The h Bar

12:54 AM

Successful detonation ignition, or creepy old man face...

3:47 AM

@SirCumference the two equations. In general if two equations are equal to zero then under what conditions can we say that the two equations are equal to each other?

Sir Cumference

@Korra what does it mean for two equations to be equal

we say "numbers" are equal, but i've never heard someone say equations are equal

satan 29

5:29 AM

@DavidZ Thank you so much for your response!

JingleBells

6:01 AM

I'm really struggling to understand the concept and purpose of marginal costs. It's the difference in total cost over difference in output. So it's basically how much I am paying to be able to produce one more output but it's not how much it takes me to produce a single output?

JingleBells

6:13 AM

(i.e it's not the per product price, it's just how much money I've paid to be able to produce with one more additional output)

Korra

@SirCumference so if two equations are equal to say a constant c, we cannot say that the two equations are equal? Ever?

user434058

@Korra No. The expressions on a certain side of the "equals" sign might be equal to another expression (similarly defined), but two equations as a whole, are never equal.

user434058

Though, if you really want to, then you could say that the equations $x^2=0$ and $x=0$ are equivalent, but using "equal" doesn't make any sense.

Korra

Then what about something like this?

user434058

@Korra Which line do you think says that equations are equal?

Korra

6:42 AM

Well, not equal

It says they are simultaneous

And both of them equal to zero

Giving us the equation in the box

@FakeMod

user434058

System of equations

In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a: System of linear equations, System of nonlinear equations, System of bilinear equations, System of polynomial equations, System of differential equations, or a System of difference equations == See also == Simultaneous equations model, a statistical model in the form of simultaneous linear equations Elementary algebra, for...

Sir Cumference

you're looking for a common solution set

user434058

In this context, it means that both the equations are simultaneously true, and can be used to find a certain quantity.

user434058

That's it.

Sir Cumference

but you used unknown coefficients $A, B$ in your first equation and $C,D$ in the second

do you mean to have $A$ and $B$ in both equations, then find out which values of $A$ and $B$ satisfy the system?

(in the Math SE question I mean)

user434058

7:08 AM

How do I explain the OP that their question is not as valuable in the physics community as they think it to be?

user434058

No, the value of speed of light is universal constant. Light can't travel at any random speed. It is us who quantify this value. This seems to be an open question in relativity theory. — Richard 3 mins ago

user434058

Or is it really important to know why the fourth digit in the speed of light is a 7, and not a 6?

Sir Cumference

well we define the meter as 1/299,792 times the distance light travels in a second

Korra

@FakeMod it is really important to know that speed of light in vacuum is universal, irrespective of the observer

@SirCumference wait a minute I will post the actual equations I am concerned with

user434058

@SirCumference that's what I said, but the OP isn't just satified with "conventions" as a reason and needs a depper reason for such a constant's digits.

user434058

7:17 AM

@Korra It is. It eefinitely is important to know the speed of light, but the digits are immaterial.

Korra

user434058

@Richard What can I say, except a mere coincidence? Choose another base system, you get different digits. There's no really deep, magical, mystical significance of these random digits. Light goes on with the speed which it wants to, it doesn't know the value of its speed. It is we, who have defined a clever way of communicating such a speed. Somewhere out there, if aliens existed, they would've definitely chose a different number system with a different base and symbols, giving them a different value of speed of light, but does that chamge the speed of light? No. — FakeMod 18 mins ago

user434058

@Korra ^

Korra

The equations I am concerned with @SirCumference

Marked 1 and 2

I somehow want to combine them because they both are for the same system but with separations

user434058

Now I am seriously doubting my answer:

user434058

7:24 AM

@FakeMod Your answer and comments seems to be misleading. You aren't even able to understand the question well. This is an open question. Please don't try to give nonsense answers to such conceptual questions. — JamesP 1 min ago

user434058

Could anyone be kind enough to verify its correctness? physics.stackexchange.com/a/567348/258881

Korra

7:58 AM

@FakeMod your answer is correct but maybe the person who asked the question is not interested in the quantitative but rather the qualitative aspect

His question is more philosophical

user434058

@Korra The last comment really put me off. Though the commenter doesn't have any apparent authority over the subject, but I just can't se why he's right. Anyways, thanks for confirming.

user434058

@Korra Yeah, I have tried a bit of philosophy in the comments, but to no avail :(

Korra

It's anyway is kind of pointless

Why is the speed of light 3 and not 4 or 5?

