The h Bar

Ryder Rude

19:00

@DIRAC1930 it depends on the goals. sometimes the goal is just to do abstraction and generalisation. but some generalisation is stupid, yes. one's intuition guides the kind of abstraction that might lead to interesting results.

Allie

orbitals were so abstract to me and also alluring, thats hwy i got into quantum chem :3

ACuriousMind

@SillyGoose sure (but then you could argue if you're engaging in this meta-study of notions of aesthetics you're again not using your own aesthetic sense :P)

Allie

literally because i wanted to know more about molecular orbitals

Silly Goose

@naturallyInconsistent which all make no sense without ap hysics background :-)

ACuriousMind

I hated these orbital pictures because everyone was always like "isn't it beautiful?" and I was like "I have literally no idea what I'm looking at" :P

Allie

19:01

it is very confusing

Ryder Rude

@Allie is classical mech relevant in chemistry

ACuriousMind

now that I do have an idea I'm more meh towards them :P

naturallyInconsistent

@Allie I've recently been binge-ing upon essentially exact solutions to few-electron wavefunctions, especially He and Li and H2, some of which are even without Born-Oppenheimer approximation.

Allie

@RyderRude i mean, like advanced classical mech? probably not to most chemists

Ryder Rude

i would say mathematics has art-ish elements to it. and a mathematician may strongly feel like they are studying something beautiful

Allie

19:02

but for theoretical chemists, i mean we're basically halfway on the way to physicist so

naturallyInconsistent

@SillyGoose it aint so bad, if the instruction is spectacularly good

Allie

@naturallyInconsistent oh wow, link?

how did they solve? CC? perturbation?

Ryder Rude

@Allie oh

Allie

@RyderRude hence why im here all the time and not the chem chat room

Ryder Rude

i am reading "theoretical chemist" for the first time :P

@Allie yes

Allie

19:03

does that mean youre meeting one for the first time?

actually i dont know if i can call myself a chemist just yet..

Ryder Rude

are there also theoretical biologists

Allie

maybe when i get my degree

Ryder Rude

@Allie yes

naturallyInconsistent

@Allie My dad was super nosy and went to switch off my computer, losing weeks worth of opened private tabs

Allie

@naturallyInconsistent NOOOO

Ryder Rude

19:04

it makes sense. people who come up with theories are called theoretical, regardless of the field

Allie

@RyderRude no clue, biology hurts my bones

Tobias Fünke

@RyderRude nah, it is a wide field in chemistry. but it has much overlap with solid-state physics, material science etc.

@RyderRude yes

Allie

(biology is incredibly important and interesting and needed, but living things creep me out)

Ryder Rude

@TobiasFünke oh

Tobias Fünke

enter Turing :p

Ryder Rude

19:05

@TobiasFünke oh

Allie

im specifically interested in computational chem so

naturallyInconsistent

@Allie Hylleraas. Just straight up brute force computer-adding terms and terms to the wavefunction until you get more than 40-digits goodness.

Allie

ah

thats cool though

Tobias Fünke

No, but sure, biology is a huuuge field too, and some areas have an overlap with other disciplines

Ryder Rude

@Allie do u find it disgusting or disturbing

Allie

19:06

both :P but i am a very squeamish person. also why i didnt go into synthesis, i am very clumsy and also very afraid of spilling some nasty chemical on myself

naturallyInconsistent

@Allie it is beyond cool. Hartree Fock cannot even compare.

Allie

@naturallyInconsistent makes HF look like a joke

Ryder Rude

@TobiasFünke biologists also have some fundamental problems like origins of life

naturallyInconsistent

And DFT too, at the cost of being too computationally expensive to do many electrons

Ryder Rude

they overlap with information theory and chemistry

Tobias Fünke

19:08

@RyderRude sure

List of unsolved problems in biology

This article lists notable unsolved problems in biology. == General biology == === Evolution and origins of life === Origin of life. Exactly how, where, and when did life on Earth originate? Which, if any, of the many hypotheses is correct? What were the metabolic pathways used by the earliest life forms? How did genetic code originate? What was the molecular mechanism that allows the association of the amino acids with their triplet codons? What were the biochemical paths from individual bio-building blocks like amino acids or nucleic acids to functional polymers such as proteins and D...

