Mathematics - 2022-12-11

12:51 AM

I've got a physics question, but I think the heart of my confusion lies in the mathematics, so I am asking it here. To my understanding there are pure states represented by vectors in a Hilbert space. Then, there are mixed states represented by operators (density matrices) defined (I am guessing) over the Hilbert space. Then, we can have entangled pure or mixed states. How do I bridge the gap between the concepts of pure states and mixed states?

I have that a pure state that lives in a Hilbert space composed of $n$ subsystem Hilbert spaces takes on the form $| \psi \rangle = \sum_{\textbf{i}_n} c_{\textbf{i}_n}|\textbf{i}_n\rangle$. I guess I am confused because the mixed state does not live in a Hilbert space, right? So how do we analogously write out an arbitrary mixed state associated with the Hilbert space of the mentioned system?

leslie townes

you will likely need a physics educated person to put this in language that makes sense to you. likely, 'states' even if represented by vectors also live lives as rank-one operators (i.e. the 'state' represented by v corresponds to the operator of orthogonal projection onto v). and there are other, more general operators that correspond to density matrices (likely, positive operators, perhaps with some normalization or other conditions, whether or not they have rank 1 or even finite rank).

vectors being vectors but also operators might make you think of bras and kets. ask someone who speaks that language for more info.

Silly Goose

1:08 AM

What structure do operators over a vector space live in?

leslie townes

well, the set of all operators on a vector space is itself a vector space. the set of operators that correspond to 'density matrices' likely isn't, probably because of things like not being closed under addition, or multiplication by -1.

so you probably have just a convex set within the set of all operators, that are the 'density matrices.'

"what structure does __ have" is a pretty vague and open ended question from a math point of view. in physics, there might be particular structures that naturally suggest themselves. for example, if you wanted an inner product between density matrices, you could probably come up with a notion of that that doesn't just satisfy the axioms of an inner product, but has physical relevance.

but i don't know enough physics to get into details.

and if you just ask 'can i put a [mathematical structure] on [something]' without caring about the physics the answer is often a very uninformative yes.

Thorgott

as for "write out", mixed states are convex combinations of pure states, so that's how you write them down

Silly Goose

hm okay i see i thin kthat helps

do you happen to know if there is a natural vector space associated with these convex combinations?

Thorgott

1:30 AM

no, not quite

consider the physicist's favorite toy example, the qubit, a quantum system with two possible states

the mixed states there are naturally parametrized by a sphere, so not quite a vector space

they call this the Bloch sphere

PM 2Ring

5:17 AM

@leslietownes & @SillyGoose The MonaLisa Twins posted a music video starring Neve the gosling:

Pretty Little Thing - MonaLisa Twins (Original)

►

Silly Goose

omg

Akiva Weinberger

5:33 AM

@robjohn @PM2Ring Hey remember the Borromean rings stuff?

I posted it as a question on Reddit...

and check this out!

desmos.com/calculator/mr3qjipolo

Karl Kroningfeld

woah

PM 2Ring

@AkivaWeinberger Nice! Unfortunately, Desmos is a bit painful to read & interact with on a phone. But at least the sliders work.

PM 2Ring

5:59 AM

Yesterday I was reading about cell rings in the 24-cell, which got me wondering if you could do some kind of Borromean structure in 4D. But I get dizzy if I try to think too hard about 4D stuff. :) Here's a cell ring that's been disconnected and flattened out to make it 3D.

one potato two potato

6:27 AM

Smooth $1$-form is exact if and only if it's conservative, i.e., line integral is path-independent. Is this can be generalized to smooth $n$-forms?

nickbros123

6:58 AM

What are your views on serge lang for linear algebra, i dont want a very comprehensive viewing of linear algebra now, i just want to equip myself with matrix/determinant/ linear algebra arsenal to study quantum mechanics

shintuku

8:36 AM

@AkivaWeinberger does reddit have a strong higher math community?

Feynman_00

9:11 AM

@nickbros123 I used that book in my freshman year. If I had to find a flaw I would add more about matrices, but the book is great and I really enjoyed it.

Silly Goose

9:30 AM

i feel like if your main goal is to study quantum mechanics you can probably briefly familiarize yourself with bases and inner products and just jump into an intro quantum book @nickbros123

maybe also vector space decomposition related concepts (eigen-stuff, spectral theorem)

nickbros123

9:54 AM

@SillyGoose @Feynman_00 thanks for the answers. I worked through the first 50 pages or so of serge lang today, so i guess I'd be able to finish it by this month end

Feynman_00

10:42 AM

@nickbros123 be sure to have a solid understanding of basic linear algebra before delving into QM to avoid being misguided by abuses of notation, terminology and extremely powerful notations that hide the math

E.g. using Dirac notation projection operators come out "naturally" but I think that's a bad first approach, which most introductory QM books use

nickbros123

10:59 AM

@Feynman_00 thanks for the advice @Feynman_00 . Do you think I should read feynmann lectures as an intro, since I'm basically going to self study qm. Or some other standard text

imbAF

@PM2Ring but the F^(-1) is -ln(1-u), and if u take values let's say from the interval 2 to 10, then the log will be negative no?

