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12:40 AM
Definition III in Newton's Principia:
> Materiae vis insita est potentia resistendi, qua corpus unumquodque, quantum in se est, perseverat in statu suo vel quiescendi vel movendi uniformiter in directum.
Is this "materiae vis insita" really a force? Does this suggest that Newton believed that it is a force?
> Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.
here in his First Law he uses "viribus impressis" (nom. sg.: vis impressus)
It's like the inherent force of matter
> Lex. II. Mutationem motus proportionalem esse vi motrici impressæ, & fieri secundum lineam rectam qua vis illa imprimitur.
The inherent potential to resist
and in his Second Law he uses "vi motrici impressae" (nom. sg.: vis motrix impressus)
@bolbteppa but it isn't a force, in the "modern" view
Not sure what that means
12:49 AM
I would say that Newton's "vis impressus" is the modern "force"
The first law is what happens when there is no force, the 2nd when there is a force, pretty sure it's wrong to try to use the 2nd law as a definition of force in any sense, if that's what you mean
then what is a force?
I thought nobody thinks that inertia is a force nowadays
at least I have never heard it referred to as a force
@LeakyNun My perspective is, a force is something that causes a change in an object's velocity. Although this is dependent on the reference frame, coordinate system, etc., which determine whether a change in velocity is observed
My point is, nobody thinks that inertia is a force, right
whatever definition of force employed
If a force acts on a body, that body will exert an equal and opposite force on the thing causing the force right
The 'inertia' is causing that force
12:59 AM
Well I think I have been told that this is not true in QM or GRT or something
Say I'm swinging a yoyo very fast. The ball breaks off. In my inertial frame, I can explain it perfectly as inertia, i.e. the ball's tendency to maintain its direction. But from the ball's frame, my trajectory suddenly spun out of control the moment the string snapped.
The ball couldn't explain this as "maintaining velocity" from its own measurements. In the ball's noninertial frame, it'd say there was a centrifugal force acting, whereas I'd say there was no force acting. We wouldn't agree on the forces.
Once you bring in quantum mechanics, even the idea of 4 dimensions might not be true :p
"In modern particle physics, forces and the acceleration of particles are explained as a mathematical by-product of exchange of momentum-carrying gauge bosons."
In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass to change its velocity (which includes to begin moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F. The original form of Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass...
Some of this is simply madness if taken seriously
Momentum exists but the idea of force and acceleration only arise as a by-product
Perhaps a more useful question would be: what did Newton think "vis" is?
What is that latin for
that's exactly the point
the common translation is "force"
1:10 AM
the Principia begins with a list of definitions before the axioms
Definition III talks of "materiae vis insita" (material force inherent)
@LeakyNun Welp not sure what your context was, I was just responding to this
Def. IV talks of "vis impressa" (force impressed)
Def. V-VIII talk of "vis centripeta" (force centripetal)
1:12 AM
'Definition III.

The Vis Insita, or Innate Force of Matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forwards in a right line.'
and in Law I he uses "vis impressa" and in Law II "vis motrix impressa"
and I as a modern reader would agree that "vis impressa" & "vis centripeta" & "vis motrix impressa" are really forces, but that "materiae vis insita" is questionable
Looks like definition 1 is defining mass, definition 2 is defining momentum (moving mass), definition 3 is defining inertia which is like the force a body would exert if interacting with another body, definition 4 is defining the idea of a force acting on a body to change it's motion (impressed force), a thing you couldn't define without having defined a body to carry with it the idea of a force in the first place, if objects are not defined to carry a force (inertia) how can forces even act
Inertia is like 'the empty set', and impressed force is like a set containing the empty set runs
user image
1:30 AM
@ACuriousMind does that mean that kind of manifold can't accommodate fermions? Or we can introduce other objects than spinors to describe fermions?
2:29 AM
This question has been on my mind for a while. Is quantum field theory an approximation, or a special case of quantum mechanics?
Before I started physics I used to think that QFT was the more general or more correct one, simply because I had heard it was harder. However, now I am not sure if this is the case.
I haven't studied quantum field theory systematically.
5 hours later…
7:31 AM
@JohnRennie hello
@user8718165 hi
For how long will you be here today?
@user8718165 I'm around for several hours more
Sup humans.
Quick question. Do electromagnetic waves curve along the Earth as they travel. (Are em waves affected by gravity?)
