The h Bar

ACuriousMind

20:00

ah, the joys of physicists not establishing what calculus of variations is :P

imbAF

@TobiasFünke I am. Yes I know what I asked is QM. Is that in what I am doing rn in condensed matter theory is QM, while theoretical particle physics is where QFT is present

Slereah

@Relativisticcucumber there is very much one, and it's called the fundamental theorem of variational calculus!

Fundamental lemma of the calculus of variations

In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf. The fundamental lemma of the calculus of variations is typically used to transform this weak formulation into the strong formulation (differential equation), free of the integration with arbitrary function. The proof usually exploits the possibility to choose...

imbAF

The question I posed had to do with the functional of a wave function $E(|\psi\rangle)=\frac{\langle \psi|H|\psi\rangle}{\langle \psi|\psi\rangle}$ @TobiasFünke

Relativisticcucumber

@imbAF r u doing a&m ch 17

Tobias Fünke

@ACuriousMind the struggle is real... I've never understood why it is introduced so poorly in most courses... (this is at least my experience)

imbAF

20:01

@Relativisticcucumber a & m?

Slereah

@TobiasFünke Well people don't like infinite dimensional spaces

Relativisticcucumber

@imbAF ashcroft and mermin

Mr. Feynman

@TobiasFünke I've come to the conclusion most physicists just have a poor understanding of it

Relativisticcucumber

@Slereah thank you v much

Tobias Fünke

@Slereah as usual: physicist often get away with discussing the finitedimensional/discrete case, i.e. directional derivatives, and then generalize. no?

Mr. Feynman

20:02

Beware that I'm talking of the physics level of rigor

Silly Goose

@ACuriousMind i thought you have recommended to not go down the rabbit hole of calculus of variations :P

Relativisticcucumber

@SillyGoose thats what i thought too

ACuriousMind

@SillyGoose well, not the formal one where we prove everything like a mathematician would

imbAF

@Relativisticcucumber I am not sure. I am not aware of the book you mention

ACuriousMind

but it would be nice if we at least wrote down properly what we mean by all the $\delta$s since otherwise we get questions like this :P

Silly Goose

20:04

i just read $\delta F[f]$ as the linear coefficient in the intuitive expansion of $F[f]$

Mr. Feynman

Actually CoV is not that bad compared to e.g. functional analysis

ACuriousMind

@Relativisticcucumber the expression there doesn't really make a lot of sense without more context - what exactly are you varying here? The $\delta$ is a notation that hides a lot of context. What might be meant here is that the integral is a functional of $f$, $I[f] = \int f$, and you vary $f\mapsto f + \delta f$. Then, of course, you have that $\delta I = I[f+\delta f] - I[f] = \int \delta f$ since integration is linear.

Relativisticcucumber

@imbAF oh what text does ur course use for condensed matter

Mr. Feynman

My biggest gripe with CoV is that I had trouble understanding why constraints must be imposed after taking partial derivatives in the EL

ACuriousMind

@Mr.Feynman for a moment I thought you were saying functional analysis is worse than getting covid :P

imbAF

20:06

@Relativisticcucumber In preparing the lecture I have consulted the following books: G. Grosso and G. P. Parravicini, Solid state physics J. M. Ziman, Principles of the theory of solids J. M. Ziman, Electrons and phonons P. G. deGennes, Superconductivity of metals and alloys

Tobias Fünke

@Mr.Feynman what do you mean exactly?

Relativisticcucumber

@imbAF oh

imbAF

yeah xD

Tobias Fünke

Or put differently: did you resolve the issue(s) already?

@imbAF Do you, by chance, discuss the Rayleigh-Ritz variation principle?

Mr. Feynman

@ACuriousMind It is; I had covid in August and it was not so bad that time. I even got my voice lowered enough to speak in the contraoctave

imbAF

20:08

Ritz variation is how we took it

Mr. Feynman

@TobiasFünke It's something I could only find in an italian book about relativity and later on a book about CoV. Let me check if I can find a question I asked in the past

Tobias Fünke

@imbAF I see.

Relativisticcucumber

@ACuriousMind hm. well im working out a derivation of the hartree equations via $\delta \langle H \rangle$ and so i have written the problem as $\delta [\langle H \rangle - \varepsilon_i(-1+\int dr \psi^* \psi)] = 0$. When i plug in my expression for \langle H \rangle, i get that i would have the hartree equation if i say that the integral in the expectation value and the variation "undo eachother" but this doenst seem great to me

imbAF

Yes it;s called Ritz variation method (principle )

Mr. Feynman

The context is related to this question. For example, in that case you should take the unconstrained action, find the E-L equation and then impose the normalization. If you start from the already constrained actions things don't work

Silly Goose

20:11

for more clarification the cucumber's problem has a Lagrange multiplier

Relativisticcucumber

Mr. Feynman

Or you should take a longer path to get the same result

Relativisticcucumber

this is the ham expectation value im working w

Mr. Feynman

By the way, it's hilarious that Tobias finally joined us just now that we've entered the condensed matter era of the hBar

3

ACuriousMind

$\langle \text{shatner}\rangle$ (a large ham expectation value)

Tobias Fünke

20:12

@Mr.Feynman haha

Relativisticcucumber

@ACuriousMind ?