Well, if you want 4 or 5 or 1 to be the speed of light change the system of units

That is completely logical

user434058

@Korra yeah. I mean we can express light in other known constants, but then nothing stops the OP form asking why those other constants take the value they take. That doesn't seem to lead us anywhere.

Korra

You cannot ask why $\pi$ is 3.14

It's in the universe, already in grained

user434058

8:01 AM

@Korra OP's expecting something deeper than units :P

user434058

@Korra right.

Korra

:-D

Anyway, I think the second long answer is enough for the OP

@FakeMod the commentator is actually the one being real rude

Saying that what you are giving as an answer is nonsense is actually nonsense because your answer makes perfect sense

Someone please help meeee

user434058

@Korra Yeah, makes sense to me. But it can be wrong. FWIW, there's a third answer as well.

Korra

@JohnRennie can you take a look at my question above?

user434058

It talks about anthropic principle.

Korra

8:12 AM

@FakeMod umaxo's answer is pretty spot on

user434058

@Korra Yeah, his answer is definitely nice.

Korra

But again it all boils down to the same thing that in this universe where we are living, the speed of light is what it is and it doesn't depend on us in anyway whatsoever

If someone doesn't like the number which is the speed of light change your units and work with your favourite numbers 😉

user434058

@Korra hmm... True. In fact, you'd always have to use some unaccounted facts, always, no matter what kind of theory you propose, you cannot justify everything.

user434058

@Korra lol :D

Korra

I remember having such stupid questions as a high schooler

@FakeMod yup

Like how we have to accept that 2 comes after 1 because we've understood it that way since millenia

user434058

8:17 AM

@Korra I also used to have them (mine was about electron's charge and mass), but then I convinced myself that it might be nothing more than a coincidence (Idc at this stage whether I convinced me right or not, I just, probably, wanted to move on, and leave such stuff for the time when I understand stuff like standard model and QFT, so on).

Korra

@FakeMod I think maybe the OP person's question arise from the way we usually define speed

That is distance divided by time

And so one

Like how someone runs at a particular speed which is contributed by various factors

Like the speed of rotation of earth which is again contributed by various factors

But light isn't something like that

So to accept a quantity like speed just

as is

Goes against our intuition of having a reason for it in most of our daily instances

user434058

True. ^

Korra

I think I will drop my project

Again

And go back to lazing around

☹️

Korra

8:54 AM

@JohnRennie can you help, pleeeeeeeeease

John Rennie

@Korra hi :-)

What's up?

Korra

9:25 AM

@FakeMod the one before this

@FakeMod and before this

@JohnRennie

so if two equations are equal to say a constant c, we cannot say that the two equations are equal? Ever?

@FakeMod So can you explain what is happening in here

John Rennie

@Korra you mean $f(x) = C$ and $g(x) = C$ for some constant $C$ ?

Korra

Yes

@JohnRennie

John Rennie

But that means $f(x) = g(x)$ doesn't it?

Korra

Yesss

Now suppose these two equations are here

@FakeMod the ones before and after this message

@JohnRennie what you told will still apply?

John Rennie

I don't understand what you are asking ...

What are the two equations.

Korra

9:33 AM

When the constant c is zero

@JohnRennie I have tagged it

I can't reply to my own messages so I can't directly point to it

Sorry

Sir Cumference

@Korra you wouldn't really say the equations are equal, you'd say the terms involved are equal

i think this is why people were confused

John Rennie

Assume I have no special desire to read back through a series of hand written notes. What are we talking about?

