« first day (2944 days earlier)      last day (2281 days later) » 

02:05
a quick question, I am trying to prove that tensor operators $$[V^{(1)}\otimes W^{(1)}]^{(1)}_{M}$$ are proportional to the three components of the vector cross product of V,W
Things are defined as follows
$$[V^{(1)}\otimes W^{(1)}]^{(K)}_{M} = \sum_{p,q} \langle 1,1; p,q| K,M\rangle V^{(1)}_{p} W^{(1)}_{p}$$

and $$V^{(1)}_{\pm 1} = \mp\frac{1}{\sqrt{2}}(V_x \pm iV_y)$$
$$V^{(1)}_{0} = V_z$$
I was able to show, $$[V^{(1)}\otimes W^{(1)}]^{(1)}_{0} = (i/\sqrt{2})(V \times W).\hat{z}$$
but, for $$[V^{(1)}\otimes W^{(1)}]^{(1)}_{-1} = (1/ \sqrt{2})V^{(1)}_{0} W^{(1)}_{-1} - (1/ \sqrt{2})V^{(1)}_{-1} W^{(1)}_{0} $$ i get terms that don't cancel out
I am using the fact that the clebsch-gordan coefficients in the sum are zero whenever p+q =/= M
as far as i know for each value of M which runs from -1,0,1 the tensor operator should be proportional to a component of the vector cross product for the the case of K = 1, M = -1, I get that it is a linear combination of 2 components of the vector cross product, I am pretty sure this incorrect, unless someone here knows otherwise perhaps?
 
3 hours later…
04:50
0
Q: Wedding and death related

ABHIJIT BAGCHIMy elder brother expired on 02-12-2017. All the rituals were properly performed. wedding of my niece is going to take place on 02-12-2018 exactly on the anniversary date of my elder brother's death..Both are related to my own father.Are the happenings of astronomical significance.Is/are these e...

Probably 5 out of 10 people confuse astronomy with astrology.
Anonymous
05:46
@SirCumference It might be on-topic on Hinduism SE
Anonymous
@SirCumference Yeah, that sucks :/
06:07
oho
 
1 hour later…
07:08
-2
Q: Should this really be put on hold?

Alpha_PiWhy are questions concerning the consistency of different parts of mainstream physics with one another considered as questions about non-mainstream physics and put on hold? I asked a question about how different parts of mainstream physics that to me seemed inconsistent with one another were con...

07:35
@JohnRennie in your question physics.stackexchange.com/questions/82678/… i believe you considered time dilation due to gravity. My question is the same but instead i am thinking about time dilation due to speeds close to c. Or maybe i understood wrong :)
@deadpool yes, I agree, your question is different.
But my concern is still that I'm not sure what you mean by the end of the universe, because I doubt the universe has an end.
@JohnRennie Why is my Fn + Brightness (F3) not working?
@JohnRennie Fn + F8 (Sound level) on the other hand, works properly.
No idea, sorry
Is the F3 key working?
Yes
And the Fn key is obviously working since it works with Fn+F8 ...
Remind me what model your laptop is. It's a Samsung isn't it?
07:53
@JohnRennie yes
Some Googling suggests the power settings can interfere with the manual screen brightness control ...
@JohnRennie what are power settings
08:15
@Abcd Open control Panel and click the Power Options link.
Then click Change Plan Settings, then Change Advanced Power Settings, and in the new windows scroll down and find the Display heading.
@JohnRennie found,then?
@Abcd I found some posts on the Internet saying these settings can interfere with the manual brightness control, but it didn't give any details. Is Enable adaptive brightness on? That would be an obvious suspect.
Turned it off right now
but still not working
What are the other settings? Can you expand them all and post a screengrab?
08:27
Ah, I meant the other display settings. I'm working on a PC not a laptop so my display settings are different. I wanted to see what the display settings were set to.
Hmm, they all look fine.
If you have now disabled adaptive brightness try restarting and see if that makes a difference.
08:44
hey dudes
@JohnRennie like a Big Freeze ? Or just like you phrased it -can he "see the universe age by an infinite time? "
it is what did this morning
Age by an infinite time
@deadpool the thing about falling into a black hole is that the time dilation becomes infinite at the event horizon, while in your example the time dilation is just a finite and constant factor. So the two situations are rather different.
For the guy falling into a black hole it takes only a finite time for an infinite time to pass outside the black hole.
In your case it still takes an infinite time for an infinite time to pass for the rest of the universe.
09:13
@JohnRennie at first i wanted to know what would happen if i travelled exactly at the speed of light. In that case the time dilation would become infinite correct?
@JohnRennie i know that is not possible but that thought experiment always made me wonder :)
@deadpool it's impossible to travel at the speed of light.
i know that is not possible but that thought experiment always made me wonder.
Can we look at it from a photon's point of view?
@JohnRennie
A photon has no point of view. It has no rest frame.
 