Allie

@naturallyInconsistent well thats the whole issue

Ryder Rude

all of science overlaps. categories don't exist in the real world. categories are a human way of speaking

Tobias Fünke

true, at least to some extent

Allie

what i think is freaking cool is that the most complex systems still have some solution out there, its just inaccessible to us

like it exists, and if you could write it down you can prove its the solution, but we cant find what it is

Tobias Fünke

@Allie hehe yeah

Ryder Rude

19:09

>
Laughter: While it is generally accepted that laughing evolved as a form of social communication, the exact neurobiological process that leads humans to laugh is not well understood.

Allie

very computer science-y problem

naturallyInconsistent

@Allie still, it is extremely exciting to see evidence that quantum theory gets > 40 digits agreement with experiment in all kinds of phenomena, from 2 electrons to 6 electrons, to Rydberg atoms, etc.

Allie

@naturallyInconsistent i know, beautiful!

thats what i love about this field

being able to calculate reality. freaking insane!

naturallyInconsistent

But that method isnt usually used to work out the orbitals. I think it would be fun to try and plot the orbitals coming from such computations

Ryder Rude

@Allie like the three body problem

Tobias Fünke

19:11

I wouldn't phrase it like this, but yeah :p

@RyderRude in Allie's field the example par excellence is the universal functional in DFT

Ryder Rude

i feel physics is computable. So we can sort of access all solutions by making lattice approximations

Allie

@naturallyInconsistent when you say orbitals would you want to look at the natural orbitals?

Tobias Fünke

@RyderRude depends on what you mean. there are undecidable problems even for those systems

Ryder Rude

@TobiasFünke oh

DIRAC1930

Man there are so many unsolved problems in physics

List of unsolved problems in physics

The following is a list of notable unsolved problems grouped into broad areas of physics. Some of the major unsolved problems in physics are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result. The others are experimental, meaning that there is a difficulty in creating an experiment to test a proposed theory or investigate a phenomenon in greater detail. There are still some questions beyond the Standard Model of physics, such as the strong CP problem, neutrino mass, matter–antimatter asymmetry, and the nature of dark matter...

naturallyInconsistent

19:12

@Allie what are natural orbitals again?

Tobias Fünke

and computationally hard problems even on quantum computers (QMA complete for example)

Ryder Rude

@TobiasFünke yes. but those haven't been shown to be physical, i think

Tobias Fünke

@naturallyInconsistent the eigenfunctions of the 1-RDM

Allie

@naturallyInconsistent eigenfunctions of the

damnit

naturallyInconsistent

@TobiasFünke What is 1-RDM?

Allie

19:13

1st order reduced density matrix

i won that one :3

naturallyInconsistent

Ah

Tobias Fünke

@RyderRude yes, sure: the spectral gap problem, i.e. deciding whether a family of Hamiltonians have a finite spectral gap in the thermodynamic limit

hehe yeah allie :p this one!

Allie

because im just curious what you mean by orbital in this case

Tobias Fünke

Allie, do you know why these are called "natural orbitals"? I cannot remember if I've read it at some point

Ryder Rude

@TobiasFünke yes. But the Hamiltonians they assumed were purely mathematical?

naturallyInconsistent

19:15

Well, the Hylleraas method means that the resulting wavefunctions have electron-electron separation as part of their coördinates, which makes it difficult to plot the wavefunctions in space

Ryder Rude

@DIRAC1930 interesting

@DIRAC1930 i am interested in foundational issues

Tobias Fünke

@RyderRude Well, the problem is of course general. The point is that you cannot decide for every Hamiltonian (from a certain class) whether or not is has a spectral gap, roughly speaking

naturallyInconsistent

@Allie But I still want to plot those wavefunctions in space just as to see, especially to see what the handling of the Kato cusp conditions causes the wavefunctions to change.

ACuriousMind

@TobiasFünke I've repeatedly told him he doesn't understand the subtleties of lattice theories and their continuum limits; it never sticks

Ryder Rude

> Locality: Are there non-local phenomena in quantum physics?[98][99] If they exist, are non-local phenomena limited to the entanglement revealed in the violations of the Bell inequalities, or can information and conserved quantities also move in a non-local way?

Tobias Fünke

19:17

Wow, quite many theoretical chemists are from northern Europe. Interesting

Allie

@naturallyInconsistent i would like to see that too!!!!

