The h Bar

Secret

2:22 AM

(Sorry was asleep at that time but forgot to log out, hence the apparent lack of response)
Yes you can (since $k=\frac{2\pi}{\lambda}$). To convert from path difference to phase difference, divide by k, see this PSE for details
http://physics.stackexchange.com/questions/75882/what-is-the-difference-between-phase-difference-and-path-difference

Physics Meta

3:55 AM

0

Q: Reasking a Question

What should I do if I find a question that I want to ask but does not have satisfactory answers, how can I make it active or re-ask the question without it being marked duplicate?

support

John Rennie

5:00 AM

@WilliamBulmer Postulate?

Noah P

8:29 AM

Morning all!

The Quantum Physicist

9:14 AM

Hi guys, I have a little issue with matrix traces in quantum mechanics density matrices that I don't understand. Based on this discussion:
http://chat.stackexchange.com/transcript/message/31105713#31105713

If I do this "element-wise" time evolution, then the diagonal elements of the matrix are unchanged

but then the trace is constant!!! right?

The literature says that the trace of some time independent operator Tr(rho.A) is my expectation value

but this is then constant!!

The solution that we use in our calculations is that we sum ALL the matrix elements, and still call it trace. Why is that correct?

Noah P

@MAFIA36790 You there?

user116211

@NoahP go on; bit busy now so maybe would be late to respond.

Noah P

Ok - just a quick question. If I'm considering a universe which is radiation dominated, in order to find the time dependence of the density of matter, can I ignore cosmological constant? It makes it a hell of a lot easier @MAFIA36790

Just want to check im not commiting a cardinal sin or anything

user116211

9:30 AM

@NoahP Well, as I said, I'm not into cosmology yet; so can't comment. But if there is maths part, I may help.

Noah P

Ahh okay, no worries. I think that im doing okay with the maths so far, thanks

The Quantum Physicist

@ACuriousMind Could you please look into my question, since you know the problem?

user116211

@yuggib: o/

yuggib

\o

Swapnil Das

hi

Noah P

9:42 AM

@MAFIA36790 I've got to $p^{-1}dp=-4a^{-1}da$

It's not going to work if I integrate that is it?

user116211

@NoahP Why not?

Noah P

Would I not get stuff to the power of 0?

user116211

@NoahP Elaborate your next step to solve the equation.

Noah P

Wait, would I get $ln(p)=-4ln(a)$?

user116211

@NoahP looking good.

user116211

9:45 AM

Although lacking the constant.

Noah P

Okay, thanks! And yeah, but I'm trying to find proportionality

user116211

@NoahP BTW, use \ln to get $\ln$

Noah P

Ohh ok thanks

Do I then use () or {} for the terms?

Swapnil Das

Hello. Any one willing to explain me about the cosmological constant?

user116211

@NoahP if there is more than one term, yes, you have to use the latter one.

Noah P

9:47 AM

Depends what you want to know @SwapnilDas ? I've learnt a bit over the past few days, I'm not sure who else here could help right now

Certainly there are people who can, I'm just not sure who is online

@MAFIA36790 Ok, thanks

Swapnil Das

Any thing you could tell, I must be indebted to you.

Noah P

:31185348 I don't get the \o stuff

@SwapnilDas What do you know, whats your backround?

user116211

@NoahP oh, man; google emoticons.

Noah P

@MAFIA36790 I've never seen that before though?

Swapnil Das

I am a tenth grader with interest in Cosmology

Noah P

9:49 AM

Like :P ;) and everything yeah, O.o, but the \o is new to me

@SwapnilDas What do you know so faR?

Swapnil Das

Cosmological constant was Einstein's biggest blunders. It predicted a 'static universe model'

blunder*

Shing

I need someone's help on probability...

If I roll three dice, what is the chance of getting a result sum up to six only?

I can't think of a way other than list all the events....

Noah P

@Shing Thats probably one for the maths chat, but a quick google would help you too

@SwapnilDas The cosmological constant refers to dark energy, most would agree

Though take everything I say with a pinch of salt - I'm not sure what grade I would be in (UK, year 13), but I'm 17 myself

And learnt all this in the past week

Shing

@NoahP um....okay, though I am just working on the problem set of statistical physics.