2 hours later…
11:26
Hey @DavidZ, how was my comment here obsolete? "This answer is misleading" doesn't become obsolete unless the answer gets fixed (which it hasn't).
 
2 hours later…
13:29
@JohnRennie @JohnRennie can we assume he can attain speed of light in this case? I am more interested in knowing your answer on what happens afterwards.
rob
rob
@deadpool It was supposedly a long-standing question about imagining the rest frame of a ray of light that led Einstein to develop relativity, in which such a perspective is impossible.
Think about how the electric and magnetic fields would look in the rest frame of a light ray: complicated and varying in space, but without any charges or currents to produce them.
The best response to your question really is that it doesn't make sense to imagine traveling at the speed of light. It is a non-trivial result.
14:15
@rob i understand :)
$\mui$
Does the equation $B = \mu_0 \mu_r H$ always hold? Even when B isnt proportional to H?
Ie is the non linear behaviour 'absorbed' by the $\mu_r$?
14:41
It can’t generically hold like that—what if the H and B fields point in different directions at a given point?
I was thinking about that too
$\mu_r$ becomes negative?
But im pretty sure its defined to be real positive
That won’t be enough. If $\vec{B}=\mu \vec{H}$, then the two fields are parallel at every point in space
Can they not be?
Is there a generilsation which takes all of this into account?
(What I said is true even if you allow $\mu$ to be itself a function of position)
There is, actually
You can generalize it by requiring that the components of B be linear functions of the components of H
Or, in other words: we should obtain $\vec{B}$ as a linear transformation of $\vec{H}$
Do you know how to effect such a linear transformation on a vector?
Not quite
@Semiclassical Can I aska tangential question
Im doing a formal report write up atm you see and Im a little confused
We'rre measuring B and H for different materials using a large solenoid to create an external H field which can be calculated, and then a smaller solenoid with a sample of the material inside to measure B
And then plotting these against eachother to produce a hysteresis loop
But why is the external H the same as the one inside of it?
14:57
Isn’t that how H-fields work? They’re generated by free currents, not bound currents
Oh yeah I see
So H wouldnt change in the material
Right.
It does make me wonder if there are materials which would produce a free current upon imposing an external H-field tho
For instance, an electron moving through a magnetic field gradient will be deflected due to its spin ie its intrinsic magnetic moment
So I could see that being relevant in a field like spintronics: en.m.wikipedia.org/wiki/Spintronics
I think I may be over-complicating it tho
15:21
though it is cool
Anybody know of a way f using latex in word without having to press equation every time?
For an operator $\hat{A}$, show that $\left(\hat{A}^\dagger\right)^\dagger = \hat{A}$
how do I prove such a thing?
The dagger represents Hermitian transpose
$\hat{A}^\dagger$ means that $\langle \alpha, \hat{A}^\dagger \beta \rangle = \langle \hat{A} \alpha, \beta \rangle$
So $$\langle \alpha, (\hat{A}^\dagger)^\dagger \beta \rangle = \langle \hat{A}^\dagger \alpha, \beta \rangle$$
$$\langle \hat{A}^\dagger \alpha, \beta \rangle = (\langle \beta, \hat{A}^\dagger \alpha \rangle)^*$$
$$(\langle \beta, \hat{A}^\dagger \alpha \rangle)^* = (\langle \hat{A} \beta, \alpha \rangle)^*$$
$$(\langle \hat{A} \beta, \alpha \rangle)^* = \langle \alpha, \hat{A} \beta \rangle$$
Badabing badaboum
Could you also do it via $(A^\dagger) = (A^*)^T$
thus $(A^\dagger)^\dagger = (((A^*)^T)^T)^* = A$
(I havent used the notation you used before, think I do it next term)
I wouldn't.
How come? Is it just stupid?
15:30
That's only really valid for finite dimensional Hilbert spaces
@JakeRose The question appears to be sufficiently silly that I'm not sure if I am even allowed to assume that transpose and conjugate are reversible.
and yeah I think I can't do that as there are infinite dimensions
@Yashas @Slereah Ahh I see
My linear algebra is still somewhat primitive
My linear algebra is non-existent
you rarely learn hilbert space stuff correctly in physics
Just for reference
@Slereah not useful?
15:31
Well you know
It's physicist math
True I suppose
It also seems that physicists mean to say $L^2$ space when they say Hilbert space (I don't understand neither though :P).