Ryder Rude

@TobiasFünke yes. im just saying that it is not established for the Hamiltonians of physical systems

Tobias Fünke

arxiv.org/pdf/2410.16532

Ryder Rude

@TobiasFünke i haven't read this paper but my comments come from a review of this paper

i will try to find it

Allie

i think its time to exercise

my neck ad shoulders hurt badly

naturallyInconsistent

19:21

@Allie I wonder if it is possible to merge the Hylleraas method with the method that treats the nuclei with Dirac equation too, i.e. fully SR in everything. However, I'm wary of trying to compute the Bethe logarithm in this context. It will be hell.

Tobias Fünke

@Allie, you did not answer my question :(

about the name of the natural orbitals

Allie

@TobiasFünke im sorry i was typing and i forgot

Tobias Fünke

hehe no problem

Ryder Rude

this philosophy.stackexchange.com/a/30479 @TobiasFünke

Tobias Fünke

?

that answer does not address the paper I linked to?

Ryder Rude

19:22

oh

Allie

im honestly not sure. the way i think of it (and it might be a complete cop-out/wrong answer, but) is that they are just the orbitals that naturally arise from an N-electron wave function

Tobias Fünke

but thanks for the link

Allie

since you're not doing HF you don't have any explicitly one electron wave functions

Ryder Rude

@TobiasFünke for some reason, i thought there was only one paper on this

but it seems multiple people have written about undecidability of spectral gap

Tobias Fünke

@Allie thanks. hmhm if I have time I will try to do some research. The work of A. Coleman is probably a good starting place

Allie

19:23

so if you want some way to extract a one-electron wave function from the N-electron one, the eigenfunctions of the 1-RDM seems like a somewhat natural way to do that. but again, may be completely talking out of my ass

@TobiasFünke definitely let me know! i am interested too

Ryder Rude

@ACuriousMind i am aware of the subtleties. i am making the statement on a speculative level

Allie

i am quite new to the density matrix stuff :p

ACuriousMind

@TobiasFünke oh no there is more than one Coleman? :P

Ryder Rude

many physicists also make this statement. it is assumed that the subtleties are handed on a case-by-case basis

Tobias Fünke

haha yeah

Allie

19:24

@RyderRude RR, Speculator-in-Chief

Tobias Fünke

This one is quite famous, too

I know three actually: S. P. and A.

naturallyInconsistent

@Allie oh yeah, this gives meow a headache. Forgot about this

@TobiasFünke in this weather, going to the spa is fun

ACuriousMind

@TobiasFünke not famous enough, if you google "a coleman physics", the first hit is Sidney ;)

Allie

ok i need to make my body feel better and then maybe play some piano. ill do some of the hardcore studying later

Tobias Fünke

@ACuriousMind true :p

Allie

19:25

bye hbarrr

Tobias Fünke

see you, Allie. Have a nice day

ACuriousMind

(to be clear, the actual cause is probably not the lack of fame but that Google interprets the 'a' as the English article and ignores it)

naturallyInconsistent

bye

Ryder Rude

bye

some people think QM is non local

they probably believe in some Copenhagen type model where a balloon collapses everywhere all at once

Tobias Fünke

Do you know Maudlin's thoughts on this?

In any case, the words "local", "realism" and so on are so overloaded...

Ryder Rude

19:29

yes. in Bell's inequality, the meaning of local and real is technical

but people use these terms in QM carelessly

@TobiasFünke i have watched few podcasts with him. he seems to believe in a non local hidden variables theory

and if i recall correctly he also says that standard QM is non local

Tobias Fünke

as I said, "(non-)local" is overloaded ^^

Ryder Rude

interestingly, Heisenberg was bothered by this non locality of Copenhagen interpretation, and he proceeded to re-define Copenhagen interpretation to mean that the collapse is not a physical process, but it only happens in the mind of the observer

this completely contradicts the previous usage of Copenhagen interpretation. it is where the idea that "no one really knows was Copenhagen interpretation means" originates

Tobias Fünke

@RyderRude idk if this is his position. he and others, mostly philosophers (but not exclusively) say that most people dismiss the two-step argument in Bell's original paper, and thus come to the wrong conclusion (they do not a priori claim that "realism" is true, but rather that "local" must be false)-- but their work is criticized, too, in a lot of papers

vey roughly speaking. Here on PSE, there are some users also holding this position. wait a sec.

4

Q: Do Bell’s inequalities assume determinism?

