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1:59 AM
@SillyGoose sorry didnt read the rest of the convo but that excerpt was from a computational quantum mechanics book basically for computer science
so dirac wasn't talking about ballentine
 
2:39 AM
For all I know, I may, over the years, end up causing people to be awarded the steward badge for Triage. Because I <3 flagging questions as 'very low quality.'
 
3:14 AM
@Obliv All those basic analytic answers are actively used in the perturbation expansions and other trickery for getting numerical answers. This guy is on some serious drugs.
@Obliv Amusingly, there is extremely little beyond quantum tunnelling that you need to understand. Something that some light reading will help bridge you the gap from quantum theory to nuclear physics. Instead, it is the plasma physics that will take forever to learn.
 
 
1 hour later…
4:18 AM
@Obliv ooh
i don't really understand why any "derivation" of BEC of a non-interacting bosonic gas is called a derivation. It simply seems to be, literally, a model of the phenomena in that it produces the empirically observed statistics.
what i mean to observe is that, e.g., the grand canonical "derivation" of BEC of a non-interacting bosonic gas is not what is happening in the lab. we are not, in real life, fixing the expectation value of the number of particles a grand canonical ensemble.
 
@naturallyInconsistent huh?
 
4:42 AM
@SillyGoose is this a q or ur just writing an opinion piece
 
it is a q
 
what is the q
 
what is a method to derive BEC
 
like do u mean to ask for a physical argument for BECs or a semantic debate about if this should be called a derivation?
i only have the former
or can point u to the former
 
a physical argument for how BECs form using only the rules of quantum mechanics and statistical mechanics and perhaps some reasonable assumptions
 
4:45 AM
@SillyGoose theres one in the book u sent but theres a gap in it that i have not filled. its section 2.2
whats ur qualm w that one
 
is this the QAO book?
 
yes by timothy
 
chalomet
 
i assume u read that section and had a qualm w it and that inspired ur commentary, so im curious what that problem was?
and btw i have another q also on this qao topic
 
what does a two state system have to do with an ensemble of identical bosonic gas systems (section 2.2)
 
4:52 AM
well i think we should back up. tbc im not like discussing as if i know the answer, just to work through it also
 
hm well maybe i should write specifically what i hope for an explaination of
 
so when we say we want to explain BEC, then we are saying we want to explain why, given a large number of bosons and multiple states to fill, the bosons will collectively occupy the lowest energy state?
@SillyGoose my understanding is that the first is just the model we are using because that is the model the author has in mind/is most interested in and we can show that using such a model, BEC will occur. i dont think its a general requirement.
 
i would like to deduce that given a finite number $N$ of free particle and noninteracting bosons, then if we take the temperature $T \to 0$ the occupation of the ground state approaches $N_0 \to N$
@Relativisticcucumber hm i guess i could see this. it is a "proof of concept" perhaps
 
right i think section 2.2 shows that for a specific model but i think a more general derivation is available in most stat mech texts
 
is the type of derivation you refer to different than as in section 2.3?
 
5:01 AM
@SillyGoose not fundamentally i think all i have looked into are different variations/levels of depth of the grand canonical ensemble discussion
but tbh 2.2 was relatively sufficient enough for me
 
but i don't get how that deduces BEC because to me that is just not appropriately modeling reality. it just happens to give a fitting result at the end of it all
 
can u elaborate
 
unless experimentally, there really is some sort of particle number fluctuation in a lab's BEC
using the grand canonical ensemble seems strange given that (at least this text) it is said that in a lab you hold the particle number fixed
 
@SillyGoose idk to me this notion of "not appropriately modeling reality, just happens to give a fitting result at the end" is inconsistent. to me physics is throwing models at a problem until something works with good accuracy.
like there is no "correct model" in my view
there is works well or doesnt work well
 
i agree but the BEC case is a little bit different to me than business per usual
we are introducing a new rule that is not related to quantum mechanics or statistical mechanics and is not an approximation all just to even begin the derivation
 
5:05 AM
well this is where i prob would have to step out bc this is what every element of stat mech felt like ot me
no rhyme or reason so i dont expect anything more. i just accept the arguments presented to me and gave up on reasoning w SM awhile ago
but maybe you can think of it like
"consider what happens if we do make this assumption" and then we find that it works so then ur q can be why does this model work despite this weird assumption
i dont have an answer to that q tho
 
an answer to that question indeed would satisfy me
 
i feel thats a tangible q
/ reasonable q
hm idk i look at stat mech backwards. like we study the ensembles that work for a lot of situations. they are confirmed to work via experimental evidence. and they model many systems even though they are idealizations
so its not that infeasible to me that slightly hand wavy use of these ensembles works to explain physical phenomena -- seems stat mech was made to match w experimental results but idk i mean this is not really v rigorous, maybe naturally knows
 
fqq
@SillyGoose that's just the fact that for T->0 you get the ground state, and for non interacting bosons the many-body ground state is trivially "all particles in the single-particle GS". It has nothing to do with BEC
 
5:24 AM
1
A: Why do we use grand canonical ensemble for BEC description?