I think the maths for physicists course here is pretty competent
Not on par with mathmos
But probably more on par with mathematics elsewhere
In QM you almost exclusively use $L^2$, $\ell^2$ or a finite dimensional Hilbert space, so just $\mathbb{C}^n$
You don't really use any fancier Hilbert space
$B = \mu_0 \mu_r h$
oops sorry
15:51
@JakeRose transpose and conjugate are basis-dependent. @Slereah's proof hinges on a basis-independent definition of the hermitian conjugate, which is almost always just plain better.
$\det(e^\hat{A}) = e^{\text{tr}(\hat{A})}$
how do I start?
I have more math questions in my QM assignment than actual QM stuff .-.
@Yashas Express the exponential as a Taylor series
@Yashas in finite dimensions?
@EmilioPisanty infinite
if not, then you're in for a world of pain, and the result will require some fairly strong additional assumptions on $\hat A$ beyond "it's an operator"
16:00
Don't you need like compactness or something
or self-adjoint
I forget
the question also said $\hat{A}$ is diagonalizable which I think is implied by it being an operator
well
at the very least you need it to be trace-class
or the whole thing makes no sense
you also need a credible definition of determinant
@Yashas $\hat A$ is an operator $\Huge \implies\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!/ \ \ \ \ $ $\hat A$ is diagonalizable
what is this for? an intro-QM class?
@EmilioPisanty Yes
@Yashas it doesn't really work as an argument, but I guess it's reasonably clear what they expect you to say
I thought all measurements of an operator are its eigenvalues. Hence, the operator must be diagonalizable?
16:05
assume for the moment that this is in finite dimension
Actually I'm not really sure how much of finite dimension LA arguments are applicable in in infinite dimensions.
@Yashas they're not
but you won't be able to make headway on that question without assuming that
if you're in infinite dimensions, then basic linear-algebra statements about matrices like "the trace exists" are no longer guaranteed
Unless your problem text explicitly says that $\hat A$ is trace-class, then it's already playing fast and loose with rigour
and, by extension, it's expecting you to do the same.
Anyways, the linear-algebra perspective is to shift to a basis where $A$ is diagonal
and where the exponential $e^A$ is therefore just the exponentiated diagonal elements
and where the determinant is just the product of all the diagonal elements
... which are basically enough ingredients to reconstruct the identity you're after
Now, as you note, that argument doesn't hold in infinite dimensions, because both the determinant and the trace would require limiting procedures with (i) no guarantee that they converge and (ii) no guarantee that the result will be basis-independent.
However, unless you've been given strict rigorous definitions of the determinant, trace, and operator exponential, then it's pretty pointless to ask about rigour.
Just write $\det A = \prod_i A_i$
It should work fine
@Slereah you're missing a step
"ignore all convergence issues"
then it works fine
That's what you do when you do physics @EmilioPisanty
16:13
@Slereah indeed
Hell I'm assuming that eigenvalues exist at all
That's already insane!
this is why I asked
9 mins ago, by Emilio Pisanty
what is this for? an intro-QM class?
if it was for a mathematician's-perspective thing, then it'd be a whole 'nother ball game
$A = P^{-1}DP$

$\det({e^A}) = \det(e^D) = e^{\lambda_1 + \ldots + \lambda_n}$
how much of scam is this?
nvm
I am probably going to ask for a clarification for this quesiton from my TA.
Consider a one-dimensional bound particle. Show that, if a particle is in a stationary state at a given time, then it will always remain in a stationary state.
is that even true?
If I find the particle to be in a stationary state at a point $t$, then if I leave the system on its own, it would again get jumbled and my next measurement could give some other E?
Anonymous
16:31
@Yashas It's fine and rigorous. Sylvester's formula. But oh, wait. Is $D$ diagonal?
In Millikan's oil drop experiment, why did he choose to measure the velocity upwards in a strong field rather than just capturing an oil droplet and measuring the field strength required to keep it stationary, in his paper he even talks about how stokes law breaks down for spheres with small radiuses.