Noah P

@Shing My solution would be to sum all the results in a table, but im sure they could suggest quicker, and its pretty quiet here atm

Swapnil Das

9:56 AM

@noa

Cool

lucas

I haven't studied probability for more than 15 years, but I think you can break your problem in two problems for simplicity. @Shing

Swapnil Das

I'm 15

Noah P

Ok

lucas

If first dice come 1, then sum of the rest dices must be 5 and so on

2, 4

3, 3

4, 2

Shing

thanks, I list all the events and then obtain the correct answer, but then I realized what if the question is 100 dice? therefore, I am thinking hard of a way to calculate it, instead of listing all the events.

lucas

10:02 AM

I don't know about mathematics solution, but I think you can create an algorithm by the method that I said.

For computer solution.

(programming)

Shing

um... that's a way to do it, thanks!

lucas

Good luck!

Noah P

@MAFIA36790 Shh, whisper it quietly, but I think I may have done Q5 without any (Apart from that quick integral) SE intervention...

AngusTheMan

11:01 AM

Hi all, just wondering about this matrix problem and require a bit of help. Here $J$ are symplectic matrices, $H$ is the Hessian of the Hamiltonian with respect to a point in phase space $x$ and $\eta$ and $\xi$ are phase space vectors. $(JJH\xi)\cdot \eta +(J\xi)\cdot JH\eta = -(H\xi)\cdot \eta +\xi \cdot (H\eta)$. I have no problem with the first term since $J^2=\mathbb I$, but how does the second term generate a negative sign! :)

*not generate a negative sign

Noah P

11:14 AM

How do I filter out tags?

Ah found it

Shing

@lucas I think there is no direct method, at least binomial distribution doesn't work. since binomial only deals with "two outcomes". I couldn't really make the question into two outcomes.

David Z

@Shing I don't know of any way other than enumerating the possible outcomes either, but you can be a little clever and use a generating function: $(x + x^2 + x^3 + x^4 + x^5 + x^6)^3$. Expand that and add up the coefficients of the terms with powers of $x$ up to 6, then divide by the total number of outcomes.

Shing

11:36 AM

@DavidZ thanks for the help, I will try that out!

David Z

Make sure you don't leave the problem until you understand what that works.

Shing

okay :)

Sanya

12:05 PM

Hey ... just a quick and stupid question (I haven't done QM in a while). For a superposition of eigenstates of a Hamiltonian in Schroedinger QM, the time evolution of the superposition is given by letting each eigenstate evolve with its own eigenfrequency, right?

Slereah

operators are linear, so yes

Shing

yes, in general, for a time independent Hamiltonian

Sanya

yep, time independent Hamiltonian, I forgot to add that; thank you both :)

lucas

@Shing I meant you can make your problem easier by breaking it. For example: 1/6 P(5)+1/6 P(4)+1/6 P(3) +1/6 P(2) and P(5) is the probability of sum of two dice to get 5 and so on. Then for P(5), ... you can do the same method.

And as I said I have completely forgotten that topic. It was a suggestion.

Jim

12:46 PM

A user accused me of behaving in a way that does not belong in the world of physics. I calmly explained why what he is arguing doesn't matter in the slightest, outlined a hypothetical (and likely true) situation that would show his motives as egotistical only, and explained that through my use of words I never had acted the way he claimed and that he should not repeat the claim without sufficient understanding in what I was saying. Then I dropped the mic and walked out of the conversation

#ThatllLearnHim

ACuriousMind

@TheQuantumPhysicist While the trace of $\rho$ is constant, that of $\rho A$ is not.

lucas

@ACuriousMind Hi :-)

ACuriousMind

@lucas Hello

@Jim Is this where I point out a flaw again? ;)

Jim

@ACuriousMind You must be a jedi, because every time I talk to you, I come out thinking I made a mistake somewhere

ACuriousMind

@Jim I'm...not sure how I feel about that

Never liked the Jedi much

(nor the Sith, for that matter)

Jim

12:53 PM

maybe they never liked you either

Noah P

More of a Princess Leia?

