The h Bar

Obliv

12:45 AM

im confused on what $e^{\beta \varepsilon}$ means here

why isnt $\varepsilon = 0.82$ meV

OH i get it

it's to keep the work neat

naturallyInconsistent

2:19 AM

@Semiclassical why is that confusing?

Obliv

yea nah that ain't it.

Silly Goose

7:18 AM

does anyone have a favorite treatment of EM? I am interested in something maybe a little quirky...not just a copy paste of jackson, zangwill, or griffiths

DIRAC1930

9:11 AM

"I asked Dirac in 1973 how he then felt about this point; he told me that he did not regard bosons like the pion or $W^{\pm}$ as 'important'"

At the top of pg 14 here professores.ift.unesp.br/ricardo.matheus/files/courses/2014tqc1/… in Weinberg QFT 1

Mr. Feynman

9:57 AM

Dirac had double standards for fields :P

Emilio Pisanty

10:10 AM

physics.stackexchange.com/questions/812781

wow, that was fast

(10k only)

any mods around -- was this deleted through an automated system, or via community flags?

ACuriousMind

@EmilioPisanty semi-automated: ultimately deleted via flags, but most of the flags were driven by Smokey

Silly Goose

10:29 AM

what is the mean value theorem used for in physics :P

DIRAC1930

@Mr.Feynman I wish the particles had better names. The proton, neutron and electron sound profound compared to all the ones discovered later

Silly Goose

"Three quarks for Muster Mark!
Sure he hasn't got much of a bark
And sure any he has it's all beside the mark.
But O, Wreneagle Almighty, wouldn't un be a sky of a lark
To see that old buzzard whooping about for uns shirt in the dark
And he hunting round for uns speckled trousers around by Palmer-
stown Park?"

DIRAC1930

Tbf, string theorists are very good at coming up with names

Silly Goose

naming something after a word in finnegan's wake seems like a very physicist thing to do :P

DIRAC1930

Murray Gell Mann should have had a disciplinary for that lol

Silly Goose

10:37 AM

i sort of wish every lecturer was a string theorist :P

Mr. Feynman

11:49 AM

@DIRAC1930 not sure about what in "neutron" sounds more fundamental than "kaon" or "pion"

DIRAC1930

12:16 PM

Probably romanticizing the history of the electron, neutron, proton, positron

and of course the photon

Currently I would put my money on all fields being emergent in the same way the fields are emergent in cond-mat.

It sounds completely weird that the same methods work and you can quantize something like a lattice vibration and subsequently get an emergent particle i.e. a phonon

Sanjana

Why are homogeneous coordinates of $\mathbb{CP}^m$ sections of the hyperplane line bundle?

Odestheory12

12:47 PM

Hello.

I have a question with the following problem:

"A platform of M kg moves with a velocity of 2 m/s on straight horizontal rails without friction. On the platform, and at rest relative to it, an elastic spring of negligible mass and a spring constant of 1000 N/m is arranged in the direction of motion, initially compressed by 20 cm, and two bodies, attached to the spring, one at each end, of masses m1 and m2 kg, respectively. At a certain moment, the spring is released, causing both bodies to start moving on the platform in opposite directions, being shot off. Assuming no frictional forces exi…

user 70432

12:59 PM

Problem Solving Strategies

General chat for high school physics. For MathJax see [here](m...

3

107

Ryder Rude

1:14 PM

"the way to study science/math is to be a frog in one field and a bird in the others"
do u agree with this

user 70432

1:48 PM

how about start out as a caterpillar and work your way up to becoming a butterfly

Ryder Rude

or a tadpole

user 70432

yup

Sir Cumference

6

I hate that this made me laugh

2

user 70432

2:07 PM

pff

4.

ACuriousMind

2:49 PM

@Sanjana what is your definition of "hyperplane line bundle" where this is not immediately obvious?

@DIRAC1930 What's profound about "the first one", "the neutral one" and "the one that has to do with amber"? :P

I wasn't trying to do that but the last one sounds like a Friends episode

Sanjana

3:29 PM

@ACuriousMind See I have attached the definition.

ACuriousMind

ah, that's an unfortunate starting point

Sanjana

resource recommendations?

Or just tell me what is going on if it is not tooo long????

ACuriousMind

4:09 PM

@Sanjana This answer should explain what you need; the idea is that the sections of the dual of the line bundle assign a linear functional to each point $[z_0:\dots :z_n]$ that can acts on the fibers of the line bundle; since the fiber above $[z_0: \dots : z_n]$ is just vectors $\lambda (z_0,\dots,z_n)$,

such linear functionals can be given by $\sum_i a_i z_i$ and for only one $a_i = 1$ and the rest zero you therefore have $z_i$ "the homogeneous coordinates" as sections

Vincent Thacker

I can finally see deleted posts

Obliv

do you guys think the speed of light is pretty slow? in comparison to the size of the universe that is

I mean in earlier age of universe it was more impressive but 186,000 miles/sec is so slow :p

Sanjana

4:31 PM

@ACuriousMind Where from the $a_i=1$ for one $i$ and zero for rest come from?