Korra

@SirCumference aren't those two things similar?

Sir Cumference

i mean equations are statements

they don't really have value

well there's logical equivalence, but that's something different

Korra

@JohnRennie it's like this. I have two trigonometric equations both equal to zero

John Rennie

9:36 AM

Then they must be equal to each other.

Korra

Then can we put them as f=g=0 or do we have to put it as f=gλ=0

@JohnRennie okay but in this one

John Rennie

Ah, good point, I see what you're saying now.

Korra

Yay

John Rennie

Because you can multiply the equations by an arbitrary constant and remain equal to zero.

Korra

Please enlighten me so that I may have a chance to complete the project

@JohnRennie so my question is which one should I use in my case

The f=g = 0 one or the other

John Rennie

9:40 AM

@JingleBells delete that please or I will flag it.

Korra

f=gλ one

John Rennie

It should work regardless of the value of $\lambda$

i.e. you can choose any value of $\lambda$ you want and you should get the same results.

Korra

@JohnRennie but which is correct?

John Rennie

I wonder if it means you have gauge redundancy in your equations ...

Korra

What is gauge redundancy?☹️

John Rennie

9:43 AM

@Korra when in doubt, try it and see. Set $\lambda = 1$ and solve the equations. Then set $\lambda = 2$ and solve the equations again, then compare your result. You should find you get the same result.

Sir Cumference

@Korra isn't the lambda one just a general case of the other?

Korra

@SirCumference Yes

@JohnRennie may I create a separate room for this discussion?

John Rennie

@Korra I don't have time for a prolonged discussion this morning I'm afraid.

Korra

@JohnRennie that's okay. Whenever you are free

Can I create one for that time?

John Rennie

Yes

Korra

9:49 AM

Btw @SirCumference I like your name! It's amazing!

@JohnRennie thank you! Can I ask again tomorrow?

John Rennie

Yes

Korra

@JohnRennie thank you!!

JingleBells

1:26 PM

I have a simple FM radio receiver here and I can pick up radio channels and listen to music. My question is, why am I not able to listen to phone calls for example? Have I not tuned it to the right frequency and isn't the device not able to decode the signal? I'm surrounded by RF signals (TV, Calls, Youtube videos...), why can I only pick up music channels?

@JohnRennie When you said you are getting paid for checking servers here and there, what exactly do you do? What types of servers?

ACuriousMind

3:20 PM

@JingleBells Because your typical radio receiver is tuned to receive in the HF or VHF band of frequencies. Cell phones, for instance, typically use UHF frequencies, one magnitude of frequency higher.

JingleBells

3:47 PM

Hmm got it, so it's all about frequencies

Stan Shunpike

4:11 PM

hello

Charlie

When we go to the position basis and do QM, as is typically first introduced, we (I think) are still working formally with the Hilbert space elements, but we talk about now the individual components of these vectors by labelling them with the eigenvalues of the position operator, which is now diagonal, for instance $$|\psi\rangle\rightarrow \psi(x).$$ Can we do this generally for any operator,

for instance if we work in the energy eigenbasis could we in principle write $$|\psi\rangle \rightarrow \psi(E)?$$

where $\psi(E)$ is an element of (the same, $L^2$?) function space.

ACuriousMind

@Charlie Yes, but for operators with discrete spectrum (like energy in bound systems), this isn't a continuous function of the reals but just a square-summable sequence you get by expanding $\lvert \psi\rangle$ in the eigenbasis $\lvert \psi_E\rangle$ as $\lvert \psi\rangle = \sum_E c_E\lvert \psi_E\rangle$.

The coefficients $c_E$ are what you call $\psi(E)$ and are a square-summable sequence

Charlie

oh that's pretty cool

user434058

Does anybody know a good dupe target for this question?

