@TobiasFünke Miao miao iz with @Relativisticcucumber on this point; teach it from the get-go and also density operators asap. But there is no reason why we must avoid old notation. That is, other than the parts where we are deriving the properties of the bra-ket notation, everything else should be done in bra-ket. But we can assert that $\left<x|\psi\right>=\psi(x)$ and use that as a crutch to derive some properties that are built into bra-ket such that it is confusing to derive in bra-ket.

And there is no reason why a treatment of QM should be sloppy just because it is using bra-ket-first. That is, there is no reason why a bra-ket-first treatment should be avoiding the linear algebra background and the cross-links to other topics. In particular, I would start with basic 3D vectors and their basis transformations, to introuce the topic, and then the link to statistics with their polynomial linear regression, then Fourier analysis. It is also a good time to discuss Sturm-Liouville

This way, the student who is learning QM for the first time, will see that all that mathematical apparatus is not conjured from nowhere. It will be both rigorous, yet also hand-holding the students to understand the subject as smoothly as possible.

@HerrFeinmann I vehemently disagree. There are situations whereby the struggle is unavoidable, but we have to always avoid dropping into the "I suffered, and now so must you" death spiral. In particular, when it comes to notation, one should always just seek to get students to convenient notation as soon as possible, because it takes so much time to get familiar with notation, even if the notation is great.

@naturallyInconsistent So long as you warn students about dangerous things of that notations yes. I was not talking about a QM course, which should start with bra ket notation. I was saying that the students should know those things in advance

I'm not sure what you mean by "dangerous things of that notations"

You know, after working with bra-ket for so long, things are so natural...

morning everyone

@naturallyInconsistent thanks for your response

I propose to drop indices in GR

That's the real enemy, not bra-kets

@GroveRover what would be your replacement?

@naturallyInconsistent the clean geometry notation

not so clean when working with tuples, 3 vectors and 4 vectors at the same time. nor when calculating observables in a particular frame.

@naturallyInconsistent For example working with non orthonormal bases

@GroveRover that is not a specification of one specific notation. Do you mean like using $\gamma$ for paths and $\varphi$ for converting that to coördinates and so forth?

Or with non-self-adjoint operators

@HerrFeinmann that is a disaster even everywhere else in physics!

Dirac notation still works but you have to be a little more careful :P

@qwerty Idk what you mean by tuples, nor by observables (maybe QFTCS?)

@naturallyInconsistent So long as the metric tensor is with me...

@HerrFeinmann in my research I have to deal with this at the moment; soooo meessy lol

@HerrFeinmann with that too lol

@naturallyInconsistent \$\phi$ are commonly omitted in geometry too, but yes $\gamma$ would be clear as it would avoid some initial confusion when trying to distinguish if x is a coordinate or a path. However I was thinking more about stuff like # and $\omega(X)$

@TobiasFünke No research done on this side, I just got badly confused some weeks ago with something about the variational method using non orthogonal states :P

@GroveRover I don't know much of GR, but do you know the lectures by F. Schuller (on YouTube)? Do you mean like this?

@GroveRover that's very malformed

@HerrFeinmann in which context? I know for example that the derivation of the Hatree(-Fock) equation sometimes are not explained very well...

but the point is that if you want to be angry at indices, you would first have to state which alternative you want to propose

@TobiasFünke YES

Aha! Then I agree

haha

@naturallyInconsistent I just did (?)

Non-orthonormality is usually treated rather well in GR indices notation, because it maps exactly onto the standard stuff in cond mat lattice vectors v.s. reciprocal lattice vectors

(but okay, as I said: I don't have much knowledge in this field, sadly. Perhaps one day...)

@TobiasFünke In my cases it was something really stupid. The SE for the variational ansatz did not appear as an eigenvalue equation, instead it was a generalized eigenvalue problem $H\vec{c}=ES\vec{c}$ and this made me go crazy. Then it was all about the fact that $\langle i \lvert H \lvert j\rangle$ is not a matrix element if the basis is not orthonormal

@GroveRover it was malformed and did not appear. Do you mean using musical notation? I'm not even sure if people can work in terms of that...