Jim

for reference, here is the conversation:

@Jim 1) Analyze the very concept that all objects are moving across space-time at c, and that in turn the only change still left for each of these objects is a change in each of their specific directions of travel within space-time. 2) The inevitable outcome of this analysis is an independent discovery of SR, along with an independent deriving of all of the SR mathematical equations. Granted, some folk of today would say that this is merely a magical coincidental outcome. — Sean 21 hours ago

ACuriousMind

@Jim The feeling might well have been mutual

Jim

On a different note.....

here's my reaction when the computer all my unsaved work is on suddenly crashes:

2

also when I come into work and suddenly realize it's Saturday

ACuriousMind

I don't want to see how you look on a Sunday, then :)

lucas

12:57 PM

May I ask what is your job?

Jim

physics lab coordinator

Noah P

Ooh la la

ACuriousMind

@Jim That particular user has been at this for years. If you actually manage to change his mind, I'm beyond impressed.

lucas

I am unemployed:-(

Jim

@ACuriousMind change his mind? Doubt it. Win the conversation? Probably

Sanya

12:59 PM

what is "winning a conversation"?

ACuriousMind

^

Jim

Stumping the other person momentarily and walking away before they have a chance to speak

without asking a question

Sanya

I mean, if there is an audience, it works; but most audiences are dumb and can be won without being right, so I question the worth of these "victories"

that seems like a very mature way of handling conversation :o

Jim

makes you feel good

ACuriousMind

@Jim So, if I say "flargharble" so that the other person is momentarily confused and then walk away, I win?

Sanya

1:01 PM

well, I'm happy it does that for you :)

ACuriousMind

Man, I've been playing this "conversation" game all wrong

Jim

@ACuriousMind I think you'd get a yellow card for that

Sanya

愛天使伝説 <- I want a red card for that

The Quantum Physicist

1:16 PM

@ACuriousMind But we've agreed that element-wise multiplication is valid with $\exp(-i(\omega_i - \omega_j)t)$, right?

skill patrol

@Sanya ::flagged as inappropriate:: suspension pending upon approval by TPTB </joke>

ACuriousMind

@TheQuantumPhysicist Wait...what do you mean by "element-wise multiplication"?

The Quantum Physicist

I mean every element from one matrix with the corresponding one in the other

ACuriousMind

No

If I said that, it's wrong. Why would the time evolution change how the ordinary matrix multiplication behaves?

The Quantum Physicist

@ACuriousMind Because the components are independent

and the evolution doesn't mix them

because the density matrix is in the eigenbasis of the Hamiltonian

ACuriousMind

1:18 PM

You can evolve each element of $\rho(0)$ to $\rho(t)$ independently, but when you compute $\rho(t) A$, you have to properly multiply these two matrices

The Quantum Physicist

Interesting...

ACuriousMind

So the first entry of $\rho(t)A$ is $\sum_{i} \rho_{1i}(t) A_{i1}$, where $\rho_{1i}(t) = \exp(-\mathrm{i}(\omega_1-\omega_i)t)\rho_{1i}(0)$.

The Quantum Physicist

How has it been working with this guy all along... I wonder!

ACuriousMind

You're telling me that you have simulation code where $\rho A$ is computed as having $\rho_{11}A_{11}$ as the first entry? oO

The Quantum Physicist

Yes... and it works, and fits experimental data

ACuriousMind

1:20 PM

...I have no words

The Quantum Physicist

Neither do I xD

ACuriousMind

That sounds utterly impossible

The Quantum Physicist

Well maybe it's possible because the guy starts with a thermally populated state along the quantization axis

populating only the diagonal element of the density matrix

ACuriousMind

Ah

Yes

The Quantum Physicist

then he multiplies that with that operator, element-wise

ACuriousMind

1:21 PM

If $\rho$ is diagonal, that works.

The Quantum Physicist

and THEN, he evolves with element-wise evolution operator

hmmmmmmmmm

Seriously... worst code I ever dealt with that produced the best result

this is horrible!