@JohnRennie Does this video make sense? youtube.com/watch?v=tU1UKn7iXv8
7:49 AM
@user8718165 the video doesn't explain exactly what measurement is being done. For example it doesn't say whether the current is travelling with or against the flow. But it is measuring some current through the flowing water, which is what I would expect.
@JohnRennie I don't know but please correct me. I think the current will pass through the water from the fence, through the body of the person and then try to go to the ground through the feet if the fence is at higher voltage...so this makes the current flow upstream.
And the person will be shocked
Possibly, though as I say it isn't clear how the circuit is completed.
8:13 AM
@JohnRennie In yesterday's problem we discussed that at one point cl- will be electrolysed to cl while at the other contact point na+ will be electrolysed to na.Though the water stream is neutral after the water leaves this point, will we get the original na+ and cl- ions?
Guys, what if only the wings on an airplane move?
You can add images by simply pretending the link with a !
@NovaliumCompany Like a bird?
@AvnishKabaj When the airplane is making a turn, instead of the whole airplane to do the rotation, only the wings will do?
@NovaliumCompany Should work
But engineering that would be
8:21 AM
Well, yep.
@user8718165 what actually happens is that we get hydrogen and oxygen produced at the electrodes. That's because if neutral Cl atoms were produced they would immediately oxidise the water to oxygen and $H^+$. Likewise at the other electrode sodium atoms would reduce water to hydrogen and $OH^-$.
So in fact the NaCl is unaffected.
9:21 AM
@JohnRennie Thanks...got it
@JohnRennie I've a last query about this topic. Yesterday I didn't understand it. Are you free now?
@user8718165 yes
Sir, we've discussed that the video I linked didn't make much sense as it was not shown how the circuit was completed. But suppose if the person was standing bare feet and the fence was hot...would he receive the shock?
I mean if he completed the circuit through his body and feet.
Suppose the fence was at +6000V relative to ground. Then the current would flow from the fence, through the urine, through the person and then to ground. The shock the person would experience would depend on the resistance of the person and the resistances of the urine stream and the ground.
9:49 AM
@JohnRennie are you here?
@LeakyNun hi
Are you familiar with the Aristotelian notion of mover?
The unmoved mover (Ancient Greek: ὃ οὐ κινούμενον κινεῖ, translit. ho ou kinoúmenon kineî, lit. 'that which moves without being moved') or prime mover (Latin: primum movens) is a concept advanced by Aristotle as a primary cause (or first uncaused cause) or "mover" of all the motion in the universe. As is implicit in the name, the "unmoved mover" moves other things, but is not itself moved by any prior action. In Book 12 (Greek: Λ) of his Metaphysics, Aristotle describes the unmoved mover as being perfectly beautiful, indivisible, and contemplating only the perfect contemplation: self-contemplation...
Aristotle said that projectile motions happen because the medium (air) moves the particle after being released by us
OK ... ?
9:52 AM
but in particular I'm interested in what Newton meant when he said "materiae vis insita" in Definition III
did he believe that inertia is a "force"?
is this an Aristotelian influence?
Don't know sorry.
oh ok thanks
@JohnRennie I got it but could you please tell me how does the electricity pass upstream through the water steam into the ground through the person . Won't the water just be electrolysed at the point where the stream touches the hot fence and the current won't flow?I'm not getting this thing.
10:18 AM
@JohnRennie Hello
I'm not sure it's helpful to start with the person peeing on a fence because there will be different conduction machanisms in the wires, urine, the person and the ground. It would be better to consider a simple setup like a battery with two wires dipping into a flowing stream of salt solution.
@JohnRennie yes sir ok
What will happen there
@JohnRennie I just gave an example. This is much better
Something like this.
I've drawn the direction the electrons move through the battery.
At A H2O is oxidised to O2 + e, and the electrons flow out of the water into the wire. This leaves H+ ions in the water.
At B electrons flow into the water where they reduce H2O to H2 and leave OH- in the water.
10:33 AM
@JohnRennie Sir thanks a lot, I got this diagram. The circuit is completed just by electrolysis at points A and B so there is no need for the current to flow thro' the water...correct?
@user8718165 well, consider the situation when the flow velocity is zero. If the H+ and OH- ions didn't flow through the water then we'd get a buildup of H+ at A and a buildup of OH- at B. This buildup would eventually stop the electrolysis.
