The h Bar

Rudi_Birnbaum

01:44

Hi all. I recently had a talk about QM of non-rel. electrons in the mag. field. I wrote down a Hamiltonian with a component $\bf A\cdot\bf p$. A colleague pointed out this was not possible since the spatial variables contained in the magnetic vector portential $\bf A$ do not commute with the momentum operator, hence the term is "illdefined".

I could not follow the argument but I saw that he was right in his conclusion and that one actually has $\bf A\cdot{\bf p} + \bf A\cdot\bf p$, but I concluded that from the term it originates from in the Hamiltonian, that is $({\bf p}-{\bf A})^2$ and the fact that they do not commute.

Is anyone able to explain what his reasoning was? I then would also wonder about the angular momentum operator ${\bf r}\times {\bf p}$ and if his argument would not have to hold for that as well. I shall add that the colleague is quite experienced, such that I would be inclined to take his comments serious.

sorry the correct term should read ${\bf A}\cdot{\bf p} + {\bf p}\cdot{\bf A}$.

Semiclassical

02:25

@Rudi_Birnbaum Consider the operator $xp$. The hermitian conjugate of that would be $px$, so $xp$ is not Hermitian and so not an observable.

By contrast, $\frac12(xp+px)$ is Hermitian. That only differs from $xp$ by the commutator $\frac12(xp-px)=\frac{i\hbar}{2}$, so in classical mechanics there's no difference between the two.

Rudi_Birnbaum

@Semiclassical @ how comes that each single component of the Hamiltonian have to be observalbles for themselves?

@Semiclassical thats a nice argument!

Semiclassical

@Rudi_Birnbaum it doesn't: $H=\frac12(xp+px)$ is an observable, but neither term in it is an observable.

Rudi_Birnbaum

right

so why would $H_0 + {\bf A}\cdot{\bf p}$ have to be wrong. Couldnt it be compensated by something in $H_0$

that does not obviously look like ${\bf p}\cdot{\bf A}$

Semiclassical

I would agree, in principle. But in practice you usually do want to be able to interpret $H_0$ itself as an observable

Rudi_Birnbaum

:-) OK!

one more thing I see immediately that $({\bf xp})^\dagger = {\bf p}^\dagger{\bf x}^\dagger = {\bf px}$. However when I calculate that explicitely by $({\bf xp})^\dagger \psi$ and ${\bf p}=-i \hbar \partial/\partial x$ and $({\bf px})\psi$ I get a different result. Is that possible or do I make an error in calculation?

Semiclassical

02:48

I think the issue is that complex conjugation of $xp$ alone doesn't capture the fact that the Hermitian conjugate should reverse the order of the operations

Maybe this instead: If $$\langle \psi|xp|\phi\rangle = \int \psi^*(x)x (-i\hbar\partial_x)\phi(x)\,dx=\int \psi^*(x)(-i\hbar \phi'(x))\,dx$$

then $$\langle \psi|xp|\phi\rangle^* = \int \phi^* '(x)(-i\hbar x)\psi(x)\,dx=\int \phi^*(x)(-i\hbar \partial_x x)\psi(x)\,dx=\langle \phi|px|\psi\rangle$$ upon integrating by parts

hnnng

Rudi_Birnbaum

Isn't $\partial_x (x \phi(x))= \phi + x \phi'$?

Semiclassical

$$\langle \psi|xp|\phi\rangle^* = \int \phi'^* (x)(-i\hbar x)\psi(x)\,dx=\int \phi^*(x)(-i\hbar \partial_x x)\psi(x)\,dx=\langle \phi|px|\psi\rangle$$ upon integrating by parts

sure? I could've expanded it, I just didn't see a reason to

the point was that $\langle \psi|xp|\phi\rangle^* =\langle \phi|px|\psi\rangle$

Rudi_Birnbaum

Yes but for that I am always left with $1+ <\phi|px|\psi>$

Semiclassical

which, since $\langle \psi|xp|\phi\rangle^* = \langle \phi|(xp)^\dagger|\psi\rangle$, means $(xp)^\dagger =px$ as expected

Rudi_Birnbaum

Sorry!

you mean $<\psi|\phi>=0$, right?