Does the method of measuring field strength for stationary droplet and terminal velocity without any E field not result in a smaller error?
By eliminating the slip factor needed in Stokes law for the upward motion at least.
@Blue Yes.
Anonymous
@Yashas If $D$ is diagonal, then it's perfectly fine.
Anonymous
$A=P^{-1}DP\implies e^A=P^{-1}e^DP$
I don't think that formula is allowed for infinite dimensions.
Anonymous
16:35
Are you studying functional analysis?
0
Q: What is the line between fiction and physics with respect to off-topic

Kurt FitznerThere is a question I believe to be well on topic, relevant, and interesting. It is asking to consider a fictional scenario and evaluate, as given in the fictional scenario, whether it is possible to achieve with classical physics. It is, I believe, people who are misapplying the prohibition ag...

Anonymous
@Yashas For infinite dimensional operators, there are a lot of other things to be taken into consideration. You'd have to precisely define the type of operators you're dealing with. I don't think just calling $D$ a diagonal operator makes sense
Anonymous
Are you sure you're being taught infinite dimensional QM with operations like these? Then you'd probably have to direct those questions to ACuriousMind
@Yashas it's a QM class. "allowed" is not a very useful word there.
the formula is certainly not allowed for general operators in infinite dimensions
but it is also valid for nice-enough operators in infinite dimensions
if it's a functional-analysis class, the viewport is to start with a rigorous foundation, define your budget in terms of how nice of an operator class you're dealing with, and then work up to see how much you managed to buy, i.e. how strong of a result you managed to prove.
if it's a QM class, you need to ditch that debit-based system and take a credit-based approach
a.k.a. asking for forgiveness instead of asking for permission
3
Anonymous
16:45
@EmilioPisanty lol :P
assume that your operator is nice enough, and prove the result that you need. Then you go to the cashier mathematician and you ask how much you owe what "nice enough" needs to mean in order for your result to be true
Or you forget about it until you later realize that interest can hurt
Howdy
How's life
whomp
@danielunderwood that's what happens when you try do solve particle on a ring
16:58
Don't suppose anyone knows how to concatenate Word files with heavy amounts of equations? Wrote several proofs and am trying to put the correct ones together to hand in, but Word crashes when I copy certain equations
@SirCumference color me surprised
@SirCumference Use links rather than trying to import everything into a single document.
@EmilioPisanty Yep I know...this is like the 15th time I've come here about word
@Slereah ooooooohhhh, indeed it does
Perhaps I ought to look at other software
16:59
@SirCumference I don't think I can recall you talking about word
but Word crashing
@JohnRennie Links?
dunno how to do work post-thanksgiving weekend
feels sluggish
@EmilioPisanty I'm always a bit amazed at exactly how awful that gets
17:02
@JohnRennie Hmm, well these are being handed in on paper
@Semiclassical yeah, I know
@danielunderwood I planned on working on the RL library over the long weekend, but I did not lol
Oh wait, it displays the actual text
I showed you the slides for that lecture, right?
watched 2 movies, played CSGO and Overwatch...ate a lot...
17:02
ya
@SirCumference IIRC when you go to print it Word correctly pulls in all the links and prints it as if it were a single huge document.
it was pretty fun watching the mounting disbelief XP
bought a new laptop for black friday...
Anonymous
@EmilioPisanty Just curious - did you ever have to take a functional analysis class with a focus on QM? Or a pure functional analysis class (like, from a math student's perspective)?
17:03
@Blue nope
never had to
never did
@enumaris New laptop!!! :-)))
never did take any formal functional analysis
yeah, it's being shipped to me
will arrive on Wednesday supposedly :D
would've liked to in undergrad, but I only made it through Analysis I and II, and functional analysis was on III
Anonymous
Ah. But you do seem to know a fair bit of it (that's the impression I get from your PSE answers), as far as QM is concerned. Just picked them over time? :)
17:04
@enumaris details?
would've liked to audit mathematician's FA on my first year of PhD but that just turned out to be unrealistic
@JohnRennie It's a gaming laptop for me to waste more time on lol. Razer 15 incher
and also I felt like a grampa next to all them undergrads