I was watching a video of Tim Maudlin where he talks about how the CHSH version of Bell’s inequalities do not assume determinism and only assume locality. He said that it is a common misconception that Bell assumed determinism and that in the CHSH version, he was explicit about not assuming that....

Ryder Rude

@TobiasFünke wowww

i thought non determinism was an unspoken loophole in Bell'e inequality

it is great that this loophole is closed too

Tobias Fünke

there is a scholarpedia article by some quite famous authors (and proponents of these "ideas")

Ryder Rude

19:38

apparently, Sabine does not know this :P. I have seen her talking about this "loophole"

Tobias Fünke

idk what you mean, sorry ^^. and I don't watch Sabine's videos too often (basically never)

Ryder Rude

lol

she believes in superdeterminism....

@TobiasFünke do u mean people trying to exploit this loophole in their hidden variables theory

Tobias Fünke

I don't understand what you mean with loophole here

Ryder Rude

i mean the "loophole" that " Bell assumes determinism "

Tobias Fünke

It is "just" a discussion about what Bell's theorem really says. This and related controversies are quite old, and do not touch loopholes a priori

Ryder Rude

19:41

oh

@TobiasFünke yes

it shows that locality and non-superdeterminism are the only two assumptions of Bell. Determinism is not a real assumption

i just saw Sabine saying that Bell's result only applies to deterministic hidden variables theories. so she wasn't aware of this result

Tobias Fünke

As I said, their approach is criticized in many papers, too. ^^ I don't have the knowledge (yet) to "decide" what I find more plausible-- I am annoyed by the overloaded words lol

Ryder Rude

yeah... i would have to look up Maudlin's criticism of this. i too always thought that Bell assumed determinism

it would be great if Maudlin is right

Tobias Fünke

as I said, check the scholarpedia article (for me the site somehow does not work right now)

he is not an author, IIRC, though

there is some article "What Bell really did" or so by him. (and also check the critique article by Werner)

in Tumulka's QM book the argument is outlined, too... and if you search a bit you will find many more articles, and also the opponents arguing against it

Ryder Rude

oh. it seems like this is still controversial. i will have to research it

@TobiasFünke thanks

Tobias Fünke

that's what I said now several times :d

no problem

I find the discussion funny, though. Each side claims the others do not understand Bell's theorem

Ryder Rude

19:51

cya

@TobiasFünke lolol

yeah.. experts are really confused about this

they will talk about free will and counterfactuals and stuff

Maudlin does not talk about that stuff. but I have previously seen him misunderstand his opponent's theory

DIRAC1930

Whats everyones favourite representation?

Mine is the spinor representation of the Lorentz group

Or the spinor representation of $SO(3)$

Ryder Rude

@TobiasFünke this paper reads like a huge essay :P

DIRAC1930

by this I mean projective representation

Tobias Fünke

well, it is, no? it was due some conference if I remember well

trivial representation for sure

Ryder Rude

oh

i just think he would need some math to shows that indeterministic hidden variables are ruled out. but he hasn't done that

but i will read his arguments

DIRAC1930

20:00

Has anyone studies representation theory from L&L 3?

He does it very differently

I guess it is a way that is more appropritate for chemists

Silly Goose

Is Wick's theorem a statement solely about free QFTs? Is this why particle number is implicitly conserved in a Wick theorem computation?

ACuriousMind

@SillyGoose it's a theorem about relations between certain expressions in c/a operators. There are no dynamics involved, so I don't understand where "free" or "interacting" would enter

Silly Goose

er i guess i am confused about the following facts

(1) a VeV by construction doesn't allow non-particle-number-conserving "scattering events" to survive, e.g. $\langle 0 \lvert a^\dagger a^\dagger \lvert 0 \rangle = 0$

(2) Wick's theorem should be general as you stated, so how do I get contributions from NPNC "scattering events"?

ACuriousMind

@SillyGoose I don't understand the question. What does Wick's theorem have to do with the VEV in (1)?

maybe it would help if you stated what you believe to be "Wick's theorem" :P

Silly Goose

this is the defiition i am working with

the $P$ is a permutation on the set $\{1, 2, ...\}$ from which $r := t_r$ takes its values. The notation $P'_r$ means apply permutation $P$ to $r$ then turn it into a primed coordinate.

ACuriousMind

20:15

...and $G$?