Which was kinda obvious if you asked me at the beginning, but I got into that because of notation

@GroveRover i mean with index notation you can easily use it to keep track of the dimensionality of the object you are working with. that's all I mean by tuple; it doesn't even have to be a vector. when you want to do it without indices it gets messier. and i dont know how "an observable" is ambiguous...

@HerrFeinmann I see. These kind of things appear over and over again in quantum chemistry ^^

@TobiasFünke is Riemannian geometry used in condensed matter? I never got further than BECs and crystals

@GroveRover ufff.. not that I know of, at least at this very moment... but I could imagine. sorry

@GroveRover you can recast a lot of its maths in terms of them

I mean, I just told yall...

@naturallyInconsistent as long as you're doing proper geometry you can absolutely do it

@qwerty the definition I know of an observable in classical mechanics is a scalar function on the tangent bundle but I'm not sure that's what you meant

@GroveRover I'd be surprised. I mean, start with an expression of EFE in musical notation and the assumptions of Schwarzchild black hole, derive the Schwarzchild metric, is that possible?

@naturallyInconsistent EFE does not have musical notation as all indices are low. However I'm not saying to abolish coordinates, but that when they're confusing/unnecessary we should drop them

You can check Straumann's book

He does GR coordinate free mostly

@Slereah Thank you I will

Do you mean Norbert Straumann?

yes

I kinda like some parts of his other books, but I've never studied any for a longer time

@GroveRover ...that conditional makes it quite different to "I propose to drop indices in GR, they're the real enemy" :P

aah you always need a catchy phrase to start a discussion :P

@qwerty hahaha cmon it was implicit, mathematicians often use coordinates as well

it is similar to the usage of bases. I avoid that if possible, but proving statements for finite-dimensional spaces often is MUCH easier in working with some (arbitrary) basis, for example

@TobiasFünke I was about to make the same parallelism

some of my supervisors almost always express their quantities/equation in a basis, and for me it makes everything much more messier, and I quite often lose the oversight... but as usual, it is a matter of taste, it seems

it probably depends on your end goal. what you're calculating, what you're trying to show, whether you want to be consistent with other fields etc. the purists will tell you anything but natural units is evil, but imho that's just... not true.

yes, true that

I'm pedantic enough to write things twice, once with everything stated, and once with stuff hidden

@naturallyInconsistent Unfortunately, most people don't and I spend days trying to figure out what they are actually doing

Doubly unfortunately for you, the "with everything stated" is index-full

@naturallyInconsistent so it already hides stuff

what do you mean by that? I'm very explicit when I'm writing all the indices down

The thing I can't stand the most about physicists is their attitude to sweep under the rug all the things they don't understand or think other people won't understand

Yeah

@naturallyInconsistent Alright a bit better, then it's just hard to read and not general

It is, however, not uncommon when I'd write $$\left.\left(\frac\partial{\partial x}\right)_y\left(\frac\partial{\partial y}\right)_x f\right|_{x_0,y_0}$$ and students would omit stuff, and then I'd get to spring onto them an example down the road of why I'm keeping all that dayum notation

@GroveRover I can make my students deal with hard to read. But I have trouble seeing what you mean by "not general".

@naturallyInconsistent was about to ask the same thing

yes! derivatives, also in relation to Lagrangians, but also thermodynamics, are so often such a mess. because one can get confused quite easily, and a precise notation is sooo important IMHO

@naturallyInconsistent I was thinking about not chart-induced coordinates on the bundles but thinking about it I believe the expressions are still the same as long as you don't use Christoffel symbols

@TobiasFünke reminds me to revisit inexact differentials one day D:

@TobiasFünke exactly; if I could get my way, I'd have forced the curriculum to go all the way to differential forms before dealing with the mess that is thermodynamics. And for Euler-Lagrange equations, I insist that students see it first as $$\frac{\partial\mathscr L}{\partial\frac{\mathrm dq}{\mathrm dt}}$$ and make sure that they understand the insanity that is differentiating by a derivative, yet keeping the original function's position constant. The extreme precision is very helpful

@GroveRover I really have no idea what you mean, especially how it relates to "not general"

yes

@naturallyInconsistent what mistakes do you see students make without that?

I mean without writing the EL equations like that / the precision...