By SymmetryIn the thermodynamic limit fluctuations in particle number should tend to $0$ and so we should get the same result whether we use the cannonical or grand canonical ensamble. If we are ultimately interested in a system with a fixed particle number, we can, if we so choose, set up the problem in th...

 
thanks that seems to be a reasonable answer
 
have u seen the def of SCS @SillyGoose
its 5.9 and defined in terms of fock states
 
let me shee
 
so iiuc, these spin coherent states can be used like coherent states and they can describe, for instance, an atomic ensemble. so im trying to understand this in the open quantum systems picture. can i use these SCSs as i do wave functions to form a density matrix?
 
i would think yes; they are states in Fock space all the same
 
5:32 AM
okay and the thing im getting tripped up about is
if u look at fig 5.2
this is a bloch sphere rep of a SCS for N = 10
and iiuc, in bloch sphere pic for OQS cases, mixed states live in the bloch sphere and pure states live on the surface
is that still true here? do i combine these visuals in the same way?
 
i don't think the parameterization of states for the SCS is the same as you would do with a qubit state
but i am not sure what this parameterization presented in the text is
i think
it is conceivable to visualize the SCS on a sphere because each state is parameterized by two parameters
 
@SillyGoose theres an equivalence to qubit case in 11.16
@SillyGoose right i get that
im just wondering if its still true in this picture that mixed states live on the interior of this new bloch sphere
 
im not sure since i am not sure what the parameterization for mixedstates would look like
 
5:51 AM
ok nuff said
 
6:39 AM
n u f f
 
7:39 AM
h o n k
 
8:02 AM
@ACuriousMind should I ask on the site? I'm scared it's so trivial that the answer is a one-liner 💀
 
☠️
if it is a one-liner, that'll be a line you'll never forget :P
> memory is the residue of thought
 
8:25 AM
om nom
 
8:36 AM
@Mr.Feynman Is it not that the Jacobian is non-negative?
 
 
1 hour later…
9:47 AM
@bolbteppa I had thought about it. The factor that appears is indeed the Jacobian. Does it have to be non negative though?
It is about orientability, isn't it?
 
9:59 AM
I think it's just that when you do integration you're summing up areas which are always positive so the factor is chosen to have absolute values for that reason, i.e. you take the absolute value of the jacobian (the Jacobian itself can be + or -)
 
Unless I misinterpreted what you mean, the factor appearing in the metric is still the Jacobian, not its absolute value
 
10:22 AM
Well, really what you're talking about is preserving $\sqrt{-g} g^{\alpha \beta} = \sqrt{-\tilde{g}} \tilde{g}^{\alpha \beta}$, your scale factor has to be non-negative because of the absolute value in $g$ and the square root right, when you go to complex functions technically you're talking about $f(\sigma) = |f(\sigma)|e^{i\theta}$...
 
 
4 hours later…
2:24 PM
@Relativisticcucumber there is an element of being mysterious there, but miao miao is not cloaking behind a whole mountain worth of mysteriousness. It should be understandable as it is written.
@SillyGoose but it is already reasonable...
@SillyGoose this is so funny, that miao miao translated it to a friend, and he still managed to catch the joke.
@SillyGoose This is actually a wrong statement. Even in a system that ought not to have BEC, as $T\to0$ the occupation of the ground state ought to $\to N$, simply the case for any system following Bose-Einstein statistics (or Maxwell-Boltzmann, though that does not actually exist IRL). Only Fermi-Dirac systems will behave differently.
@SillyGoose what new rule?
@Relativisticcucumber no, stat therm is a very well-established theory and everything had been reduced to maths. If there is a theory of physics whereby you can ask every question you wish to ask of it, it ought to be stat therm.
@Relativisticcucumber This is actually a pretty bad answer.
Anyway, my answer to the wacky fowl's question would be the following: You have failed to remember an extremely important fact: Thermodynamics is only well-defined and well-established in the "thermodynamic limit". That is, one first starts by proving that, in the thermodynamic limit, microcanonical, canonical, and grand canonical ensembles always agree in all their predictions. In that way, you can then swap questions that you want to ask in micro- or canonical ensembles, for questions in
the grand canonical ensemble, and obtain correct predictions. Since it is vastly easier to derive things in the grand canonical ensemble, it is thus the case that we would swap a fixed $N$ problem for a variable $N$ problem with fixed $\left<N\right>$, safe in the knowledge that, in the thermodynamic limit, the results are necessarily the same.
Remember that, unless we are talking about surface effects themselves, we are always dealing with thermodynamic systems by imagining an infinitely large system, i.e. the universe is just the infinitely large system we are considering, and then mentally carving out a volume precisely the same size as the actual system we are considering. Again, in the thermodynamic limit, the difference between the infinitely large system thusly considered, and the actual system, differ only by surface terms,
and the corrections matter less and less as the system size increases.
(to be fair, that was a good answer for the question it was trying to answer. It is a bad answer for the question the wacky fowl is asking.)
 
 
4 hours later…
6:19 PM
Ellen & Yoyoka are both great musicians, but when they get together they're a formidable rhythm section. You'd think they've been playing together for years. The rest of these kids are very impressive, too.
 

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