I mean: $\int \phi^*(x)(-i\hbar \partial_x x)\psi(x)\,dx=\int \phi^*(x)(-i\hbar) (\psi(x) + \partial_x \psi(x))\,dx = \int \phi^*(x)(-i\hbar) \psi(x) \,dx + \int \phi^*(x)(-i\hbar) \partial_x \psi(x)\,dx$

$=-i\hbar \langle\phi |\psi \rangle + \langle\phi|p|\psi\rangle$, no?

Semiclassical

03:07

no. $\partial_x (x \psi(x))=\psi(x)+x\partial_x \psi(x)$

so you'd end up with $\langle \phi|-i\hbar + xp|\psi\rangle$

which is what you expect: $px=-i\hbar+xp$

since $xp-px=i\hbar$

Rudi_Birnbaum

@Semiclassical Ah yes, thats it!

Great! thanks!!

Semiclassical

np

Rudi_Birnbaum

I am so stupid! Sorry but its 5:09 over here

Semiclassical

no worries

Rudi_Birnbaum

As we are happily chatting about $xp + px$, why is that thing non-hermitian on $l^2(\Bbb R^+)$?

Semiclassical

03:12

Probably has to do with the integration by parts step. In integration by parts, you really have $\int u v'\,dx=uv-\int v u'\,dx$

If you're working on the entire real line, you can say that the boundary term vanishes at $\pm \infty$

and so that can be ignored

but if you're working on $[0,\infty)$ then the boundary at $x=0$ probably creates issues

Rudi_Birnbaum

Ah OK, I just see its then already $p$ for which its unclear if its self-adjoined on that space.

See number 4 in the best answer here.

Ryan Unger

whats happening

Rudi_Birnbaum

Correction: $p$ is not self-adjoint on $l^2[0,\infty)$.

Ryan Unger

use $L^2$ for these spaces

Semiclassical

yeah

l^2 is for sequences

Rudi_Birnbaum

03:22

@RyanUnger ok!

Sir Cumference

Yo does anyone remember protractors

Those things that where everywhere in 5th grade and then suddenly vanished

Who'd have thought you'd never need them in real life

Semiclassical

ehhh

if i want to measure an angle accurately, I wouldn't use a protractor. I'd probably be more likely to use trig

Sir Cumference

it's just weird that i've never really seen one since elementary school, yet i always used to see them back then

well it's midnight so i'm probably not making much sense

Semiclassical

the high-school equivalent is the compass, I guess

Sir Cumference

well compasses were something you'd use maybe 5 times in the year, but protractors were on like every 5th grade math hw

Semiclassical

03:36

I have compasses in mind for high-school geometry

Sir Cumference

oh man high school geometry

i'm getting flashbacks of these stupid proof columns

> All proofs are separated into two columns. There are statements on the left-side and reasons on the right-side. See the diagram below to see how they look.

Semiclassical

ugh

Sir Cumference

i'm starting to suspect the people who set up these curricula have never worked in math

Semiclassical

i do wonder about the history of it

there's a whole article on the evolution of 2-column proofs here: jwilson.coe.uga.edu/Texts.Folder/herbst.pdf

3

Sir Cumference

oh wow

yeah it's pretty incredible how far removed the curriculum is from what mathematicians do

and no one likes those column proofs. it just makes the class more miserable for students

Semiclassical

03:46

interestingly, the timing of the emergence of the two-column proof seems to be around 1900

Sir Cumference

there's a gaping hole in my memory regarding high school geometry. it's the only class i can't remember anything from, probably because it was horribly taught

but oddly enough it's the only class where recalling nothing from it has had (seemingly) no effect on my pursuits in math

Semiclassical

lol

Sir Cumference

i mean it's a lot of useless terminology like "locus"

Semiclassical

04:06

eh, I like locus. but yeah, lots of words which don't matter

my favorite one is from parabolas: the latus rectum of a parabola

roobee

anyone know how the enthalpy of glucose is measured? i imagine they burn powdered glucose in an o2 rich environment. but when i saw youtube videos burning powdered sugar on a table was difficult. they had to make it into a dust cloud form before it ignited

John Rennie

04:26

@roobee with a calorimeter.

roobee

it seems like a calorimeter uses electrical ignition? so electrical ignition is a reliable way to ignite difficult to ignite materials then?

John Rennie

04:46

Typically a calorimeter heats the material until it spontaneously combusts and it measures the heat produced.