@naturallyInconsistent chart-induced coordinates on the bundles are just a subset of all possible coordinates and have special properties. However as I stated I think that it makes no difference as long as you don't write Christoffel symbols

@qwerty not everything is calculating something, it is also useful to have a deep understanding of what you're doing

@qwerty It is not about mistakes in this case, but that students will nod and tell meow that they understand, yet it is tremendously obvious that they don't, and there are sooooo many possible misconceptions until you tease them out with exercises and whatnot

@naturallyInconsistent It becomes very clear when you think about it in terms of bundles

@GroveRover I include conceptual mistakes as a mistake

@GroveRover But why not Christoffel symbols and all? I don't understand. I can also go into vierbeins and make them work with non-coördinate bases. I don't see what you mean at all

@GroveRover yes, but you do not have to make it that complicated, actually. You just have to view the Lagrangian as a function of three variables, and then construct $L(t):=L(q(t),\dot q(t), t)$ or so (and of course, we abused the notation, so perhaps it is even better to use different symbols)

The important point, IMO, is that when you take the derivative of $L(t)$ and apply the chain rule, you take partial derivatives, which I'd write as $\partial_1$ for example, which you then evaluate at $(q,\dot q(t), t)$; then one cannot be confused about taking derivatives with respect to a function and its derivative or so.. I think we have a handful of questions on the main site about exactly this misconception/issue

the same holds for problems in thermodynamics, which is, I think, even worse, due to the incredible abuse of notation

The transition matrix element, when some measurable observable is considered is written as: $|\langle f|\hat O||i\rangle|^2$. Where O is the operator representing the physical quantity and f and i are final and initial states. Can someone tell me, what is the arguement made behind the claim that $|\langle f|\hat O||i\rangle|^2$ is the probability of transitioning from i to f?

I mean, there must be some way of arguing the formula. It doesn't simply fall from the sky

I know that if you consider a state of the form $\psi=\sum c_n|\\rangle$

the prob. of measuring some eigenvalue of an operator is $P(a_n)=|c_n|^2$

yes, that is described in any textbook on qm, no?

But they are not the same thing

e.g. in the context of time-dependent perturbation theory

I asked in the lecture, if this is somehow related to fermis golder rule

to a degree

I mean fermis golder rule has to do with transition rate once you divide with time

But the idea, should have been the same. And I was told no

Even, when we did time dependent PT

We made the same claim i.e. $|\langle f|\hat O||i\rangle|^2$ represent probability of transition

but I think the worst part in a physics curriculum is how functional derivatives are introduced

We never argued why is that

How the formula actually represents that

so many weird things happen, instead of simply defining it properly, which is not too hard actually. at least this is my experience, and also talking to colleagues

We simply started the derivation of fermis golder rule

with the premises that $|\langle f|\hat O||i\rangle|^2$ represent probability of transition

Please. Have you consulted a textbook already??

I have Griffiths QM

He just takes it as such

And Griffith's Intro, takes to many things as granted

in numerous occasions

And I really don't like the book, and that's my opinion

take another book then

Which book explains how the formula represents what we claim it to represent

Which book tries to give a logical/intuitive explanation as to why the expression indeed represents probability of transition

I'd say all, but I don't know all and you say Griffith does not, so: Mostly all? You can just look online, too

just google "Fermi's golden rule derivation" or so

Wikipedia has the same derivation you should find in most textbooks.

I know the derivation

I said that

it will show you how the transition probably per unit time results in your expression (times a delta function) for e.g. harmonic perturbations

I also said that the derivation begins with the claim that $|\langle f|\hat O||i\rangle|^2$ represents probability of transition

If you know the derivation, and understand it, then it should be clear

@imbAF which is incorrect - that's the result of the derivation, not the starting point.

exactly

@ACuriousMind We start with that, I mean I have the notes right infront of me

you start by considering, in your notation, $c_n(t)$, expand it order by order, and then compute e.g. for first order the absolute square

@imbAF And I just linked a Wiki page that derives that statement.

