3:25 AM
Might be of interest:
12

Similar to: What are the different types of charge analysis?, What are the types of bond orders?, and What are some recent developments in density functional theory?, I would like to ask: What are the different variations/flavors of DFT (density functional theory)? I ask users to stick to one of ...

3:54 AM
Sorry if this isn't the place for this, but the astronomy chat room is dead. Can anyone help me setup a location to view Neowise? Absolutely nothing I've tried has worked. And I'm assuming that my area is just a complete dead spot for the comet.

4:18 AM
@BioPhysicist it's just terminology. Suppose you are pushing a heavy object across a rough surface then it's you doing the work i.e. you are the one supplying energy, not the friction. Personally I have no problem with saying the friction does work. Whichever way you phrase it you are just describing an energy transfer.
@ABC the battery cannot absorb or create charge. It can only move charge around. The reason we have two spheres in my example is so the battery can move charge from one sphere to the other. If we only had one sphere then the battery couldn't charge it because it would have nowhere to move the charge from.

5:04 AM
how is quantum computing related to quantum gravity?
by invoking the likelihood of formulating the both in terms of quantum field theory?

1 hour later…
6:32 AM
Hmm, if I'm going to have to ask here, then let me start with this: from whatever I've gathered on this topic, a single unit of light--and of microscopic particles in general--can be modeled as either as a hard sphere, or as a "wave". This also generalizes to a collection of said "particles". Is this correct?

1 message moved from Problem Solving Strategies
@Dunois hi :-)

Hello.

"particles" are much more complicated objects than you think.
This applies to photons and electrons, protons, etc
We describe particles using a theory called quantum field theory, and this describes particles as packets of energy in an object called a quantum field that fills all of spacetime.
These packets can look like particles, i.e. like little balls, or they can look like waves. It depends on what they are doing.
As a general rule if a particle is interacting with something else, e.g. if two particles are colliding with each other, then they behave like little balls.
If a particle is just travelling along and not colliding with anything then it looks like a wave.
This is the reason for the "wave particle duality" that you've probably heard of.

What aspect of the "particle" looks like a "wave"?
What does it even mean to look like a "wave"? I presume some amplitude is oscillating about some mean value, and this amplitude itself is moving about space and time (i.e., "the particle is moving")?

We're all used to the everyday world where balls are ... well ... just balls. They have a position and they have a speed.
But when you get down to elementary particles they are fundamentally different. A particle like an electron does not have a position. It is more like a fuzzy cloud that is spread out over a region of space.
Likewise it doesn't have a speed. It has a range of speeds.
The difference between behaving like a particle and behaving like a wave comes down to how spread out the particle is i.e. how big the fuzzy cloud is.

6:52 AM
@JohnRennie could you elaborate a bit on this?

@Dunois it's hard to say much more without diving into the maths. The problem with QM is that it's inherently a mathematical theory. When we talk in general terms, like calling particles "fuzzy clouds" this is only a metaphor.

Okay, let's stay with the fuzzy cloud idea. What would diffraction and interference mean in this context? You also stated that particles generally behave like little balls when they collide (interact?) with one another. But wouldn't interference (I gather diffraction is a form of interference) require that the particles actually interact like waves? (I hope what I am trying to say makes sense.)

No, because in diffraction the particles are not colliding.

7:10 AM
What actually happens in diffraction then?

The functions that describe particles have two properties, an amplitude and a phase.
When we have two particles interfering we add together the functions that describe the particles.
If particles only had an amplitude then the sum would always just be a positive number and we wouldn't get any interference. This is what classical mechanics predicts. However quantum particles also have a phase and it is the phase that causes the interference.
Now you're going to ask what a phase is, but this is going to get us into some maths.

Oh no, I know what amplitude and phase are (at least at a simplistic level). So this I can understand.
But coming back to my question, quantum particles are always fuzzy clouds and also always "balls", right? So what governs whether they collide (like balls) or whether they interfere (like waves)?