For sugar that would only need a few hundred degrees C.

dmckee

@Semiclassical Those appealed to my compulsive tendencies when I first encountered them. But that was in middle school.

roobee

@JohnRennie thanks, didn't realize spontaneous combustion is a thing. found for glucose in a divided solid state somewhere over 716 for the temp if anyone else was interested

Slereah

05:43

morning

Sir Cumference

yo does anyone remember when the symbol ÷ was used for division

all right i'm going to sleep

huh it's actually called an obelus

Slereah

06:08

@SirCumference jeff560.tripod.com/operation.html

The obelus (÷) was first used as a division symbol by Johann Rahn (or Rhonius) (1622-1676) in 1659 in Teutsche Algebra (Cajori vol. 2, page 211).

PM 2Ring

06:41

@SirCumference I think I still have a protractor somewhere, but I don't really remember using them all that much at school. OTOH,

@SirCumference I think I still have a protractor somewhere, but I don't really remember using them all that much at school.

And like Semiclassical said, using trig to measure or construct an angle tends to be more efficient, assuming you can construct a right angle. And when constructing angles you can use continued fractions to find a nice rational approximation for the tan of the angle, or if you're really keen, find a Pythagorean triad (of integers).

Let $z=x+iy$, where $y/x=\tan(\theta/2)$, where x & y are integers. Then $z^2=x^2-y^2+2xyi$ and $(x^2-y^2, 2xy, x^2+y^2)$ is a Pythagorean triad, and $2xy/(x^2-y^2)=\tan(\theta)$.

@roobee An old magic trick is to discreetly rub a small amount of cigarette ash onto a sugar cube, and then set the sugar cube on fire with a match. The ash acts as a catalyst.

Electronic calculators started appearing during my high school years, but we weren't allowed to use them until my final 2 years. We spent a lot of time learning how to do calculations using logarithm tables, and slide rules. That's now (mostly) archaic technology, but it did give us a good understanding of significant figures.

Physics Meta

07:58

0

Q: Why my recent answer was downvoted?

Why does frictional force cause a car to move? Also, is friction a reaction force? There is the above link for question to which I answered.I don't find the reason for the answer to be downvoted.Is the answer a low quality answer?

discussion

yuvraj singh

@JohnRennie hi

John Rennie

@yuvrajsingh hi

yuvraj singh

So I have issue with this chap unique. Why see the link I am giving sir. @JohnRennie

There is no need of upvoting I already informed the moderators — Unique 22 mins ago

John Rennie

@yuvrajsingh Loong isn't one of the site moderators. What you should do is flag the comment. Click on the little flag symbol to the left of the comment and leave a message for the moderators. They'll get the message and have a look at the comment.

yuvraj singh

OK sir,.

But sir he deleted his all. Comments. @JohnRennie

John Rennie

08:12

@yuvrajsingh the moderators can see the deleted comments.

Physics Meta

12:22

0

Q: Regarding down voting a question

Actually u was watching answer of one of the user, and saw that many of his answer either need update or correction but instead of this he makes the rude comment to me, threatening me for a permanent banned over site, making my image as a false voter. I belive the moderator should be fare. I won,...

feature-request user-accounts

Ezze

What?

^

Semiclassical

@Ezze your guess is as good as mine

grovesNL

12:51

@ACuriousMind interestingly WolframAlpha seems to support these "temperature rate" conversions by assuming a context (temperature becomes either the dimension "temperature" or a separate dimension "temperature difference", though both have the same result for these cases): wolframalpha.com/input/… So I wonder if there is a general way to reliably infer whether offsets should be applied solely based on the units being converted

skullpetrol

14:28

@dmckee did you first encounter the two-column proof style in geometry or algebra during middle school?

dmckee

14:47

@skullpetrol I first encountered it during a unit of boolean algebra that was part of a summer enrichment program (bwtween 7th and 8th grade, I think). The first time I saw it in school was in the 9th grade course on Euclidian geometry.

skullpetrol

Thanks for replying @dmckee

it looks like it originated in geometry, but is perfectly suited for a step by step approach to elementary algebra

the enrichment program must have softened the blow for when you saw it in Euclidean geometry

Semiclassical

15:19

the article I linked seems to set the context as: teachers were trying to get students to learn how to write proofs, and so conceived of the two-column proof as a way of accomplishing that

the issue with this being that you treat 'proof' as a subject unto itself, and so divorced from the task of learning math

skullpetrol

yes, in geometry first

thanks for linking the article @Semiclassical

Semiclassical

google-fu serves me well, lol

skullpetrol

:-)

dmckee

15:34

@Semiclassical I think that it does one thing well: it provides a structure to enforce the requirement that you are able to say why every step is OK. I've seen in my own teaching that not every student finds that a natural way to think.