@imbAF what you do in your courses is not standard.

and we pointed out that basically everywhere else the standard derivation is given

I don't care what your notes do, but I just linked literally the derivation you want and instead of looking at it you complain that we're not answering your question

I am not complaining

@TobiasFünke Indeed, I ended up in QC references saying weird stuff like "Löwdin transformations" :P

@HerrFeinmann haha Löwdin is the goat

And you linked it, and I was typing as you linked it, I didn't see the link like a minute later. Have to go through hoops here and explain every action

@TobiasFünke Subtle way of correcting the ö :P

haha, did you just changed the o to an ö? :P

ah no, this was not my intention ^^ but yes, looks much better now lol

but no joke Löwdin made huge contributions, particularly in perturbation theory and so on

@TobiasFünke Indeed

I just copy-pasted yours. I don't remember the ASCII code of that symbol

The only ASCII I remember are those for ~ and È

...it's not an ASCII symbol

I think you mean Unicode codepoint

I knew you would say that :P

@HerrFeinmann then why did you say it wrong :P

I think we had this conversation in the past, I'm going to check. Well, it's one of those things I don't care about enough to bother even understanding correctly, so long as I can type the right symbol :P

well, at least I'm consistent :P

I think we have found a normal mode of the chat

People always mistake ASCII and ANSI :p

@HerrFeinmann Poincaré recurrence is real

lol

If you can find a reference explaining the difference and the relevant things shortly, I will consider learning the difference :P

ASCII is 7 bits

Otherwise you will have to correct me again in about a year and a half

It only has the first 128 characters

ANSI is 8 bits and contains accented characters

although today realistically every "ANSI" character is just the first UTF8 block

@Slereah So ~ is ASCII (?)

(It's 126)

I mean it's both, but you can write it in ASCII yeah

Any ASCII code xxxxxxx can be written in ANSI or UTF8 as 0xxxxxxx

Damn, then I'm using unicode because I actually type 0126

wait you can type the character code and it will render? U+0153

it's the future, everything is in unicode

nup :(

@Slereah I wish

(I mean, it would be unicode even if I type 126, I understand that)

@ACuriousMind Dealing with old code? :p

@ACuriousMind Which makes me understand that your remark now and in the past was deeper that it seemed :P

@qwerty ™

@Slereah I do sometimes have to work on code that communicates with systems using non-Unicode codepages. It's not fun.

Non-unicode web pages these days are extra annoying because most browsers don't have an easy way to switch the encoding now

(the software has been Unicode-only for like 15 years but business companies are extremely bad at upgrading software if you don't force them to :P)

&#169;

One time at software school we had to make some software as a project, and part of it was testing the other teams' softwares according to their documentation

To be evil I tested it by using a file with bad encoding because those fools never specified the file encoding in the documentation

@qwerty I don't think the chat has functionality to turn codepoints to symbols, it's something on his input device that does it for him

@Slereah I mean, that's what a good tester does!

being evil when testing is good

booo. this is actually one benefit of using a mobile device as the keyboard lets you input extended latin alphabet easily

well, compared to anglophone keyboards.

Get one of these

@qwerty under windows you hold Alt while typing +XXXX on the numpad, where XXXX is the hex representation of the UTF16 codepoint number. That is: Press and hold Alt, press +0169, then release Alt.

@Slereah omg that's amazing. does it have a name?

that's the Space Cadet

@qwerty I use a Chrome addon to convert vaguely Latex like sequences to unicode symbols.

It is very useful when you can't be bothered to type the MathJax :-)

© :O

now I wish my laptop inbuilt keyboard had a numpad :P

but cool trick, thanks for sharing!

@ACuriousMind being evil or testing all the possible things?

didn't I say right after?

I should increase my FOV

@Relativisticcucumber This made me laugh hard

@HerrFeinmann are you accessing chat from one of those old electronic typewriters that could only display one line at a time?

No, I'm accessing it with the same FOV I have when I read a physics book :P

ah, so it's a software not a hardware problem :P

I guess that's what evolution did to save me from getting spoilers. My brain only processes one line at a time

@HerrFeinmann watching movies must be hard hehe

@qwerty Well, kind of. I'm so used to processing one line at a time that I sometimes go in a loop and watch the same scene many times until I actually process everything they said

It's a mix of OCD and that

And that's why I rarely watch movies :P

ahhhh you were being serious?