Tobias Fünke

@Mr.Feynman thanks, I'll have a look

Mr. Feynman

Ham because you're sandwiching states?

ACuriousMind

@Relativisticcucumber but what are you varying?

@Relativisticcucumber "ham" is a term for overacting (see also "chewing the scenery", etc.)

Relativisticcucumber

@ACuriousMind oh it also has another meaning

like "going ham"

Tobias Fünke

20:14

@ACuriousMind I guess they vary the set of one-particle orbitals/states, i.e. the set of $N$ $\psi_i$

@SillyGoose yes

Mr. Feynman

@TobiasFünke Incidentally, the book that I found useful in describing this thing was Analytical Mechanics for Relativity and Quantum Mechanics by Oliver Davis Johns. Chapter 16 and I think also chapter 5 was useful. It's been a long time

Relativisticcucumber

@ACuriousMind er i think the wave functions?

Tobias Fünke

@Mr.Feynman thanks

@Relativisticcucumber You are studying the Hartree approximation, no?

Relativisticcucumber

yes

ACuriousMind

@Relativisticcucumber so you're trying to do this derivation?

Tobias Fünke

20:18

Have you tried to just vary with respect to $\psi_i^*$? This should be do-able, i.e. it is standard textbook stuff. There shouldn't arise too complicated problems.

Mr. Feynman

Some lore: I had been searching for a book discussing this point for months, probably over a year and I was on the bus, still desperate to find something. My phone battery was quite low and eventually I found this book in the list of references of a page on libretext. I managed to save the name of the book while I still had 4% and then, after securing the kill (FPS jargon; not a killer yet), I used the rest of the battery to search the book, I think.

Relativisticcucumber

@ACuriousMind this is for hartree fock right? so similar but different

Mr. Feynman

I don't remember if I could download it though, since in that period lib*** had been banned in Italy, so I could not access it via mobile

ACuriousMind

this is where I show my ignorance of any kind of practical physics :P

Relativisticcucumber

@TobiasFünke er i am not sure what this means. this variation stuff is always smth i was told to never learn rigorously

so these stationizing problems have always troubled me

Tobias Fünke

20:20

@Relativisticcucumber Ok... but then the problem is "deeper": You should learn the very basics of functional derivatives etc.

Relativisticcucumber

@ACuriousMind i think the only difference is hartree fock enforces antisymmetrized states via the slater determinant

Tobias Fünke

and then only later come back to this specific problem. But don't worry, the basics are really not hard

Relativisticcucumber

@TobiasFünke ok i will try. the problem is finding the right resource i guess

ACuriousMind

anyway, the problem is probably that you're not pulling the $\delta$ "through" the integral; if you aren't comfortable with how the $\delta$ behaves as an operator, try computing it from the definition I used above: You have a functional $E[\psi]$, and you vary to get $E[\psi + \delta \psi]$. The $\delta E$ you want to compute is the difference of these two terms, throwing away everything that's more than first order in $\delta \psi$

Tobias Fünke

I think Wikipedia has a nice summary... i.e. they give a definition and list some properties

ACuriousMind

20:22

If you do it that way you can just use all the "normal" rules without having to think about how the $\delta$ on its own interacts with anything

Mr. Feynman

I recently had to learn about Hartree vs Hartree-Fock with Feynman diagrams

Hartree-Fock neglects the correlations for more particles GF: via the Wick theorem you have that correlation functions are just sum of products of (dressed) propagators but there is no "genuine" more particle term, if I understand correctly. On the other hand, Hartree is just the usual mean-field approximation, that neglects a diagram in the one-loop self-energy, but in some cases it turns out to be a better approximation

Tobias Fünke

@ACuriousMind exactly. let me add: everything is just like directional derivatives here in the end. so if you are fine with this concept, functional derivatives should not cause too much trouble.

Mr. Feynman

That's all the understanding I could get from Rickayzen :P

imbAF

Is parity the same as an even or odd function? (Defined as it is in the picture I uploaded)

Silly Goose

20:36

@Relativisticcucumber I think you would have $\delta \langle H \rangle[\psi] = \alpha \frac{\partial \langle H \rangle}{\partial \psi} \delta \psi$ where $\alpha$ is the parameter you're using to vary the argument of the functional.

imbAF

20:58

what is positive and negative parity?

imbAF

21:51

If parity operator changes the sign of all 3 spatial coordinates, what is it when only one coordinate change? Reflection?

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