Not that I'd want to write or read such proofs anymore.

Semiclassical

yeah

the way the article puts it is that the 2-column proof brought 'stability' to school geometry

but, well, rigor mortis is also a kind of stability :P

skullpetrol

lol

> The form can function as either a straightjacket or as a safety net, depending on how it is deployed.

generally, novices need a safety net

barlop

if you have two people, one a strong old man that can pull and push heavy weights while in a chair, the other a young fast guy that can't pull and push heavy weights while in a chair. Both sit in a chair. And each push a light object. They are each in a chair so they can't use their body weight. A) Is it possible that if there's no weight there, the young one can push his arm faster than the old one?

B) Is it possible that the old one will push the light weight further than the young one?

i'm wondering if there is a distinction between speed that you can move your arm, and strength. (if mass is equal so they are in a chair).

skullpetrol

18

A: Is there any evidence about the effectiveness of "table proofs" in pre-college mathematics education?

The two-column proof form has been the dominant mode of presentation for proofs in secondary geometry in the United States for most of the past century. You ask about its effectiveness; unfortunately, I think that question is ill-posed, because the goal state isn't clearly defined (effective at ...

Semiclassical

@skullpetrol same author (Herbst) involved in the paper quoted there as the one I linked

so I guess they're the expert, lol

skullpetrol

15:48

yup

Semiclassical

oh, lol, they cite the linked paper at the end

skullpetrol

also, more generally this

@dmckee what do you suggest to those students who don't find it natural to think that way, sir?

skullpetrol

16:13

debate.org @NovaliumCompany

Novalium Company

@skullpetrol thanks! (for saving me time and energy)

skullpetrol

np

have fun pal

Novalium Company

@skullpetrol I was seriously thinking of programming such a website and I'm glad you found this, cuz... I would have been so pissed if I programmed it and then found out such one exists. How did you found it? I searched so much with keywords like public opinion polls and stuff... I guess I didn't thought of the word debate.

skullpetrol

yup, "debate" is the key word

Novalium Company

I thought such a website would be kinda like a way to see what people want and vote on different topics, so the world can become what the people want it to be.

skullpetrol

16:21

> debate website for students

try that^

Novalium Company

Like, whether drinking should be allowed over 18 and not 21. People can vote for a side and state why they think that way. They can share their opinions in Facebook and so on.

skullpetrol

debate.org is a good start

Novalium Company

Yeah, thanks. I'm not looking to use such a site, but to make one. And thank you for stopping me again! I can't believe I didn't think of the word debate.

skullpetrol

yeah, no need to reinvent the wheel :-)

Novalium Company

@skullpetrol :D

barlop

18:24

if an olympic weightlifter accelerates a large mass, even if it doesn't accelerate that much(not like a cricket's arm with no load), the force is large because the mass is large and accelerating, say speed is 50cm/s and acceleration is >=1 unit of distance per second squared. Now suppose the mass is suddenly reduced from 150kg to 80g. There will be a moment straight away where it'd accelerate . What'd the new acceleration be?

Physics Meta

0

Q: Disrespecting words, and behaviour,

Another user of the site continously, disrespecting by using ungrateful. Comments, he used to call me a serial down voter, and his previous comment on the main site too irritate me, he also warn me. As I said this site main ethos is that here each one is respectfully except this mistake. And im...

feature-request user-accounts

Novalium Company

18:52

Do you guys think AI will take over the world?

dmckee

@NovaliumCompany Let me construct a big-data/machine-learning model and I'll get back to you ... ... ... No. But you need to buy toothpaste on your way home.

Novalium Company

@dmckee Wut?

dmckee

The boundaries of "AI" are debatable (and they creep over time), but big-data is using things that used to be considered AI on you right now.

A lot of it look innoculous enough that you might invite it into your home. Things like helping with your shopping list and predicting where you'll want to drive in time to tell you if you need to leave early because of traffic.

Novalium Company

AI seems to do things pretty differently but it can't "think". It needs a human. And we need an AI. Let's merge AI with humans and boom.

sorry, not "differently", "better than humans".

donno

world is a weird place

ACuriousMind

How do you know it can't "think"?

Novalium Company

19:03

@ACuriousMind I love how you randomly pop up :D

@ACuriousMind plz let's not start a long long LONG conversation on this topic.