About the chat? Partially. I often read only one line, yes. About movies, totally

I learned about this maybe a few years ago en.wikipedia.org/wiki/Monotropism#Characteristics and I felt like it explained why I did not like lectures at all at uni

Mhhhh, during lectures I don't really know. In most of them I didn't care enough to listen; when I listened, I would usually follow without many problems, although if something I didn't understand popped up, all my attention was lost and I couldn't think of anything else

I'm now reading the wiki page

juggling copying notes from a blackboard and actually following what the lecturer was saying verbally alongside actually trying to understand the content was very difficult for me, and many lecturers did not provide notes so it was essential to write and copy fast.

Interesting. I do not have the problems mentioned in the wiki page examples, but I definitely experience something similar

> For example, some students have trouble taking notes in class while listening to a teacher and may find it difficult to read a person's face and comprehend what they are saying simultaneously.

This, for example, only if it's a very complicated discussion, which I guess is normal. Taking good notes is not easy (and I prefer not to take notes if they're not good). Regarding facial expressions, those are already difficult on their own :P

@HerrFeinmann "normal" is a dangerous word but plenty of people actually find it easier to listen to others if they can see their face

ugh why does the quote formatting always get messed up

@ACuriousMind I don't. For me looking someone in the face is a very conscious act. An additional task. I can handle it but it doesn't make anything easier :P

@qwerty I mean, it's always an option to mess up your future self. I did :P

Oh, by the way, since we're talking about personal perception, there is something I've been meaning to ask here. I (not-so) recently learned about aphantasia and I was wondering how strong your mental images are.

reasonably strong, I think? I think aphantasia is quite rare, no?

Yes, it is. The problem is quantifying what's strong. Maybe your strong image would be weak for me and so on

the classic is my red your red problem

Yes

true aphantasia is rare but there appears to be wide variance in what people can "see" in front of their inner eye - e.g. apparently some people have trouble imagining colour or other details even though they "see something"

I can imagine virtually everything, with colours and so on, but still images do not seem to be an option

well, if I ask you to imagine a rubik's cube, and imagine it rotating, could you do that?

I have full scenes - e.g. I would describe the experience of reading a novel as seeing an internal movie generated by what I read

@qwerty I imagined a rubik cube. It spinned for an instant and it was gone

ahh. i guess maybe im somewhere in between the two of you then

my experience of reading a novel is not at all like an internal movie, and i never give the characters faces. they are like shadows or suggestions. scenery and objects, even animals, might be quite detailed, but human characters are not

@ACuriousMind I think I do experience that too, but only if I'm immersed enough that I do not realize. Also, it's not like a proper movie. For example, I can only picture short portions of the rooms in the novel or only close-ups. A detailed general picture is not viable for me. In other words, I would call it more a sensation, a feeling

I suppose it's like a series of vignettes

@qwerty Yes, exactly, I imagine the concept of it, mostly

The best kind of movie in my head would be a stop-motion movie :P

yes - as you describe. i would call them vignettes

What I can't stand the most is the lack of control, though

Probably I just don't have a fervid imagination, even my dreams are the most illogical and unphysical thing possible :P

my memory is generally strongly tied to visuals - for instance, when I think "QFT", the first association is an image of the lecture hall where I had my first QFT course

Okay, in that we're similar. When you mention a physics argument I remember the layout of the page. If you show me some notes, I may even remember details about the day when I wrote them or where I was

But again, they are flashes, rather than still and stable images

do you all have strong mental images when it comes to abstract maths? or does it depend on the topic

With maths not at all. I'm not much of a visualizer in those things, because you'd need still images

I think for me it depends on the topic, but I'm always impressed by a nice or intuitive visualisation

@qwerty yes, but I'm not sure the images are helpful :P (like, when I think "coordinates" I see a grid overlaid on some curved surface)

@ACuriousMind haha, because they can be misleading?

more because they're just abstract images that represent the things I'm thinking about - I can't manipulate them to "reason" about the math most of the time

some mathematicians seem to be able to use their internal visualizations to reason about stuff, but I can't do that (except for very concrete geometrical questions)

@ACuriousMind I mean, I think it's also important to see how useful that is. I visualize a grid too, but as soon as I try to do something with it, it collapses or something chaotic happens. Some people can actually use their mind like a blackboard

Mathematicians just visualizing things in their heads when they solve problems is one of those fairly annoying movie trope :p

i think this was the roadblock I had when we had that discussion about the einbein - i could visualise the 1+3 but not the 3+1 lapse function