OK, so if at a point in space the two particles have opposite phases they will cancel each other out. While if at some other point in space they have the same phase they will reinforce each other. This is all that interference is.

1 hour later…
8:37 AM
@BioPhysicist I think this depends on whether you think of "work" as something "real" in the world or just the quantity computed by $\int F\mathrm{d}s$. Mostly there's no difference, but someone who thinks about work as being more than a bookkeeping device in our calculations will find applying the definition to something like friction absurd, like Charles argues there.
It's a variation on the theme of whether or not energy is "just a number" or "real" - people believing the latter often have difficulties e.g. seeing where it is "stored" in an electromagnetic field.

2 hours later…
10:28 AM
@JohnRennie Work is not only done by forces that supply energy. Work has a precise definition, not a subjective, qualitative one. I feel like talking about work in the latter way just makes it more confusing for introductory students.
@ACuriousMind I'm not talking about whether or not you think energy is real. I just don't see how you can ignore the definition of work entirely and say that friction does not do work. That seems to me that you are saying the line integral is 0, which isn't true. Unless one is saying if work is non-positive then the force doesn't do work, which I think is extremely confusing.

1

My Followed Posts list is littered with Deleted questions, which I cannot remove off the list, as they've lost their "unfollow" option. How do I prune off these entries?

10:45 AM
@BioPhysicist I agree.

1 hour later…
12:12 PM
Wow am I not allowed to send photos anymore?
Everything I know about the summation convention is telling me this is wrong
How can you have a covariant tensor being equal to a contravariant tensor?

12:40 PM
Should the bounty be revoked here? Thoughts?

1 hour later…
1:47 PM
In Deep Q Networks, the output of the neural network is basically the q values for each action. My question is on what data and how do we train and do backpropagation on the neural network? I mean, the nn spits out q values but how do we get the target q values for each action?

1 hour later…
3:14 PM
@JakeRose What the text is trying to say is that the co- and contravariant components of a tensor in default Euclidean space don't really differ because they're numerically equal

@Qmechanic yes, close it as homework and refund the bounty.

3:40 PM
I was thinking, instead of explicitly setting the rewards in a RL system, can't we make a system that figures out the rewards on it's own? For example, no one set the rewards for us humans in the real world, evolution has carved them in our brains.

@Qmechanic It does seem pretty crazy that it's at 4 close votes and now held open by a bounty. It seems like several if not all of the close votes will also expire during that time... That doesn't seem right to me.

@JohnRennie & @JMac : Done.

4:00 PM
@JakeRose $\delta_{ij}$ is the identity matrix.

Ahhh I see. In my head it feels like it’s neglecting the summation convention by not balancingnindices. Is this an abuse of notation? Or am I just not familiar enough?

It's not abuse of notation, the metric tensor defines a 1-to-1 map (in finite dimensions spaces) between a vector space and the corresponding covector space, there's no reason the vector in $V$ and the covector in $V^*$ can't have the same components.

Notation is not that universal - some would say it's okay, some would say it's terribly wrong

It's not a general relationship, it is only true in the specific case that you are dealing with the Euclidean metric, which is the identity matrix

@Charlie I don’t disagree. The textbooks and notes I’ve used previously just often use the idea that “indices should always be balanced” as a core part of the Summation convention.
To me it feels like an equivalence relationship between a vector and it’s dual

4:08 PM
Oh I see what notation you're talking about now

god I’m rusty, I think it can be called a dual

it can, $V^*$ is the "dual space" of $V$.

@Charlie I get what it means, and I get what you’re saying. I’ve jsut been taught a more “summation principle is law” type of way
*rule not principle

This is a very, very specific case where the rule can be broken without ambiguity

but saying this, I’m not sure how I would say the same relation whilst respecting my idea
maybe by using brackets

4:11 PM
In general $x_\mu \neq x^\mu$

btw, I’m trying to do a bit of reading into geometry and physics. Any recommendations for good texrbooks?

differential geometry?