@Blue yep
Anonymous
@EmilioPisanty Hehe :P
@JohnRennie Neat, thanks!
17:05
but my formal FA is pretty limited
pretty good specs, but nothing super amazing. Not as powerful as my 1 year old desktop, but should be good enough to play the games I want on :D
one bit of mathematica notation which I can never seem to remember: picking out specific elements from an array
@enumaris yeah I played a bit of Overwatch too. I started the RNN course which has been pretty interesting so far. Though I've gotten sidetracked trying to write a music generator a bit better than the homework assignment
also, Razer makes an external GPU holder thingy...so if the GPU ever gets outdated, I can upgrade to an external one
Take almost works, but not quite
17:05
@SirCumference I am the Ubernerd!
4
@Semiclassical list[[i]]?
@danielunderwood I only have like 5 hours on overwatch so far, I just sort of picked it up...trying to get better at the snipers
that's fine for one element
widow and ana
@enumaris through a Thunderbolt port?
17:06
@JohnRennie yep
@Semiclassical what do you need, exactly?
i guess i can just do {list[[i]],list[[j]],list[[k]]}
Trying to pick out the unique matrix elements from a triangular matrix
@Semiclassical if that's what you're doing, I would write list[[ {i,j,k} ]] instead
@JohnRennie I used to have a Sony that had an external GPU and that worked out pretty well so I'm thinking this one should be fine if I ever need it :P
Anonymous
17:07
@EmilioPisanty Oh, btw Analysis I and II were like Rudin type - baby Rudin and papa Rudin?
or, if they're consecutive, list[[ i;;k ]]
@EmilioPisanty For real tho, I'm noticing Word can't handle equation heavy documents. Maybe it's time to go to LaTeX
ah, right, it's Part
@Blue not quite. Analysis I was papa rudin, I think, and II was measure theory
@SirCumference yes, yes it is.
@JohnRennie this one is gonna run Windows 10 so if I ever brick it (e.g. by trying to do deep learning on it or something) I'll come to you for tech support XD (just don't tell Bernardo)
Anonymous
17:08
@EmilioPisanty Gotcha. Thanks :)
@enumaris has it got an SSD?
yeah 512gb SSD
Only thing I'm hesitant about is the lack of quick keyboard commands for equation editing, e.g. on Word, alt+V will do a square root (on Mac) rather than needing to type \sqrt{}
(Bernardo will install some weird variant of Linux on it :-)
which then makes a nice solid base for FA on III and god-knows-what on IV
never saw what's in there, though
17:09
lack of end parenthesis...*triggered*
But :-)) just looks like I have a double chin!
@Semiclassical yeah, if you know which elements you want (by index), then it's always Part
yeah. my brain just never remembers the word
otherwise you're looking at some combination of Position and/or Select and/or other dubious ingredients
@Semiclassical I've never seen Part laid out explicitly in code and thought "yeah, that syntax is clearer than [[ ... ]]"
17:11
@EmilioPisanty Hey btw
what do I do if I want to try to apply to the PhD you're offering
do I send a mail to the other guy
of course, since I"m dealing with a symmetric matrix, I probably could've just tossed in a DeleteDuplicates call after Flatten-ing it
whoop, I got the names mixed up and thought semiclassical was asking about applying to a PhD...was like...yo bro you just finished one...
@Slereah I'm not sure I could fairly describe it as us offering a PhD project. We don't currently have funding specifically assigned to it. But we're certainly interested in students.
There's central ICFO funding which would require you to make an application to the main pool
@EmilioPisanty Well you know
Who has funding really
The only one with financing are the boring ones
On like nanophysics or solid states
From BIG MONEY
and there it's always a plus if you have a specific project in mind already talked-about with a potential supervisor
17:15
I don't have a specific project in mind
I mean I have ideas of things to do in physics, y'know
9 pages of proofs. Imagine a world where math courses satisfied Writing Intensive requirements
@Slereah fully develop Einstein-Cartan theory?
@Slereah so basically yes, write to Lewenstein
But they're just stupid ideas that will not be very interesting to financing people I would guess
Alright physics nerds: is Oumuamua an Alien spacecraft or not?
17:16
@enumaris I think Einstein and Cartan did it first
@Slereah quantize it :P
@enumaris I think that has been done too!
@enumaris Yes
@Slereah so is it a quantum theory of gravity?
Sure, there's tons of quantum theories of gravity
