The h Bar - 2021-10-24 (page 2 of 2)

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6:01 PM

$$|\psi^2(x)| = \lim_{\Delta x\to 0} \frac{1}{\Delta x}\int_{x}^{x+\Delta x}|\psi^2(x')|\,dx' = \lim_{\Delta x\to 0} \frac{1}{\Delta x}\text{Pr}(x\leq X\leq x+\Delta x)$$

that's about as good of a definition as any

it captures the idea that, as you shrink an interval of with $\Delta x$ to zero, the probability of finding the particle there will shrink as well. but it'll shrink at a rate proportional to $\Delta x$

the point is that this is not an especially "quantum" thing

this is exactly how probabilities have to work for any continuous distribution

M. Çağlar TUFAN

Hmm

i mean, how do you "define" a mass density?

according to nlab, the actual standard model is $$ G_{SM} \;=\; \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6 $$

Is the $Z_6$ due to the flavors?

if you think it through, you have to determine the mass of a very small portion and divide by the mass of said portion

M. Çağlar TUFAN

I think mass density is mass per unit volume, but that's not a marhematical sefinition of course

6:06 PM

One thing that schools don't really teach and that would be a nice way to introduce such notions is to consider probabilities for classical physics

@slearah agree

closest we have is mass/charge distributions

You can consider some distribution of your initial conditions and then just use the Hamiltonian to evolve it

mass distributions are a pretty good analogy insofar as they can only be positive

I wish someone wrote an updated series of books in a similar vein to Landau's

Imagine a book that was the combination of Dirac's and Landau's QM books but for QFT

In what way?

6:09 PM

Are you talking to me or semiclassical?

To you

Also I will write the best QFT book if you want

I just need an advance of one million dollars

That would be great

@M.ÇağlarTUFAN i'm being somewhat literal in my description here

I just feel like everytime I read one of those books I learn something more and it's formulated in an exact way with no analogies

which is in contrast to books such as Peskin for QFT which I hate so much

the way you quantify the mass density at some point in an object is to extract a sample of said object that includes said point, and is "small enough" that the mass density is approximately constant in that sample

then the mass density at that point is just sample mass / sample volume

or $\left(\int_{\delta V} \rho(x)\,dx\right)/\delta V$

M. Çağlar TUFAN

6:12 PM

Alright

it's the same idea for probability density: you pick some small interval in which to look for the particle, and divide the probability of such by the size of the interval

One thing most QFT books lack are dumb examples

Few of them will just do like the hydrogen atom for QFT

@Slereah so much this

though i can sorta understand it, insofar as "dumb examples" make for good HW problems

M. Çağlar TUFAN

$\frac{1}{\Delta x}Pr(...)$ part means that right?

right

6:14 PM

Well yeah, but it also makes for good pedagogical examples!

the dumb example i've been thinking about lately

suppose you have a spinless free fermi gas and you add a contact interaction $V(x)=\lambda \delta (x)$

M. Çağlar TUFAN

@Semiclassical Well, to be honest I don't really know what to do right now. I think I will have some break for now. But I'm very thankful for your effort in explaining me. Thanks so much! This chat was more helpful than my instructor's notes.

you know what's a cool QFT case that few books talk about

@M.ÇağlarTUFAN mmkay. one passing analogy I'll leave you with

Scalar field with a step potential

6:16 PM

in classical mechanics, you deal with three integrals a lot in order to predict behavior of an object:

Is quantum statistical mechanics essentially just classical statistical mechanics where the microstates are quantum states?

@DIRAC1930 I mean classical statistical mechanics is basically a lie

Microstates in "classical" stat mech are just quantum states

Okay thanks

that is why their size is coincidentally $\hbar$

the mass $M=\int \rho(\vec{r})dV(\vec{r}),$ the center of mass $\vec{r}_{CM} =M^{-1}\int \vec{r} \rho(\vec{r})\, dV(\vec{r})$, and the moment of inertia $I=\int r^2 \rho(\vec{r})\,dV(\vec{r})$

the claim i want to make is that those are not actually so different than the expectation values $\langle 1\rangle$, $\langle x\rangle$, $\langle x^2\rangle$

with the middle case being essentially a one-to-one correspondence

6:29 PM

Is there a book that starts at quantum statistical mechanics?

probably?

@DIRAC1930 That's what Greiner's series is attempting to do

Ah yes, I have heard about them

Of course it doesn't end up doing that but they are very useful to check, apart from qft the amount of updates L&L need for what they do is relatively small

e.g. they do quantum stat mech from first principles at the beginning (they set up the classical version first but you can literally ignore that the way they wrote it, after the beginning they do both simultaneously)

Yeah L&L some L&L are perfect but I didn't find the relitavistic QFT books very useful by them

but Landau wasn't involved in the writing of those

6:39 PM

I can understand why people do classical mechanics in their first chapter but I am so sick of reading that intro QM chapter

They are still based off his teaching notes, one of the biggest differences is they set up qft for fields without starting-from/setting-up the $[\hat{\phi}(x),\hat{\pi}(x')] = ...$ type commutation relations, which is actually an unbelievable virtue

@Slereah I've skimmed/read it (their intro one) at least a hundred times and I'll have to look at it again and it was worth it :p

7:04 PM

@DIRAC1930 they derive the grand canonical distribution in two ways, section 35 and 36, the latter can be done after skimming section 4/6 in a few lines

Thank you

I missed that section when looking through that book

So it turns out average number of particles in a subsystem is conserved

Does average number of particles in a subsystem mean that total number of particles is conserved as a consequence?