Silly Goose

I have heard that the Feynman lectures are generally better to read after having already learned the material they cover. They are apparently good for getting another perspective on content you already know. I have also heard that the books are more heuristic (as opposed to rigorous), which is in line with my personal experience. Some introductory quantum books at the undergraduate level that to my understanding are often used include: Griffiths's QM, Townsend's QM, Sakurai's QM.

I have used all three to some degree and found that Griffith's was a nice introduction to familiarize myself a bit with QM, but it became quickly insufficient. Townsend is like a soft version of Sakurai and it follows Sakurai very closely at least for the first couple of chapters. I would recommend reading the first chapter of Sakurai (some of the basic formalism of quantum mechanics) over reading the formalism chapters of either Griffith's or Townsend if you choose to use any of these books.

Griffiths becomes quickly insufficient because in my opinion there comes a certain point in which it is a waste of time to learn quantum mechanics at the heuristic level that griffiths sometimes presents the material. Townsend and Sakurai thus step in with more, to my understanding, rigorous and rich treatments of QM that are still accessible at the undergrad level

user4539917

I'd concur that the Feynman lectures were written for the "best and brightest" category of undergraduate.

Silly Goose

11:14 AM

also, if you are interested in learning about the density matrix formalism of quantum mechanics, neither of the three textbooks mentioned cover that topic very thoroughly.

There's also this book that some people recommend. It's title "Quantum Theory for Mathematicians" should give you an idea if it is what you are looking for: staff.ustc.edu.cn/~shmj/Reference/…

Amin Radbord

Is there anyone interested in probability, who can help me in solving this problem,
https://mathoverflow.net/questions/436305/probability-of-multivariant-gaussian-random-variables-in-different-areas?noredirect=1#comment1124274_436305

nickbros123

@SillyGoose thank you for the pdf @SillyGoose

imbAF

@PM2Ring I tried to generate initially random variables in the interval [CDF(xmin),CDF(xmax)] and use these values in the -tau*log(1-x)

but I still do not get values within the desired range

@PM2Ring I was able to do it, thx

Feynman_00

11:37 AM

@nickbros123 My nickname suggests you shouldn't trust me on this. :P I think the FLP are an excellent resource, I wouldn't say you can even compare them to a standard textbook as the purposes are different: FLP volume 3 is Feynman's unique way of presenting what makes QM different, for this reason a lot of standard topic are either not covered or covered in a very different way. I consider the FLP as a complementary resource to develop insight.

TL;DR go through a standard QM book and after you've got acquainted with the basics start reading FLP vol III

Jakobian

$0^0 = 1$, $0$ is a natural number, and people should write $\subseteq$ instead of $\subset$. You don't write $0<0$ do you? And I don't think you write $0\lneq 1$ either

at the end of the day it doesn't bother me whatever convention one uses, but those are just morally right

nickbros123

11:53 AM

@Feynman_00 @Feynman_00 thank you for the insight. Do you know of any book that also covers, on top of all the formalisms, the motivation for every assumption, or postulate? I've read a teeny bit of quantum chemistry from macqurie and there are no explicit statement for why something is taken to be in the way it is taken

imbAF

anyone knows what is non-binned maximum likelihood function?

Feynman_00

@nickbros123 I think every book covers the postulates of QM.

The ones I know best are Griffiths, which is reader friendly, and Sakurai

nickbros123

12:13 PM

@Feynman_00 Will give sakurai a shot, as suggested by you and @SillyGoose .{ In that quantum chemistry book i was talking about, many operators, like the momentum operators were poorly motivated (just like most of chemistry haha) that's why I was wary. }

Feynman_00

Well, a proper rigorous mathematical treatment needs functional analysis which is something I lack too :P

Peter

Does anyone know a nice expression being a prime or a semiprime involving the numbers $2022$ and $2023$ ? Something like $2022^{2023}+2023^{2022}$ , which is however no semiprime.

IamKnull

12:38 PM

Here is a mathematical puzzle I'm stuck with:
A card dealer has 8 cards, four of which are red and the other four are black. The cards are shuffled and put on the deck with the back side on. You have got £10 to enter a gambling game as follows:
- At each round, you can bet £x, where $0 \leq x \leq$ your total available money, on the colour of the card on top of the deck. The dealer would then turn the card, and if you are right, you receive £x if you are wrong, you have to pay the dealer £x.
- You continue the game until all the cards are turned over.

Here is a strategy that can be followed: if you bet £0 in the first 7 rounds, you can easily double your money by just betting it all on the last round, when you know for sure what the colour is. So with this maximum is £20.