I have no idea what I'm looking at here :P

to me Wick's theorem is a purely combinatorial statement about the relationship between time-order and normal-order, as in e.g. this answer by Qmechanic

Silly Goose

well maybe i should just ask

certainly NPNC interactions are a thing

Signor Feynman

@ACuriousMind The GF (?)

Silly Goose

if we ultimately reduce every scattering computation into an LSZ like computation, which is heuristically the VeV of some time-ordered set of fields, how do you ever get NPNC contributions

Tobias Fünke

if someone is interested, there is also a "Wick's theorem" for stat. mech; a starting point is Une démonstration simplifiée du théoreme de wick en mécanique statistique by Gaudin (1960)

DIRAC1930

@SillyGoose How did you find the explanation of Fermi-Liquid theory in L&L 8?

Silly Goose

20:21

@DIRAC1930 do you mean L&L 9?

DIRAC1930

Maybe lol

ACuriousMind

@SillyGoose You have to be wayy more explicit about what you mean

Silly Goose

okay maybe i will save this question for when i actually do a relevant computation

Tobias Fünke

what is NPNC? just out of curiosity

Silly Goose

@DIRAC1930 i didn't get too far in :P it seems like it is kind of neat for taking the theory from a phenomenological perspective. it is a bit terse for me. i found online notes by fradkin and etc. more illuminating.

ACuriousMind

20:23

@TobiasFünke an abbreviation they coined for " non-particle-number-conserving", it's not a standard term

Silly Goose

i felt like i was wasting time reading L&L9, but perhaps it would be good to go back to after already knowing the theory so that time is not wasted thinking if a step is being skipped or etc.

Tobias Fünke

I see, thanks

Silly Goose

i did get my hadns on a physical copy though :)

DIRAC1930

Yes it is unfortunately very terse

fqq

@ACuriousMind some people call Wick's theorem the fact that in free theories correlation functions factorise into two-point functions

Allie

20:33

Hiii

So it looks like the EL equation follows directly from minimization of the action

fqq

Which I guess can be seen as a corollary of what you call wick's theorem. I think it's a statphys/condmat thing

ACuriousMind

@fqq Yeah, I would call that something you can derive using Wicks theorem!

just like you use it in interacting theories to derive the Feynman rules, it's just that the Feynman rules for free theories are very simple :P

Allie

But in cases where the action has multiple minima, do all of those paths satisfy the EL equation? Im having trouble making sense of this in regards to how this relates to the initial

value problem

Sorry i meant extrema

Or stationary paths, whatever

Tobias Fünke

mhmh naively I'd say it does not matter. A path making the action stationary must obey the ELE.

ACuriousMind

@Allie The E-L equations are local differential equations you can solve as an initial value problem. Each path you obtain in this way is a stationary point of the action with respect to variations adhering to the boundary value problem where the start and end point of that path are the given boundaries

Allie

20:38

Oh, wait

I was just about to ask “if you have only one specific initial condition, how can you get 2 different paths”

But theres nothing stopping both from satisfying the diff equation

Right?

Tobias Fünke

it depends what you mean here. under some hypotheses the solution of an IVP is unique

just take $\ddot x=0$. get the general solution and check what different initial values (for the position and velocity at some time, say $t=0$) mean

idk if I understood you correctly

Allie

And that would be the case where there is only one path of stationary action right?

Tobias Fünke

scholarpedia.org/article/…

I think I've posted this already a while ago. Perhaps it helps

Allie

Thanks!

I think ya girl has to study diff eq more

So so much to study. I should just keep chugging on and try to absorb as much as possible

DIRAC1930

Plug & chug

The quickest way to annoy a physics professor is by writing $x^{0.5}$

Tobias Fünke

20:54

btw, @Allie, I think Löwdin (who else?) introduced "natural (spin) orbitals" in 1955

DIRAC1930

instead of $\sqrt{x}$ or $x^{\tfrac{1}{2}}$

Tobias Fünke

@Allie he says that the diagonal form of the 1-RDM "is characterized by the fact that all bond orders are vanishing" (and argues that thus the name is appropriate) -- do you know what this means? (the quote is from his paper "Quantum Theory of Many-Particle Systems. I", right below eq. 74)

DIRAC1930

Or $2.718^{i x}$ instead of $e^{i x}$ lol

Had to delete the last one as it was getting me angry just looking at it

Tobias Fünke

@Allie see also "NATURAL BOND ORBITALS
AND EXTENSIONS OF LOCALIZED BONDING CONCEPTS" --but I still don't get it lol

Allie

Ill check it out when I get home, I think it might have something to do with the occupation numbers or the population analysis

qwerty

21:07

morning

DIRAC1930

Has anyone studies Principia Mathematics by Whitehead and Russel?

qwerty

it just occurred to me if the maths chat was called "the nbd" they could say "welcome to the nbd"

imbAF

What it's meant with quantum fluctuations?