Including a bunch where the connection isn't symmetric
17:18
@Slereah It depends how much you care about doing something specific. I'm looking for students for the quantum-optical HHG project I linked earlier, and for extensions to the work in arxiv.org/abs/1808.05193
there's obviously other people doing interesting things in the Lewenstein group
Oh @EmilioPisanty, I think you mentioned reading Spivak's Calc on Manifolds once right?
the more specific you can get when writing, the more likely it is that Lewenstein will be interested
Well you know
I'm not well versed in quantum optics
So obviously I don't really have a project in mind
@Slereah if it's for a master's project it's better if you have more background, because there isn't a lot of time to get up to speed
on a PhD it matters more that you're able to learn quickly
u know me I'm great
17:22
@Slereah that definitely counts.
any good recommendation on a quantum optics book btw?
Walls and Millburn is the one I usually plan on buying
Gerry & Knight
also Scully & Zubairy
that project is a bit demanding in that it melds together two parts of optics (quantum optics & strong-field physics) which haven't spoken to each other very much over the past 25 years
which means that you need to get up to speed on two independent subfields
but
@SirCumference well then...
it also means that you're one of the first working on non-perturbative non-linear quantum optics
@enumaris ?
17:25
that's pretty cool
I mean, as of now there are three papers on the subject, basically.
non-perturbative stuff in general is something I'd like to see more of, though of course that's an overly broad classification
@EmilioPisanty Do I have to solve QED
@Semiclassical well, any field that defines itself as non-X is always going to be pretty broad
@Slereah no.
17:27
Phew
Might have been tricky
Is it gonna be
The solitons
if it's nonlinear, it's probably related to solitons somehow
@Slereah not particularly, no.
@Semiclassical solitons are quadratic nonlinearities
we're looking at (what would count as) nonlinear orders in the tens or hundreds
17:28
though again, the concept of nonlinear order is pretty meaningless since perturbation theory breaks
@Semiclassical well, how else do you get a 200th harmonic?
when we say high we mean high
it reminds me of something out of the book Watership Down, where the counting system for rabbits goes as: one, two, three, four, hrair (i.e. "many")
is that even measurable?
@Semiclassical u embleer frith
lol
good book. haven't read it in a while
17:31
Movie's pretty good too
also Netflix is making a reboot
this is the most I've seen of watership down
@Slereah hmmmmmmm
Have the intro then
subtitled in spanish, coincidentally
The intro is very cute
and then it's 1 hour of rabbit holocaust
The horror
17:35
@SirCumference how long until the creators of Oumuamua make contact with us?
@enumaris I'm no visionary so I can't be certain. But perhaps some astrology will help us find out
who says they haven't already?
hmmm
so hard to find a timeline where Tensorflow and pytorch will support cuda 10...
looks like Tensorflow 2.0 will support it
not sure when that will drop...there has apparently been no talk in the pytorch community about an official version that supports it
yuck
17:55
you could always implement it yourself...sounds fun doesn't it?!
Does intern mean something different than it used to? I'm seeing internships that are looking for people with previous industry work or an advanced degree
well I can build them from source
probably just have to modify a couple lines of code to include "cuda10.0" flag instead of "cuda9.0" flag...but
that's a giant pain and I'm not really gonna get into it
@danielunderwood It means "slave".
Nothing wrong with looking for slaves with experience :P
@danielunderwood gotta love them unpaid internships that require 5 years experience and SME level knowledge.
@enumaris think back to that when your compiler explodes from errors
At least grad school has stipends lol
17:58
can I ask a question here?
@danielunderwood I'd rather not, thx
(So more like “indentured servant” maybe)
@WantingtobeanAndroidDevelor see top right
my grad school experience was pretty chill tbh
I never had to write grant proposals
17:59
Well most of them that I've seen seem to be paid at least. But I have seen some that did indeed look like they wanted slaves
I met my adviser like once a week for my entire time there
so Im thinking of writing a program that simulate chemical bond from quantum mechanics view

« first day (2944 days earlier)      last day (2281 days later) »