ACuriousMind

@Sanjana I just mean that a general section is given by $\sum_i a_i z_i$ for arbitrary $a_i$. If you choose the $a_i$ in that specific way, then you just have $z_i$

and I assume that by "the homogeneous coordinates are sections" you mean that "$z_i$" (for each i) defines a section, no?

Sanjana

Yes

@ACuriousMind Just to confirm one thing...A section is a smooth assignment of elements of the fibre over the base space, right?

ACuriousMind

yes

Sanjana

@ACuriousMind How are $\sum_i a_i z_i$ linear functionals?

I mean the $(z_0,z_1,...,z_n)$ are the vectors. How do we know that the covectors are just linear combinations of the coordinates?

ACuriousMind

the point here is that the tautological bundle consists of points of the form $([ z ], \lambda z)$ (for some $\lambda \in\mathbb{C}$, and so a section of the dual has to assign essentially just a number to every $[z]$ (since the fiber is one-dimensional, the elements of the dual are just numbers) in such a way that the number scales by $\lambda$ when the $z_i$ scale by $\lambda$. $\sum_i a_i z_i$ does that.

Sanjana

4:42 PM

Oh ok...got it

@ACuriousMind Is this canonical line bundle the same as the tautological line bundle defined in that other text?

ACuriousMind

this particular definition seems to be, but be careful that the canonical line bundle in general means something different

The intro on Wiki even mentions that the tautological bundle used to be called canonical bundle but "has dropped out of favour"

Sanjana

Thanks. Previously I mistakently thought that the canonical line bundle as defined in wikipedia and most other places is the same as tautological bundle via some theorem or something...

ACuriousMind

no, in the twisting sheaf notation the tautological is $\mathcal{O}(-1)$ and the canonical is $\mathcal{O}(-n-1)$, i.e. it's the $\mathrm{dim}+1$ power of the tautological

(where by dim I mean the dimension of the projective space)

Sanjana

sheaf :)

Such a cute name

Is there any good easy intro to algebraic geometry for physicists? The aim is to computer Chern classes for CY manifolds defined as projective varieties

ACuriousMind

the words "easy" and "algebraic geometry" don't go well together :P

as I already said to @lucabtz you probably want something that actually focuses specifically on complex geometry and not algebraic geometry over arbitrary rings/fields

unfortunately I learned the general version and don't know any such resources :P

Sanjana

4:55 PM

These are the kinds of things I am trying to understand

ACuriousMind

also by GAGA in principle you would not need to do any algebraic geometry at all - this should in principle be all also accessible via complex analytic geometry

Sanjana

@ACuriousMind Oh okay...there is a convo with Luca...lemme find it then

ACuriousMind

that one was in the context of monodromy and local systems, it wasn't about the exact some topic

Sanjana

@ACuriousMind Actually I looked into Nakahara's treatment of complex manifolds but it doesn't cover these sheafy stuff

ACuriousMind

I mean in analytic geometry you would remain talking about bundles and not sheaves but that's just terminology

Obliv

5:49 PM

am I going insane?

this while loop goes on forever, no?

the solution is apparently "1 1"

naturallyInconsistent

@Obliv no

@Obliv yes, this code will give this result

Obliv

How? $c++$ occurs only if $!(c\%2==1)$

but at c=1 it'll be true.. and $c++$ doesn't occur again

OH

ya nvm i forgot x goes to 1 so the while loop stops

jk

Claudio Menchinelli

6:41 PM

Today we introduced the Poincaré's sphere in relation to stokes vectors in Optics: it was a very brief and qualitative approach

briefly, you can identify a certain polarization state with a stokes vector

then you act on it via a transformation, mainly rotations or translations or compressions and so on

my point is: if a transformation is represented by a 4x4 matrix acting on a vector of the form $\mathbf{s} = (1,s_1,s_2,s_3)$

I need to make sure that traslations do not take the trasnformed sphere outside the original one (which has radius 1 by construction)

is there a way to ensure this condition?

my professor told me that a matrix lets me perform via a compression and via a translation simultaneously

how does this translates into matrix language though?

I need to go now, Ill come back later and explain things more carefully

in case someone knows anything, feel free to give me any hint

Sanjana

6:56 PM

@ACuriousMind How do the sections of the tautological line bundle itself look like? Are those also $\sum a_i z_i$ themselves?

ACuriousMind

@Sanjana there are none

see e.g. math.stackexchange.com/a/488155/143136

also math.stackexchange.com/a/2576402/143136

Sanjana

@ACuriousMind I think I understood the proof in this...but why aren't the coordinates a counterexample?

ACuriousMind

...because they are sections of the dual?

how would they be a counterexample

Sanjana

Cant something be sections of both a bundle and its dual?

ACuriousMind

I don't know how

note that the argument for the section of the dual relies on figuring out it has to be a linear function since it has to scale with $\lambda$

the same kind of argument shows that sections of the tautological bundle scale with $\lambda^{-1}$

Sanjana

7:09 PM

@ACuriousMind Oh...I previously misunderstood why it had to scale as $\lambda$

I thought that it follows from the definition of a linear functional's "linearity"

@ACuriousMind But now I see that it has to do with property of the base space considered here

@ACuriousMind Why is the holomorphic tangent bundle over the projective space spanned by vectors whose components are sections of the hyperplane line bundle?