So I guess to make this generally correct, I must do proper multiplication, and then evolve with element wise

ACuriousMind

But for diagonal $\rho$, I would say that $\mathrm{tr}(\rho A)$ is constant. Because diagonal $\rho$ in the Hamiltonian basis means that your statistical mixture is one purely made out of stationary states

The Quantum Physicist

or alternatively, multiply with the operator AFTER the evolution is done

ACuriousMind

And expectation values in stationary states should be constant.

So time-evolving such a diagonal $\rho$ seems pretty pointless to me

The Quantum Physicist

Actually it's diagonal along the quantization axis

ACuriousMind

1:24 PM

Because it doesn't evolve at all

The Quantum Physicist

but then is converted to the eigenbasis of the hamiltonian

that still makes it invalid, right?

ACuriousMind

Oh

The Quantum Physicist

Everything is in the same basis

maybe this is what makes it valid

ACuriousMind

Well, if he does the element-wise multiplication in the basis where $\rho$ is diagonal and then switches to the Hamiltonian basis and then evolves it element-wise there, I guess that works

0celo7

@ACuriousMind morning

@ACuriousMind Did you know that $N(X\times\{a\};X\times X)$ is trivial?

The Quantum Physicist

1:27 PM

Not really... he:
1. Populates rho thermally
2. Converts the rho basis to the Hamiltonian basis
3. Converts the operator basis to the Hamiltonian basis
4. Multiplies rho with the operator, element-wise
5. Evolves using the element-wise evolution
6. Calculates the sum of all elements as the expectation value

And that works some how...

@ACuriousMind

ACuriousMind

@0celo7 I have no idea what that notation means

@TheQuantumPhysicist oO

The Quantum Physicist

@ACuriousMind Dude... I'm as surprised as you...

0celo7

@ACuriousMind Relative normal bundle

ACuriousMind

If the sequence was 1-4-2-3, I'd believe that it works, but that way it just seems wrong

The Quantum Physicist

Well, it agrees with experimental data for some reason

it's probably a special case...

ACuriousMind

1:29 PM

@0celo7 Yeah, I have no idea what that is, so no, I did not know it is trivial :P

@TheQuantumPhysicist Have you checked that it actually does, i.e. the code you have produces the correct predictions?

The Quantum Physicist

Yes. In fact we wrote an extremely fast and efficient fitting code

and we use experimental data with it

and we fit data with mHz accuracy!

some data reaches sub-mHz!

which is a revolution in NMR

0celo7

@ACuriousMind Let $Y\subset Z$ be a submanifold, then $N(Y;Z)=\{(x,v)\in Y\times \Bbb R^n\mid v\bot T_xZ\}$

ACuriousMind

I meant do you know that the source code you're looking at actually belongs to the program you are using? I'd suspect being given a wrong version or something seeing this strange code.

The Quantum Physicist

Well, I took his Mathematica, inefficient code

and I translated it to extremely efficient and optimized C++

and I fitted experimental data

and it works...

ACuriousMind

I see

The Quantum Physicist

1:31 PM

and I'm unable to understand it

ACuriousMind

this is very strange indeed

The Quantum Physicist

Agreed...

What would you do in this case?

ACuriousMind

Ask the author

The Quantum Physicist

He left... I'll have to try to reach him...

@ACuriousMind Thanks for the help though. I appreciate it!

skill patrol

What does "Mainz" mean in your profile? @TheQuantumPhysicist

The Quantum Physicist

1:33 PM

Actually converting my code to be 100% legal, by multiplying with the operating after the evolution isn't that expensive

@skillpatrol It's a city in Germany?

0celo7

Proof?

The Quantum Physicist

@0celo7 proof of what?

0celo7

Where even is Germany

The Quantum Physicist

On Neptune

0celo7

huh

ACuriousMind

1:34 PM

@TheQuantumPhysicist Well, it might be worth a try just to see if you get the same results

The Quantum Physicist

I'm using quantum entangled signals to communicate with earth faster than light @0celo7

@ACuriousMind Yep! That's my next step!