ACuriousMind

21:22

@qwerty would it be Hausdorff so that you can split it into smaller disjoint nbds, each user in their own? :P

@imbAF we've been over this:

Jan 21 at 17:36, by ACuriousMind

it's a not particularly precise phrase that just means you're treating the field classically, not as a quantum field so that photons could be generated or whatever

you've literally asked the same question about the same situation (scattering off an external field)

qwerty

21:44

@ACuriousMind haha then you'd have to call it the Hausdorff-Raum

Silly Goose

is the particle density for a free fermion field just $\langle \Omega \lvert \int dk c_k^\dagger c_k \lvert \Omega \rangle$?

In particular, are there any funny normalization factors?

i could have included a $(2\pi)^{-3}$ but would this just come from one's fourier convention?

ACuriousMind

@SillyGoose You keep being strangely vague about what setting you're talking about. The standard "fermion field" in relativistic QFT is the Dirac field, which has two sets of c/a operators. You've written down a number operator for one set. What exactly is "the free fermion field" you want to ask about here?

Silly Goose

i am just considering one set of fermion operators with this hamiltonian

ACuriousMind

and where's the field?

Silly Goose

22:00

i guess there is no field defined in this situation

Tobias Fünke

@ACuriousMind in some condensed matter applications you also have two fields, so to speak. "particle hole picture"

@SillyGoose the particle density operator in first quantization is $n(x):=\sum\limits_{i=1}^N \delta(x-x_i)$

in second quantization this is just $\psi^*(x)\psi(x)$ in position space

nothing about being free or so

the density with respect to some state then is just the expectation value

Silly Goose

is it not generically $c_k^\dagger c_k$ in momentum space as well?

ACuriousMind

but anyway, of course the number density operator for the particles created by some creation operator $c^\dagger$ is just $n = \sum_\sigma c_{k\sigma}^\dagger c_{k\sigma}$. That's how number operators work.

Silly Goose

at least intuitively i would expect the density operator to be the total number operator

Signor Feynman

@SillyGoose are you still doing BCS? :P

Silly Goose

22:02

@SignorFeynman i have been doing 2d ising for about 3 weeks :P

Tobias Fünke

what is "the density operator"?

Silly Goose

@TobiasFünke i mean the continuum version of the number operator--is one way to put it i believe

Tobias Fünke

what I've written is the standard definition and terminology in many-body condensed matter

Signor Feynman

@SillyGoose No, I asked that because I see that expression all over the place in BCS

Silly Goose

the operator that counts the density of particles

Tobias Fünke

22:03

yes, that is what I've written

Silly Goose

@SignorFeynman ah lol

Tobias Fünke

@SignorFeynman well, that's just everywhere in condensed matter :p

DIRAC1930

I think it's best just to do $\psi^\dagger \mathcal{O}^{1} \psi$ where $ \mathcal{O}^{1}$ is the one particle operator to find all these things

Silly Goose

is there any use in using position space?

ACuriousMind

@TobiasFünke that might be, but so far I haven't seen any "field" here and the goose switches confusingly between obviously talking about condensed matter stuff and then referencing Schwartz (which is a hep-th book as far as I know :P) so I really find it hard to tell what's going on

Silly Goose

22:04

it seems awfully harder to do computations in

Tobias Fünke

@SillyGoose sure

Silly Goose

anything can be a condensed matter textbook if you try hard enough ;)

Tobias Fünke

I mean, compared to what

@ACuriousMind yeah ^^

DIRAC1930

The field operator in these things satisfies the Schrodinger equation

Signor Feynman

@TobiasFünke Then can you tell me why the chemical potential is considered as a part of the Hamiltonian. In BCS, for example they add it before minimizing as a "constraint", which mathematically is ok, but I don't really see the physical meaning. It's very reminiscent of the exponent of the density matrix in the GCE, but I would like a stronger motivation

Silly Goose

22:05

@TobiasFünke well to momentum space

Signor Feynman

Measuring energy from the chemical potential?

