf Whals We missed you during the stream, I'm glad you liked it!! To address your first question: it sort of depends on what you mean by "physical." Perhaps the best way to view it is from the point of generators of transformations. For example, let's take the translations in coordinates; in the classical case, we saw that the action of such a translation was generated by the momentum conjugate to the coordinate we are shifting in through the relation to the Poisson bracket. So we are essentially just asking how else we can generate such transformations, and a natural choice is by using these Fourier relations. I suppose this isn't so much physical as mathematical, though, but I guess the physical motivation is that we want to keep all of the same transformations (translations, momentum shifts, time evolution), but find a different way of realizing them and see what the result is. Of course, it is easy to write down equations, but the import part is to see whether or not the predictions coming from the equations are actually physical!
Ahh yes, time. Time is a very interesting topic. In this formalism, time is "special" because of the fact that it is not a phase space coordinate like the q's and p's. In particular, the q's and the p's *both* depend on time, while also being entirely independent of each other. So the p's and the q's often follow different "rules" due to this independence than time does, hence why we often consider them separately.
However, in GR for example, there is absolutely nothing special about time (after all, we can freely mix time with spatial coordinates as much as we like through whatever transformations we choose). But of course, we can do GR using Hamiltonian mechanics (it isn't pretty, but some interesting stuff has come about from it), so how does this play out? The general story is that we want to look at the evolution along a "spacetime interval" (think of just a path in a x-t plane) and we choose some arbitrary parameter, call it s, to define how far along this interval we are. Then, all of our coordinates, including time, are functions of s: x(s), y(s),..., t(s). Then, everything can be re-framed in terms of this s parameter and we have now included time as a dynamical "phase-space" variable.
So why don't we do this for standard Hamiltonian mechanics? ...honestly, I'm not really sure. I've actually messed around with it a bit recently (after doing Hamiltonian mechanics in these videos in a way much more reminiscent to GR than I have typically seen it done) and to me, it actually seems a bit more clear to do things this way, and seems to be much easier to generalize to non-conservative Hamiltonians which are functions of time. The details are a bit cumbersome to put in a UA-cam comment, but the end result is that you essentially just get a trivial relation t(s) = s, so your time "coordinate" is basically interchangeable with the parameter along the path, sort of by construction. This allows you to now include time in your phase-space transformations and G matrix to get a time-dependent Hamiltonian. In the cases I have looked at, it seems to reproduce the correct equations of motion, but I am still very skeptical about it (hence why it never appeared in any of my videos!).
@@zapphysics "Then, everything can be re-framed in terms of this s parameter" That sounds very reminiscent of coordinate time vs proper time to me. So are we deriving Special Relativity from Hamiltonian Mechanics next time? :)
Oh also: You showed in this stream that every canonical transformation that is a symmetry of the Hamiltonian comes with a conserved quantity. And that seemed to follow very quickly from just the Hamiltonian formulation. Is there anything more to Noether's Theorem and is that somehow more clear in the Lagrangian formalism? Or why did it take 200 years and Emmy Noether to prove it?
I'm vaguely aware that there is a geometric interpretation of the Poisson bracket and that it's intimately connected with Lie algebras and differential forms and other things I never studied. Is the "geometrical" aspect of this interpretation related to the link between Poisson brackets and generators of translations and rotations? Or is it deeper than that?
The Poisson Bracket with blank entries is just a bivector in phase space right?
Edit: correcting, in the tangent space of phase space, my bad
This has clarified a lot for me. Thank you
Glad to hear it!!
Are you following a textbook? The way you're covering topics is so unique!
@Zombot I am not strictly following any one textbook, I am kind of using a mixture of resources. For a lot of the classical mechanics stuff, I used Taylor's book for inspiration and also some of the MIT open courseware if I got stuck, but for a lot of it, I tried to come up with examples and problems myself that seemed interesting to me.
The ordering is sort of what I personally feel like flows the best logically. I try to motivate as much as I can because I know that I would always be bothered learning the stuff and I was just handed, for example Schrodinger's equation without having any idea where it comes from. Again, in my personal opinion, I think that many programs are too focused on teaching things in historical order that they gloss over many of the beautiful connections between the different branches of physics!
@@zapphysics completely agree. Although I really do like how using action-angle variables and the resulting Hamilton-Jacobi equation makes even classical mechanics have wave-like properties, which then also leads to Schrodinger's wave mechanics