We don't know what thinking means so...

I believe the human brain is a closed-loop one where it constantly gets info from the outside world and makes connections. When we say we have free will and let's say think about something "we want" or do something "we want" it's basically a stimulus that triggered that set of patterns that lead to me doing that thing, thinking that I "chose" to do it.

ACuriousMind

This AI did a pretty convincing impression of a high schooler desperately trying to meet their word count when I asked it "Does AI think?" and ended with "I've argued that the very idea of intelligence is an illusion." Guess that settles that.

Novalium Company

I believe we are puppets of outside stimulus.

@ACuriousMind what's that website?

ACuriousMind

@NovaliumCompany It's basically a text box connected to the latest released GPT-2 model, a so-called transformer network trained on a large part of the internet to predict text. You give it a prompt and it gives you back what GPT-2 predicts comes after it.

Novalium Company

According to that model, a pretty popular name on the internet it scanned is William Puhl.

I typed "My name is"

How does it decide what to put after what I've typed? Does it output the most often found words or? The most popular page?

ACuriousMind

19:19

@NovaliumCompany There is some randomness involved in the specific choices, but basically it has learned how to "pay attention" to words that are important for the current context, which words are appropriate for which context and it can remember the context for long enough to form a few coherent sentences (but when it gets longer you'll note it tends to lose focus and coherence).

Novalium Company

@ACuriousMind oks cool

ACuriousMind

There's a longer but still mostly readable explanation here

It's great at writing fake news and surreal fan fiction

(Also, the model behind that site I linked is only the most powerful publicly released version, there still exists a better model OpenAI hasn't yet released)

Novalium Company

I'm still a bit confused by how the output relates to the input.

Oh nvm

I get it

Does it form the sentences or does it get the whole text from a web page?

If it forms sentences then it's not a very reliable source of information.

ACuriousMind

@NovaliumCompany Sometimes it repeats some text it read verbatim (usually if the number of things with that context was very low), but usually it creates "new" text

Novalium Company

I tried inputting some text from our discussion here and it spit out some jibber-jabber :D

ACuriousMind

19:29

AI weirdness is a funny blog that throws this sort of generative network at many different tasks

Novalium Company

@ACuriousMind So not a reliable source of info?

ACuriousMind

@NovaliumCompany Well no - but the amazing thing is that it can be a source of information at all! (E.g. if you ask it to explain some technical term, you'll usually get a response that actually sounds as if it's explaining it. There's no guarantee the explanation will be right, though it usually isn't totally wrong)

For example, I asked it "What is General Relativity?"

Novalium Company

ACuriousMind

I got:

The theory of General Relativity (Gravitational-Einstein Theory), discovered in 1927 by the American scientist Albert Einstein, is a general theory of gravity that describes all of the forces, forces of nature, and particles that interact with a mass.

As you might imagine, gravity is the reason why things have mass – it is the cause of gravity's existence. But how could something be so incredibly important as gravity, if it wasn't gravitation? In GTR, if you have a ball, for example, and you throw it in the air, and then let it fall down, what does it do next?

Novalium Company

@ACuriousMind How does it form that answer? I mean, we don't know? Just like we don't know the how the brain forms thoughts and stuff?

ACuriousMind

19:34

It understands that there should be a date of discovery in the introduction, but it gets the actual year wrong. It also knows GR is about gravity, but the rest of the details are a mess - so not very reliable, no

Novalium Company

@ACuriousMind How does it know to form the sentences? Sure, it scans the text from the web, but how does it analyze and understand the text to form sentences?

ACuriousMind

@NovaliumCompany Well, there is of course a detailed technical explanation of how transformers work. But in the end it's really little more than "it's learned what kind of character strings (=words) follow other kinds of character strings by looking at millions of internet sites"

Novalium Company

So it's more complicated than I think :P

It's like, with AI and QM, humans are entering an era where things work, but we don't understand how exactly.

ACuriousMind

For instance, it will often intersperse its fake news articles with "[Advertisement here]" in between paragraphs, because that's what paragraph breaks in news articles look like to a text-only reader

CR Drost

19:50

can I get feedback on the sort of phrasing that I put here: physics.meta.stackexchange.com/a/12485/73490 ? Any ways that others would want me to amend that sort of thing?

DanielSank

21:14

Hi, everybody.

DanielSank

21:43

::crickets::

Emilio Pisanty

23:04

@DanielSank hi Dan

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