Tobias Fünke

@SillyGoose your expression is identically 0, btw

DIRAC1930

The Hamiltonian you wrote down is just $\psi^\dagger H^1 \psi$ where $H^1$ is the one particle hamiltonian

I think i may be missing an integral

Tobias Fünke

@SignorFeynman it is not part of the Hamiltonian, no. It is when you consider the grand canonical ensemble, i.e. situation where the number of particles is not fixed

Silly Goose

@TobiasFünke for the "vacuum" i mean a many-body vacuum e.g. all occupied below fermi surface

Tobias Fünke

22:06

?

Silly Goose

it is not zero then, right?

Tobias Fünke

you've written the expectation value of $\hat N$

ah okay, yes

then it is just the number of particles in the ground state

but it is still not the particle density operator

which is what I've written

I don't see what you're trying to do here, though

Silly Goose

oh i think i see my confusion

i think i am conflating just counting up all the particles (number) and actually keeping track of their locations in addition to counting them up (density)

Tobias Fünke

and I am not sure with the pre-factors, but I am too lazy to work it out

yes, exactly

Signor Feynman

I mean, in the GCE I know how it appears. On the other hand, in SC books they often just "manually" define $\mathcal{H}=H-\mu N$ and use it e.g. in the variational principle for energy. It is not as "natural" as the way it appears in the GCE density matrix, or in the entropy variational principle 🤔

Silly Goose

22:11

but actually now i am more confused because i got the correct answer by just computing the expval of the number operator

Signor Feynman

@SillyGoose your $\Sigma$ and $\int$ are something else. Can I hire you to be my human LaTeX? :P

Silly Goose

@SignorFeynman for the price of two cups of coffee a day ;)

Signor Feynman

@SillyGoose I see that you got used to the PhD salaries :P

2

Tobias Fünke

@SignorFeynman hmhmhm. I mean it appears when you e.g. want to minimize the energy but also let the particle number vary (as you seem to know), where BCS is, I suppose, an instance of that. but I've never seen a claim that this is the Hamiltonian

well, wait a second

at least the point is true when $H$ and $N$ commute

which shouldn't be the case for the BCS hamiltonian

Signor Feynman

@TobiasFünke Given that it's an energy variational principle, I think that it's enough that it commutes with the free part...? 🤔

Tobias Fünke

22:16

hmhm, SF I have to think about it, sorry

cannot help you right now :/

Signor Feynman

Someone asked that!

Tobias Fünke

well, for now I would stick to my interpretation that you really just introduce it to fix the average particle number during minimization

Signor Feynman

That's the standard motivation, yeah

DIRAC1930

Btw one has to be careful with using $\int \mathrm{d}x \psi^\dagger j^\mu \psi$ when it comes to the Dirac theory since it gives the wrong expression. However one can add an extra term allowed by an operator ordering ambiguity so everything works out

Tobias Fünke

ok. so where is the problem with that? :d sorry, I am tired, and probably should go to bed

Silly Goose

22:23

hm okay i think there is a rather simple explanation for my wrong computation getting the right answer. physically in the situation of a simple fermion gas, the particle density actually is just constant within the fermi surface. so if i count all particles and divide by the volume i will get the same answer as properly computing the density.

but indeed in no other circumstance will just counting up the particles work.

Signor Feynman

@TobiasFünke Nah, nothing if you're talking to me now. Don't worry at all =)

Tobias Fünke

hehe ok+

maybe we can talk at a later point again

Signor Feynman

I'll need to think about it on my own

Yeah

Tobias Fünke

let me know if you made progress!

yeah, I think the idea is to get the gap equation, and then extract $\mu$ from the desired average particle number; this probably has to be done iteratively or so; once you have determined those quantities, you get the coefficients $u_k,v_k$ and can determine the BCS state

This state by definition yields the minimal energy and the (desired) average particle number

and in contrast to the "usual" CGE cases from e.g. stat. mech, the hamiltonian and the number operator do not commute, so we expect that the ground state does not have a well-defined particle number (which we know already for the BCS state)

imbAF

Is this statement redundant :"gauge theories where the symmetry is local" ?

More Anonymous

22:46

@ACuriousMind what is your opinion on the hilary putnam argument on eternalism?

(random question)

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