My reasons are that if the environment and the subsystem have 2N particles in total and they are both the same size etc, the subsystem should on average have <N>=N particles

if this average rises to say <N>=N+1

then that would imply that the exact number of particles in the whole system is 2N+2

I'm not sure if this is rigorous however

I'm not sure what you mean, grand canonical is set up to allow for $N$ to be variable

7:19 PM

Variable between the system and the environment but I thought the number of particles in the system + reservoir is a constant

The same for energy

The system + environment is a closed system right?

sure, but usually you should think of the reservoir as just containing infinitely many particles - the system can't "drain" the reservoir, it's an idealization of the notion of environment

@ACuriousMind just to fill you in, I'm talking about the constraints required to maximise the entropy via the Lagrange multiplier method

Average energy and average particle number seems to be two of them

Maxwell–Boltzmann statistics

In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible. The expected number of particles with energy ε i {\displaystyle \varepsilon _{i}} for Maxwell–Boltzmann statistics is ⟨ N i...

Lagrange multipliers are only in the microcanonical section there

So for the microcanonical ensemble $\sum_i N_i$ is a constant

I'm not sure for the GCE $\sum_i p_i N_i$ is a constant

that would give rise to a $\mu$ term I believe

Now I'm confused, how does it make sense to apply those derivations to relativistic systems where particle number is not fixed

7:31 PM

That's what I'm confused about too

Just to clarity, I'm not sure anything I'm saying is correct

I'm hoping someone can help me point out if I'm wrong

7:52 PM

There are comments on this in chapter 3 here, it's going to take ages to figure out what's really going on here in this special case...

There doesn't seem to be a derivation relying on maximising entropy from conservation laws there however

8:07 PM

I think maybe if a quantum system has a conserved charge $Q$, for the GCE, we enforce that that $\langle Q \rangle =c$ is a constraint on the system

I think I remember the rationalization I came up with before, if you are studying these kinds of relativistic systems, they are not really thermodynamical in that the effects of changing particle number are instantaneous and local, but as long as the average particle number change is zero you can apply thermodynamical thinking to them, they basically do this as part of setting up black body radiation

if we do that then we get a term $p(Q_1,Q_2,\dots)=\exp{\sum_i \alpha_i Q_i}$

where $\alpha_i$ are the lagrange multipliers

So basically this is a complete edge case already and you need to be careful

8:21 PM

For relativistic systems, I don't think it makes a difference

you just find all the conserved charges

i.e. what commutes with the quantum Hamiltonian

enforce that the average is a constant

and the operator will appear in the probability expression like

$p(Q_1,Q_2.\dots)=\exp \sum_i \alpha_i Q_i$

for QED, electron minus positron number is conserved

I think that's why GCE is used

there seems to be an easy way to transfer exactly conserved charges to the probability distribution, at least that's what I think anyway

But e.g. the grand canonical distribution is not set up using that

Why not?

Because look at the derivation it doesn't use that

There are many different derivations

web.pa.msu.edu/people/pratts/phy831/lectures/lectures.pdf

Which one uses that

8:27 PM

section 1.3

What do you think?

As it says there, most books just use $N$, which I'm used to... I guess it makes sense and you can view it as an extension of using $N$ and we now have $[\hat{Q},\hat{N}] = 0$, and $Q =0$ for photons so it obviously agrees with the black body case, otherwise it looks pretty much the same right... that's a bit of a shock

there was a question earlier about whether one needs $[H,N]=0$ as well, wasn't there?

But $Q$ isn't conserved in the subsystem

but $N$ isn't conserved in the subsystem because particles are free to move between the subsystem and the resevoir

Right, that's usually just assumed, but it's not true for qft systems

$<N>$ is conserved I believe

8:42 PM

I'm not sure about that at all, I don't think so for these qft cases

though, come to think of it, one of the exercises my students had to do this semester was show that $[H,N]=0$ still held when doing second quantization

Yeah $[H,N]\neq 0$ for rel. systems

Yeah it holds for the non-relativistic second quantized Hamiltonian

right

I don't think it's true in general for interacting non-rel systems

8:43 PM

Basically this is one of the most important things one should learn before doing qft and I just didn't appreciate it for ages

i think it does hold in some generality there---your Hamiltonian can typically be expressed as a polynomial function of number operators

for example, there is a model for superconductivity

yeah, that'd break it

but there are non-particle number conserving terms in the Hamiltonian

right

8:44 PM

however the GCE expression takes the same form

$a^\dagger a^\dagger$ gonna mess you up

i dont' know how one does grand canonical ensemble in such a context, or if it even makes sense there