Yes

in the context of GR?
most introductory textbooks cruise fairly quickly over the basics of differential geometry before starting the actual gr
And if your ultimate goal is to learn GR you're probably better off using an actual GR textbook than bothering yourself with a really heavy mathematics textbook on differential geometry first
but others may disagree

originally I was just looking for a book that used normal geometry to show some beautiful ways of doing physics (e.g. Newton’s derivation of the orbits) but then I got to reading some of the “geometry of physics” and they had a mix of differential geometry, Lie groups, etc and went into both gr and quantum.
I’ve already done a course in GR
taking David Tong’s course next year too
feel like a more pure mathematics perspective on the physics might be interesting reading over summer

4:16 PM
If you want pure diff geo you could ask in the maths chat chat.stackexchange.com/rooms/36/mathematics

Geometry and Physics
Textbook by Jürgen Jost
Has a really interesting mix of subjects, probably the most what I’m looking for
@Charlie I might eventually, but I still want a physicists take.

someone more mathematically knowledgeable than me will have to suggest something

4:52 PM
Hello agents
How is everyone doin'?

@JakeRose MTW, hundreds of pages on diff geom for physicists in there
Gravitation is a textbook on Albert Einstein's general theory of relativity, written by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. It was originally published by W. H. Freeman and Company in 1973 and reprinted by Princeton University Press in 2017. It is frequently abbreviated MTW after its authors' last names. The cover illustration, drawn by Kenneth Gwin, is a line drawing of an apple with cuts in the skin to show the geodesics on its surface. The book contains 10 parts and 44 chapters, each beginning with a quotation. The bibliography has a long list of original sources and...

5:14 PM
no one has ever read that book

I was going to look at it after schutz
it long or something?

It is the longest book ever written by man
Some say that it has infinitely many pages
But no one ever got to the end to check

countably many?

hopefully

:o

5:32 PM
although rly I'm being facetious
there are way longer books in physics, unfortunately
I recall some solid states reference book that was about 10.000 pages long
It's the longest book I own, though
Even the Particle Data Group isn't as long
Handbook of Physics and Chemistry is longer, but it is expensive

6:25 PM
How's everyone doing?
why am i being ignored :( cry everitim
Is there hbar for AI?
I need some chat-like place to discuss Machine Learning and AI stuff

6:59 PM
You could check Artificial Intelligence and see if they have or can recommend an active chat room.

7:24 PM
yesterday, by FakeMod
Why does the Legendre transform work? I mean, to me it seems that we are just applying an arbitrary transformation, just to change our dependent variables, changing the original function completely. How is this any different from just arbitrarily choosing a new function with the desired dependent variables?
Anyone? 👆 I am just looking for the motivation behind introducing the Legendre transform, because it seems quite unobvious to me.

There are lots of arbitrary transformations possible, but only some of them make the problem easier to solve

@tpg2114 How do we know that this is the one?

You do the transformation and check if it's easier -- at least until you develop intuition to know what to look for in a problem and what might make it easier

@tpg2114 Hmmm... So why not just transform $f(x)$ as $f(x(s))$ (where $s=f'(x)$)? Why the (apparently) unnecessary $xs-f(x)$?
The former will also probably retain the function's characteristics more than the standard Legendre transform.

@Slereah hey!

7:32 PM
The Legendre transformation is basically just a trick to turn the problem into a first order differential equation
At its core, you basically just define a new variable to be the derivative
There's a more physics-oriented way to do it, if you wish

So after all, is it just a matter of "it works"? Or is there any way to reason it out by oneself?
@Slereah what kind of way?

Is the hang up "why do we want to do this?" or is it "why does it have to be done this way?"

@Slereah Then why the extra terms?
@tpg2114 the latter.

Basically what we have at the start is the equation of motion $$\frac{\partial L}{\partial x} - \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} = 0$$
But what we want to have instead is
$$\frac{\partial L}{\partial x} - \frac{d}{dt}\frac{\partial L}{\partial v} = 0$$
Something like that
so that everything is first order
So we need an equation of the form $\dot{x} = v$

Hmm... Makes complete sense, what next?