ACuriousMind

You get "true" functions for $x$ and $p$ because they're unbounded operators and so the spectral theorem that says that we can expand in a countable eigenbasis doesn't apply to them - their "eigenvectors" $\lvert x\rangle$ are not vectors inside the Hilbert space.

Charlie

4:25 PM

Maybe this or this? @FakeMod

ACuriousMind

Note also that if the operator is degenerate, this doesn't work quite so straightforwardly - generically you want to expand w.r.t. a CSCO, not a single operator.

Charlie

is it even possible for the position operator to have degeneracy?

I've never thought about it, but that sounds strange to me

ACuriousMind

@Charlie if you have spin it is degenerate!

user434058

@Charlie Ah, thanks. The first one will work.

Charlie

oh yeah that would make sense

np @FakeMod

ACuriousMind

4:28 PM

You usually write that as having a vector-valued $\psi(x)$, but that's equivalent to expanding as $\psi(x,s)$, where $s$ are the eigenvalues of some $S_i$

user434058

Damn! I have run out of close votes :D

ACuriousMind

Whether you write a spin-1/2 function as a pair of components $\psi_1(x), \psi_2(x)$ or think about it as a pair $\psi(x, 1/2),\psi(x, -1/2)$ is just convention

Charlie

@ACuriousMind Out of curiosity when you say the eigenvectors $|x\rangle$ are not vectors of the Hilbert space, where are they? Assuming "where" is even a well defined question here

ACuriousMind

@Charlie the game is rigged

Charlie

ah this rigged stuff again ok

ACuriousMind

4:30 PM

But no-one really uses the formulation of rigged Hilbert spaces in my experience for this, if you want to be formal just avoid talking about the $\lvert x\rangle$ at all :P

Charlie

is there much to be gained from going to the trouble of understanding the rigged spaces beyond curiosity?

ACuriousMind

I don't really think so

Charlie

hmm ok, I will live without it for now :P

apt45

4:45 PM

I don't understand something regarding Effective Field Theories in expanding/contracting universe. I

In flat space, EFTs can be written as a derivative expansion of the low-energy fields, and this expansion breaks when the energy of particles in the EFT is roughly the mass of the heaviest particle that has been integrated out.

Imagine an EFT evaluated on a gravitational background which is contracting (for example, FRW with a cosmological constant). In this universe lengths contract, so waves would experience a blueshift as the time flow. Does this mean that the EFT approximation breaks down after a finite period from the emission time?

For example, assume that you are sending a wave with high enough frequency $\omega \lesssim M$, where $M$ is the particle that you have integrated out. In flat space, at $\omega \sim M$, the EFT breaks. In the contracting universe, after a finite period the wavelength decreases (thus the frequency/energy increases) and could reach the value $\omega = M$.

Does this blueshift occur? Is there a time after which I cannot describe the dynamics of the wave with my EFT?

apt45

5:14 PM

Is my question unclear?

Charlie

I can't personally answer your question but if you want a larger audience to see it you could try asking it on the main site

apt45

sure, I will eventually consider asking on SE.

Stan Shunpike

5:52 PM

how does one draw adding torques?

for velocity, its easy and intuitive

but i'm having trouble finding a picture of adding torques in this manner

ACuriousMind

@StanShunpike What do you mean? Since in 3d torques are vectors, too, why can't you add them just like the velocity vectors?

Stan Shunpike

@ACuriousMind I guess you can. I have to do a project almost entirely based on angular quantities so I have to review a few things :') That's helpful. I'll go read a textbook then

JingleBells

@StanShunpike hello

Sir Cumference

6:14 PM

@Korra thanks lol

Stan Shunpike

9:41 PM

Are there any examples involving torque where both things rotating are confined to a hyperboloid?

like this

JMac

@StanShunpike Reminds me of the "falling cat problem" but I don't think that's actually a hyperboloid.

Stan Shunpike

@JMac that's really interesting! i actually read a book that talked about how cats fall. i'm surprised they didn't mention that

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