0celo7

I figured it was in Uranus

The Quantum Physicist

No, no. You're mistaken

It's Neptune

ACuriousMind

@0celo7 Thanks for answering a question I didn't ask :P

Slereah

Also you can't use quantum entanglement to communicate faster than light

skill patrol

1:35 PM

^

0celo7

@ACuriousMind Did you see my message about Milnor

The Quantum Physicist

@Slereah Well my people did it! We have a super-modern civilization

0celo7

My prof didn't understand the proof either

The Quantum Physicist

You "earthians" are still immature :P

ACuriousMind

@0celo7 I wasn't really here the last few days, so I'm not sure which of these you are referring to

@TheQuantumPhysicist I think we're supposed to call them "humans"

The Quantum Physicist

1:36 PM

@ACuriousMind Well we call them earthians on our planet

skill patrol

Thanks for stopping by mr alien @TheQuantumPhysicist :-)

The Quantum Physicist

Calling them humans is racist

0celo7

@ACuriousMind You were right, he hasn't opened the book in 10 years

ACuriousMind

@0celo7 Ah, that one. Yes, I read it, I'm not surprised :D

The Quantum Physicist

No problem! Always glad to take a peek at earhians @skillpatrol

ACuriousMind

1:38 PM

@TheQuantumPhysicist I'm in Germany, so it should be my planet too ;)

skill patrol

^

ACuriousMind

Unless you somehow only transported Mainz to Neptune...

The Quantum Physicist

@ACuriousMind Oh, nice to meet you here mate! We're communicating over a few light-hours

@ACuriousMind No, no. Whole Germany is involved.

ACuriousMind

Yeah, I guess it is a rather inefficient way of communication

The Quantum Physicist

@ACuriousMind xD

skill patrol

1:41 PM

How's the fly trap working? @0celo7

0celo7

they seem to like it

hope it kills them

and they're not bothering me any longer

link?

skill patrol

nvm

0celo7

huh?

why did you argue with shog

skill patrol

because I wanted to

0celo7

1:56 PM

@skillpatrol holy crap there's two dozen dead flies in the trap

skill patrol

It does work.

Hi @JohnRennie

John Rennie

Afternoon

skill patrol

When is afternoon tea usually served?

Noah P

ACuriousMind

@skillpatrol In the afternoon.

@NoahP ?

Noah P

2:06 PM

It seemed more humorous when I thought of it than now

And I cant seem to delete it

Oh wel@JohnRennie Managed q5 and q7 with no intervention!

John Rennie

Cool :-)

Though perhaps I should reserve judgement until after they've been marked :-)

Noah P

Well, one was a proof

And the others agreed with results I found online

In a book actually I think

An introduction to modern cosmologu, by Andrew Liddle

Itachí Uchiha

@Secret Thanks a lot.

skill patrol

@JohnRennie If you have a spare $8,888 going spare, you could always head on over to the Ritz Carlton in Hong Kong, as they serve the world’s most expensive afternoon tea.

@ACuriousMind :P

Slereah

but tea is gross

John Rennie

2:12 PM

@skillpatrol The things I enjoy spending my money on include phones, tablets and laptops, but exclude trips to Hong Kong for tea.

0celo7

@ACuriousMind how does "plead the fifth" get dubbed in German

ACuriousMind

@0celo7 Probably like vom Recht zu schweigen Gebrauch machen

skill patrol

They don't have "the fifth" ;-)

Noah P

@JohnRennie Do you think you could point me in the right direction for q6? Last one!

John Rennie

Right to silence

The right to remain silent is a legal right recognized, explicitly or by convention, in many of the world's legal systems. The right covers a number of issues centered on the right of the accused or the defendant to refuse to comment or provide an answer when questioned, either prior to or during legal proceedings in a court of law. This can be the right to avoid self-incrimination or the right to remain silent when questioned. The right usually includes the provision that adverse comments or inferences cannot be made by the judge or jury regarding the refusal by a defendant to answer questions...

@NoahP Oh go on then :-) Are you going to upload a picture?

Noah P

2:15 PM

Sure

John Rennie

Well 6b is a simple consequence of 6a, so getting 6a solves both. Have you worked out the relationship between $a$ and $z$ yet?

Noah P

I havent derived it but I know what it is

$z=a^{-1}-1$?

John Rennie

OK, I don't know the answer to this offhand, and when faced with a new problem I find the best way is to poke at for a bit and see what falls out.