7:36 PM
Well, what we need here is a Lagrange multiplier
So we add a Lagrange multiplier of this form to the action, ie something like
$$S = \int [L + p (\dot{x} - v)]dt$$
This will have $\dot{x} = v$ as an equation of motion

Ok so $p$ here is the $\lambda$ or the multiplier, right?

yes
Then you can rearrange things a bit with a function $$H = pv - L$$
The action becomes $$S = \int [p\dot{x} - H] dt$$
That way the equation of motion remains the same

@FakeMod one way to interpret it is that it's an old trick to replace a curve with a curve possessing the same properties regarding it's tangent

Oh, I see. Now how do we physically relate the multiplier $p$?
@bolbteppa But then using $f(x(s))$ (where $s=f'(x)$) would make more sense, wouldn't it?

@FakeMod section 35 and 36 of this discuss it
See the bottom of page 68 where they discuss it, setting the more general discussion up in the previous page or so

7:43 PM

the definition of $p$ comes from the EoM again
Consider the EoM for $v$ :
$$\frac{\partial L}{\partial v} - p = 0$$
Then you can see that $p$ is indeed what we usually define it to be

Generalized momenta?

This does correspond to the more abstract notion of a Legendre transform, but I don't think it's strictly necessary to use it to understand it

@Slereah But where did this exactly come from?

@FakeMod Well, as I said, the equation of motion
Apply the Euler-Lagrange equation to the rewritten action, and this is what you get

7:46 PM
@Slereah Oh, ok. I get it.

At its core $p$ is just the Lagrange multiplier we used to make the velocity be equal to some variable

Hmmm... I would need some time to wrap my head around it :-) Anyways, thanks a lot for walking me through the derivation :-) Have a good day!

Hey all!

Does this mean it's possible to derive the zeroth law of thermodynamics (in this case) as the answer suggests?

https://physics.stackexchange.com/a/558882/150174

Basically we just added a lot of variables and shuffled them around so that, if we worked out the EoM, and replaced everything with their proper values, we would get the original Lagrangian equation

"This remark enables us to replace the curve $c$ by any other curve which is tangent to it in questions which involve only the tangent to the transformed curve $C$". The derivative of $L$ with respect to $v$, $\partial L/\partial v$, is tangent to the Lagrangian $L$, so we can replace $L$ with an equivalent curve when discussing properties of the tangent to $L$ (i.e. $p = \partial L/\partial v$)

7:51 PM
@bolbteppa Whoa, spooky! I was just reading that line when I saw you post a message in the h bar :-)

You can also do the usual classroom demonstration of the Hamilton formalism
"It works, trust me"

@Slereah Reminds me of my university days. I wish sooo many things were better explained

@Slereah hehe. I would have compromised for that after all, if I didn't get a satisfying explanation

Or you could use the proof using the variational bicomplex, I suppose

@Slereah what? Wait, nvm, don't explain, I am too scared :P

7:56 PM
Simple enough

Found a pdf which tries to explain the discussion of the Legendre transforms in this book physics.mcmaster.ca/phys3c03/notes/contact.pdf

Is it a big deal to derive the zeroth law of thermodynamics (in any case)? Or am I just thinking something is bigger than it actually is

2 hours later…
10:05 PM
@Slereah is that supposed to be 'the justification' for $H = pq - L$?
Or is 2.16 saying something else
Amazing how such generality still ends up returning to the concepts people invented nearly 200 years ago:

'canonical pre-$n$-sympectic $n$-$\infty$ jet bundle bordism EULER-LAGRANGE EQUATIONS Yoneda lemma $\mathrm{Hom}$ ambient n-category ...'

1 hour later…
11:10 PM
111

Approach0 is a math-aware search engine. “Math-aware” means you can add math expression(s) as some of your keywords to have search engine help you find similar expressions. Check out here: https://approach0.xyz This is my side-project, hopefully it can be useful in some cases to help Math SE use...

This is a good idea

11:34 PM
oh that's a nice idea

Doesn't seem to search the physics one

the mathematicians are always one step ahead