But in that case, you wouldn't get that as a conserved charge so it doesn't matter

i.e. the chemical potential would be $0$

Yeah but that's probably just modelling electromagnetic (relativistic) effects

on an entirely different note: there's something i'm trying to remember with Clebsch-Gordon for three particles

But $N_e - N_{-e}$ is conserved for QED

8:46 PM

am i right in thinking one has to be careful about the labelling of states in that case? like, it matters in what order you couple the basis states

It's zero for photons (which are strictly neutral)

Anyways, I assume there is some sort of argument used for rel. systems that just works

e.g. "couple j1 and j2, then j3", or "couple j2 and j3, then j1"

Yes for photons, particle number is not conserved so there is no conservation law or constraint that comes out of it

I'd say this simple change from $N$ to $Q$ is all you need to do and you can then just repeat the rest of a stat mech book, I can't believe I didn't think of that well done

They start waffling at the start of the black body chapter to rationalize this, but what they say seems correct

8:50 PM

i remember at least one source arguing that writing stuff like $|JM\rangle = \sum c^{JM}_{j_1 m_1}|j_1m_1\rangle$ was wrong b/c it has the wrong number of quantum numbers on both sides

but durned if i can find it again

I still haven't read their 3j qm section yet

i knew this briefly at one point

For photon production, is it equally likely for 2 photons to be produced with an energy each of E as it is for one photon to be produced with energy 2E?

and then quickly forgot again, b/c who wants to remember 3j stuff

Not sure that photon production thing can be answered

9:01 PM

Okay, because if something like that were true, it would be impossible to enforce $<N_p>$

I think

9:11 PM

Three big stat mech books don't mention this issue about charge vs particle number being conserved, if you find one that does make this distinction like those notes do let me know

Complex SYK Model - Alexei Kitaev

Alexei Kitaev mentions something about charge here

At 5:47

SYK model is not exactly the 'usual' case tho

Thats true

but in the usual case average energy is conserved

there is a derivation in Leonard Susskinds lectures

Lecture 4 I believe

I'm not sure which ensemble he considers

I think it may be more instructive to consult books on thermal field theory

I think other conserved charges may play a bigger role and maybe the texts cover those cases more

But I don't see why not N shouldn't be considered the same as Q

9:54 PM

$[\hat{N},\hat{H}] \neq 0$ e.g. for Klein-Gordon, that's why

Having the expression $p=e^{\beta H -\mu N}$ is a statement about $<N>$ being conserved

You see the same expression in rel. systems where $[H,N]\neq 0$

And I'm not sure why

I think perhaps its just QED where you can make the statment $[H,Q]=0$ where $Q$ is $N_e - N_{-e}$

Yeah I'm not sure that's even correct to write in such cases

There must be some non-rigourous assumption or something for rel. systems

Yeah the qed thing is right, you can see $Q$ is basically like $N$ it's just $N_e - N_{-e}$ where $N_{-e} = 0$ in the non-relativistic case

For photons it's $N_p - N_{-p} = 0$

Interesting

So would the probability distribution for the KGE just be $p=e^{\beta H}$?

10:03 PM

So that's why the black body thing is going to be the same if you replace $Q$ by $N$, but if you use $N$ you need to invent a rationalization for why $\mu =0$, whether it's really valid to even use the $N$ version is a question now

You'd have a $Q$ in the KG case too

I think maybe it doesn't matter because wouldn't you usually calculate the correlation functions in a perturbative expansion with the coupling constant

so $[H_0, N]=0$

anyway

If you use $Q$ is it even right to call it a grand canonical ensemble

They frame it as applying to 'charges' where $N$ is considered a 'charge' when it's conserved, but...

I don't know

Grand canonical ensemble

In statistical mechanics, a grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium (thermal and chemical) with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible...

They say $N$ is what we called $Q$ when you allow anti-particles, so yeah...

Why would we expect $<Q>$ to be a constant for any conserved charge?

Is there a more rigorous derivation?

equilibrium is described as a state where no macroscopic change occurs

if the $Q$ is built up in one area, would that imply macroscopic change would occur?

Actually, I think the average is the most you can talk about because each state, has a classical probability of occuring.

10:16 PM

The point is, in thermal equilibrium such microscopic build-ups can be ignored and do not affect the overall average behavior

10:29 PM

I can't get over the fact that I'll never be able to do a Ph.D. in String Theory or something fundamental

AdS/CMT correspondence

In theoretical physics, anti-de Sitter/condensed matter theory correspondence is the program to apply string theory to condensed matter theory using the AdS/CFT correspondence. == Overview == Over the decades, experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors and superfluids. These states are described using the formalism of quantum field theory, but some phenomena are difficult to explain using standard field theoretic techniques. Some condensed matter theorists including Subir Sachdev hope that the AdS/CFT correspondence will...

Aren't those things just coincidences though? As in, aren't they just using it as a technique to solve CMT models?

Well the point would be to use it to go both ways hopefully and to 'research' on what's possible

Thats true

If there's actually something to this it could be a blessing in disguise so who knows

10:42 PM

That's true. Kitaev's work seems like something that concretely helps both condensed matter physicists and high energy physicists

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