ACuriousMind

at this point you just need to learn projective geometry :P

Sanjana

Where do I learn that from :)

ACuriousMind

2 hours ago, by ACuriousMind

unfortunately I learned the general version and don't know any such resources :P

Sanjana

Ok fine...I will also try to learn the general version then. Give me a name :/

ACuriousMind

The usual reference is Hartshorne

or Mumford, I guess

Mr. Feynman

7:39 PM

@ACuriousMind ACMacGyver starter pack:

Claudio Menchinelli

7:55 PM

lol

More Anonymous

I wanna rejoin academia ....

One day

I'd do a math degree this time

imbAF

Regarding the free electron model, in wikipedia the following is said:
Each quantum state of the system can only be occupied by a single electron.

This statement is confusing. There is a distinction between the state of the system (entire metal) and the states of electrons in it.

Can anyone explain to me what is being said ?

naturallyInconsistent

@imbAF It is not; Including the specification of the spin of the electrons, each electron state, which is spread throughout the metal, can only be occupied by a single electron.

imbAF

True

but the system, is comprised of all electrons

it's the metal no?

the totality of all electrons aka the electron gas, and an individual electron should represent two different systems

or rather, are not the same

naturallyInconsistent

8:12 PM

Well, you are now going into serious pedantry. Yes, the system would have all the electrons. But it should be clear that they mean each single electron's state in such a system, not the whole system at once

imbAF

ok

Claudio Menchinelli

also, I want ro remind you that two identical fermions cannot be in the same individual state, this means that you must associate a single quantum state to each electron

imbAF

Yes, when it comes to an electron as a system, the fermi principle applies

Claudio Menchinelli

Yeah I think that's what the book is referring to

imbAF

One more thing

In position space, the lattice points are the places where ions are located

in K space, the points are what?

Claudio Menchinelli

8:19 PM

This goes beyond my current knowledge sorry

what do you define as K space?

imbAF

the reciprocal space

for example for a simple cubic lattice

Claudio Menchinelli

Ok this is solid state physics, you mean like bravais lattice and stuff

I havent covered the matter yet

imbAF

in reciprocal space the brillouin zone is a cube with dimensions $2\pi/a$, where a is the constant

aha ok

yeah that is what I mean

Claudio Menchinelli

yeah I studied this a bit on my own, you must work with a particular Fourier Transform

Anyways, this is very important stuff so a quick search on the site would probably solve your doubts

or even wikipedia

or your textbook if u have one

naturallyInconsistent

@imbAF you need to be much more precise here. What do you mean by "points"

imbAF

8:26 PM

in position space the structure of the solid is that of a crystal lattice, an abstract construct, a repenting pattern of points in space. The points are the most probable locations where the ions are located

Claudio Menchinelli

en.wikipedia.org/wiki/Reciprocal_lattice This seems very clear

naturallyInconsistent

@imbAF And they don't map well to anything in k space.

imbAF

They do not. I didn't say they did

naturallyInconsistent

Ok, so what are you asking for?

imbAF

I said that in position space the abstract points have a meaning

while in k space, we also have a similar abstract structure

but in here, what do the points represent

naturallyInconsistent

8:30 PM

Well, in this k space, are you interested in k points or G points?

imbAF

G points? I know the $\vec G$ vector, which is the equivalent of the position translation vector but in, now, the k space

G points, I do not know these

naturallyInconsistent

Good. You are correct to focus first on the G vectors = G points. These have a really nice interpretation. If you study solid state physics, the textbooks will teach you: G vectors correspond one-to-one to Miller indices, which means that each one refers to a set of planes, that when you do crystallography, are what makes Bragg's law work.

Laue's equations look mathematically extremely different from Bragg's Law, but you can prove that Laue's equations reduce to Bragg's Law and have exactly the same physics in them.

imbAF

Ok

I am kinda familiar with the Millers indices

but What I was aiming at was the fact that one has discrete K values

in a crystal

as many as you have ions . If in 1D you have N ions, you have N modes

And I wanted to know if this was somehow related to the nr. of points one has in K space

naturallyInconsistent

8:49 PM

Yes, it is precisely that relationship. To have a k point very close to Gamma (origin), you must have bigger and bigger crystals, i.e. many multiples of the unit cell. k points in the 1BZ means multiples of unit cells each waving at different phases, whereas G points are always unit cells are in phase with each other

As for number of modes, you are thinking of something else. The number of possible electron excitations refers to the number of bands.

Again, if you properly read a solid state textbook, such examples are directly covered. For example, in phonons, you have the optical v.s. acoustic branches, and they are extremely clear examples of what we could mean.

Vincent Thacker

9:40 PM

@imbAF Not really. In the idealized version both the lattice and its reciprocal are infinite

If you have finitely many points then it is no longer periodic

Silly Goose

10:58 PM

i would like to know projective geometry too :P

seems nice to know for finite dimensional quantum mechanics ~

Problem Solving Strategies

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