Can someone give an idea if this can be done more smartly to increase this amount?

one potato two potato

1:18 PM

If $\omega$ is a nonvanishing smooth $1$-form on a smooth mfd $M$ and $X,Y\in\ker\omega$ then $[X,Y]\in\ker\omega$?

Peter

1:32 PM

@IamKnull We can do slightly better : We wait until three cards are remaining. If the remaining cards all have the same colour we can collect 80 pounds. Otherwise, we bet 3 dollards on the colour that must come two more times. If we lose, we can collect 28 pounds and if we win, we can collect 26 pounds with the last bet. This need still not be the maximum.

Thorgott

2:33 PM

@onepotatotwopotato certainly not

I'd suspect the answer is "never", even

Ted Shifrin

4:41 PM

@Thorgott That certainly would be false.

shintuku

5:47 PM

how much caffeine and lack of sleep can the human body sustain before beginning to fragment and disintegrate into a cloud of dust?

certainly much less than it needs to stop functioning, you say

haha, what a funny person you are

well, whatever the answer it, find out soon with shintuku's nth burnout just before his finals' very last exam

™

Magnus Alexander

7:31 PM

Stupid question: is $\mathbb{Z}/ 3\mathbb{Z}=\mathbb{Z}_3$? xd

Lukas Heger

no, unless you use dumb notation

but I don't want to unnecessarily confuse you

in your situation, probably yes

Magnus Alexander

math.stackexchange.com/questions/1296833/… I'm looking at this solution by Andreas, his group H is a set of matrices with $a,b,c \in \mathbb{Z}/3 \mathbb{Z}$, logically it probably should be $\mathbb{Z}_3$, but I just want to make things clear

Lukas Heger

what is your definition of $\Bbb Z_3$?

Magnus Alexander

$\mathbb{Z}_3=\{0,1,2\}$

Lukas Heger

uhh, I really don't like this definition

$\Bbb Z/3\Bbb Z$ is a set of equivalence classes

Magnus Alexander

7:36 PM

Well I mean you just view $\mathbb{Z}$ with modulo $3$, stuff like that

Lukas Heger

okay, $\Bbb Z$ modulo 3 is better

$\Bbb Z/3\Bbb Z$ is $\Bbb Z$ modulo 3

Magnus Alexander

Thanks, then it's the same thing

Lukas Heger

yes. Just a small warning, there's a completely different thing also denoted by $\Bbb Z_3$

it's the 3-adic integers, but I would advise you to not look into that at this point

Magnus Alexander

By the way, as far as I remember you helped me with the task: "Can all comples irreducible representations of the group be: 5 one-dimensional, 1 five-dimensional", where you told that I should look for the subgroup of order 15, but it seems like it was a bit wrong since it might turn out that there are no subgroup like that (with this order)

Lukas Heger

no, it's actually true that every group of order 30 has a subgroup of order 15

of course you need to justify it, I can do that if you want

Magnus Alexander

7:40 PM

Of course not to offend or anything, I greatly appreciate your help, but still I figured that you might be curious about it:)

Really? How do I justify it? xd

Lukas Heger

yeah, it's good to have feedback. I should have been clear about it

Magnus Alexander

I think the idea is enough, i mean it works only for certain orders or smth?

Lukas Heger

every group of order $2k$ with $k$ odd has a subgroup of order $k$

and the trick is to use the Cayley embedding

Magnus Alexander

Hm, I'll read it about for sure, thanks Lukas!

Cayley embedding you mean that every group is isomorphic to some subgroup of permutation group (of corresponding order)?

I wanna double check since in Russian we have a slightly different notation

Lukas Heger

yes, it's a concrete monomorphism from $G$ to $S_n$, where $n=|G|$

Rubi Shnol

7:45 PM

0

Q: Finding parallel vectors inside a ball

I am reading through a paper that uses a geometric construction not quite clear to me. Namely, let $B$ be a closed ball in $\mathbb R^n$ centered at $x$. Let $s \in B$. Next, let $w$ be an arbitrary vector. Then, for a sufficiently small $\varepsilon > 0$, there is $z \in \partial B$ such that $$...

Anyone?

Lukas Heger

@MagnusAlexander I have written about this argument recently in answer here:

3

A: Group of order $630$ is solvable

As I have explained in the comments, a trick using the Cayley embedding and the Feit-Thompson theorem imply that every group whose order is not divisible by $4$ is solvable. Here's a more elementary solution: we apply the same trick as in the comments: let $G$ be a group of order $2k$, where $k$...

Magnus Alexander

Thanks!

Bob

8:06 PM

Hello

If somebody who is good in probability could look at my post, that would be good. Here is the link:

Integral is wrong. Should be $2y-y^2|_0^{1-u_0/2}$ — herb steinberg 21 hours ago

Bob

8:35 PM

Integral is wrong. Should be $2y-y^2|_0^{1-u_0/2}$ — herb steinberg 21 hours ago

Ted Shifrin

You forgot to subtract the antiderivative at $0$. Too used to pretending it’s always $0$.

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$Mathematics$ Mathematics