Noah P

The question is already very sore if we're using that analogy

John Rennie

Firstly if we look at (6) it's a lot simpler than it seems. Do you know what the physical meaning of $\Omega_i(t_0)\rho_c(t_0)$ is?

Noah P

2:28 PM

(Incase 5 or 7 help)

And I recognise the critical density and density parameter there?

John Rennie

Yes, $\rho_c(t_0)$ is the current value of the critical density, so what is $\Omega_i(t_0)\rho_c(t_0)$?

Noah P

The current density?

Assuming $\Omega=\frac{\rho}{\rho_c}$

John Rennie

The current density of ... ? Remember that $\Omega_i(t_0)$ is the current fraction of component $i$.

Noah P

I'm not sure what $i$ is

John Rennie

Ah, OK, fair enough, the question oesn't make that clear.

We have three components, light, matter and dark energy.

And the subscript $i$ is a just a convenient way to label them.

Noah P

2:33 PM

Oh, so its all of them?

When you say light, you mean radiation?

Just checking

John Rennie

For example we can (arbitrarily) choose $i=0$ for radiation, $i=1$ for matter and $i=2$ for dark energy.

Noah P

Ohhhhhhhhhhhhhhhh

I get it

Slereah

it is best to use letters

Noah P

So its basically the same equation for all three, but using $i$ instead of writing out the same equation with different letters

Normally it would be $r$, $m$, and $\Lambda$

But $i$ can represent any of them?

John Rennie

Yes. Actually I agree with Sam - I would have used letters not the subscript $i$.

But then I didn't set the paper :-)

Noah P

2:36 PM

I'll pass on the complaint ;)

John Rennie

OK, so what is the physical significance of the term with $i = 1$ then: $\Omega_m(t_0)\rho_c(t_0)$

Noah P

so that would be the density of matter

John Rennie

Yes, lets call that $\rho_0m$ (rather than the unnecessarily fussy $\rho_m(t_0)$.

Obliv

@slereah what's the name of the theory that poses there are infinitely many world-lines?

Noah P

Ok, sure

Obliv

2:38 PM

instead of the closed loop world line of CTC

John Rennie

And you've already told me that $1 + z = 1/a$. So take the $i=1$ case, substitute for $1+z$ and what does the equation become?

Remember that ordinary matter is pressureless, which sets the value of $w$ to be ... ?

Slereah

Branching spacetime

it's not much of a theory

0celo7

@ACuriousMind btw, Hirsch has a really cool definition of Morse functions

A function is Morse iff it's exterior derivative is transversal to the zero section of the cotangent bundle.

exterior derivative/tangent map/pushforward

Noah P

That would make $\omega=0$?

Obliv

@slereah thanks

John Rennie

2:45 PM

Yes, so what does equation (6) simplify to?

Noah P

Giving $\rho_i(z) = \frac{\Omega_{i0}\rho_{c0}}{a^3}$?

Where $i=m$

John Rennie

And we've already decided that $\Omega_{m0}\rho_{c0} = \rho_cm0$ i.e. the density of the matter.

So the equation is just telling you that $\rho = \frac{\rho_0}{a^3}$.

Noah P

I derived that in q5

John Rennie

Which should be obvious.

Noah P

So surely I can go from the equations I got in q5 then?

$\rho_m \propto a^{-3}$ and $\rho_r \propto a^{-4}$

0celo7

2:52 PM

are we still doing this

Noah P

@0celo7 'this' could mean many things

John Rennie

Basically yes. For radiation $\omega=4/3$ and if you put that into equation (6) you're going to get $\rho_r = \frac{\rho_{0r}}{a^4}$, which reproduces what you already know.

Noah P

And for $\Lambda$, $\omega=-1$?

Sanya

physics.stackexchange.com/questions/269218/… can someone highrep please close that threat for being inappropriate, badly formulated and stupid?

John Rennie

For dark energy $\omega=-1$ and that's going to give you $\rho_\Lambda = \rho_{0\Lambda)$ i.e. the dark energy density is constant (which it is).

Noah P

2:58 PM

Yeah, okay

So, how would one go about proving the equation 6 then?

ACuriousMind

@Sanya 3 close votes and counting, it'll be closed soon.

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