@Slereah sure it's an exaggerated trope, but I've met several who seemed to be able to it, at least to a much greater extent than me

yeah but that's just how everyone does it in movies

Quite possibly the stupidest one in recent memory

cos movies are visual media I guess. same reason you dont see much smartphone usage in movies

doing science is just a superpower

tiktok.com/@astudiodigital/video/7287008677360405790

me going to my memory palace

@Slereah this is just dreadful film making :p

me thinking about atoms

@Slereah Still a better trope than mathematicians understanding things and notation they had never seen before because "this shit is natural for me, duh"

Really the most realistic math scene I've ever seen was

Or even worse, non-mathematicians geniuses just seeing blackboard

Damn you Will Hunting

more like bad will hunting

@Slereah This shit is real

aw I liked good will hunting. I saw it for the first time earlier this year

The movie is not bad, it's just annoying to see their portrayal of "geniuses"

i need to reponder what i said whoops

ok i am back to what i originally said. i do not see how this middle term is zero. what we should really do imo is $\nabla \cdot \text{Re}[E(r,t)]$ and then the real part is a cosine and then we take the gradient of that to get something that is nonzero, no?

sorry i meant divergence

@Relativisticcucumber the real part and the gradient/divergence commute

simply because for $f = g+\mathrm{i}h$ you have $f' = g'+\mathrm{i}h'$, i.e. differentiation is linear

@ACuriousMind yeah the thing is i made this observation and was trying to prove to myself that both ways work

but i only get what those note say if i do the differentiation first so i guess smth must be wrong

but actually neither way leads to zero when i do it... i mean the gradient brings down an $ik$ then the real part of this is still nonzero ? we can rewrite the exponential as $cos + i sin$ and so the i from the differentiation kills the i on the sine, so that's the real part?

I mean, the point here is not that you take your wave and differentiate it and get its zero, right?

You know it has to fulfill Maxwell's equation, and so you get $E\cdot k = 0$ as the condition for this to be zero

for waves for which $E$ is not transverse to $k$ you don't get zero

kms yes

ok so this means only transverse waves can be a solution to the maxwell eqs in vacuum

ty

@Slereah there is a recollection by feynman or someone else about von neumann or someone else about how they count numbers. The mathematician counted numbers supposedly by visualizing a measuring tape like object unraveling

i could certainly not do that but perhaps it is possible

kind of going off of this q physics.stackexchange.com/questions/415855/… and the answer from emilio, we study plane wave solutions in great depth, but are there other solutions that are of large importance? essentially, why do we only treat plane wave solutions?

@Relativisticcucumber Because the equation is linear and the plane waves form a convenient basis for the space of solution - every solution is a sum or integral of plane waves (via Fourier series/transform). It's not that we "only treat plane wave solutions", it's that in principle you can reduce everything to plane waves + the superposition principle.

in specific situations you will certainly consider specific solutions such as wavepackets or the Bessel functions or whatever

@ACuriousMind oh i see. i guess this is what we do in quantum when we find eigenstates?

@Relativisticcucumber yes, the Schrödinger equation is also linear

interesting. the more i think about it the more similarities i find btwn e&m and quantum

this is interesting bc id think someone who had studied ode and pde would breeze through physics

it's just because they're differential equations with wave solutions :p

Broad category

@Relativisticcucumber after months of observations I have concluded that you are clearly ACM's favorite student :P

Oh no, the cucumber is offline :(

@HerrFeinmann no I love you all equally

aaaaw 😭

Then it's time to mention the daily diffeom-

@Relativisticcucumber another way of putting it is that linear eom $\implies$ solution space is a vector space $\implies$ convenient to find a nice basis of that solution space

Is the transformation law for passive transformations from coordinates $x$ to $x'$ $A^{\mu}(x) \rightarrow A'{}^{\mu}(x') = \Lambda^{\mu}{}_{\nu}A^{\nu}(\Lambda x)$?

Nevermind I've found it

Is this physics.stackexchange.com/a/644062/310229 correct?

It seems a bit strange

Surely you need the Jacobian to be $1$

in his equation

@ACuriousMind

In his first equation I mean