What happens is the H+ ions generated at A flow downwards towards B and the OH- ions generated at B flow upwards towards A. This flow stops the charge buildup so we get a continuous current. In the wire that current is carried by electrons while in the water it is carried by H+ and OH- ions.
@JohnRennie ok sir...fully got this. So the flow velocity won't interfere with the direction of ion flow in the water. Is it so?
@user8718165 the flow velocity will affect the ion motion, but the H+ and OH- ions flow in different directions. So the flow will speed up one ion and slow down the other. Regardless of the durection of flow we always get some current flowing through the water.
@JohnRennie so in this case if the voltage is high enough there will be some current to shock the person right? I think I got it now:-)
@user8718165 yes. The current depends on the voltage and the conductivity of the water. The conductivity of the water depends on the salt concentration.
10:46 AM
Thanks a lot sir
Good bye:-)
what does "The eight gluons are identical except for their colors" mean? Aren't there only 3 fundamental colors, red, blue, green? But there are 8 gluons!
Gluons carry two colour charges
A gluon () is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. In layman's terms, they "glue" quarks together, forming hadrons such as protons and neutrons. In technical terms, gluons are vector gauge bosons that mediate strong interactions of quarks in quantum chromodynamics (QCD). Gluons themselves carry the color charge of the strong interaction. This is unlike the photon, which mediates the electromagnetic interaction but lacks...
@JohnRennie But $C^3_2$=3
@CaptainBohemian read the article I've linked.
11:13 AM
@user400188 Yes, a QFT is just a big ol' QM system.
The undergrad QM you know is largely just the stuff people figured out before hitting on QFT, which is why it seems a little easier, but they're not different subjects.
1 hour later…
12:17 PM
That doesn't quite answer the question though. Is QFT a more general version of QM, or is QM a more general version of QFT? Are there any approximations in QFT that do not appear in QM?
@user400188 it depends what you mean by QM.
If you mean the basic axioms then they apply to QFT as well as to non-relativistic QM.
QFT and non-relativistic QM are different because they quantise different things. QFT quantises fields while non-relativistic QM quantises particles. But they are both based upon the same principles. It isn't the case that one is a subset/superset of the other, but rather starting from the same basic principles they are two different forks.
I think QFT is used to describe fields while QM is used to describe particle dynamics. In classical mechanics, there is also such distinction.
I have seen books whose title is something involving "classical field theory", which should be the classical counterpart of quantum field theory.
12:37 PM
@JohnRennie That's true when the NaCl concentration is low. But at high concentrations you get chlorine at the anode, and you get hardly any oxygen if the voltage is low. If you do the electrolysis with concentrated brine in 2 half-cells, you get NaOH solution on the cathode side, otherwise you get sodium hypochlorite. Or if you use a mercury cathode, you get sodium amalgam.
@CaptainBohemian yeah I believe QFT is just the quantization of those classical fields. Or at least that's where it seems to start as I haven't done much in detail. Amusingly, my understanding is that you can chain together QHOs to get a quantum field as an alternative to chaining together classical harmonic oscillators to get a classical field then quantizing it
what to do when the user reverts edits and commits the same grammatical mistakes and deformats mathJAX
A: Difference between electrical potential and potential difference

user8718165You are mixing the properties of a point charge and that of a capacitor. The formula you've written works perfectly for point charges and intuitively for two entities holding the same nature of charge and to be very precise, you should use $r$ instead of $d$ when dealing with point charges as th...

@danielunderwood Does your "chain" mean to make a great number of oscillators connect together to form a chain?
@user8718165 You can roll back their reversion. Or come here & explain the situation so someone with edit privileges can do it. If a post turns into an "edit war", after a few rollbacks, it will automatically raise a moderator flag, and the mods can lock it to prevent further edits.
@user8718165 Have you tried asking them in a comment why they rejected it? (They rejected, not reverted it, by the way)
12:46 PM
Yeah you take the limit as the number of oscillators goes to infinity and their separation goes to zero. I'm feeling less certain about the case of putting QHOs together like that working out now that I think about it
@CaptainBohemian sorry that was in response to this
No, I didn't. However user Qmechanic edited the post again. This time it should be fine:-)
@danielunderwood For historical reasons, we have the slightly confusing terminology: second quantization.
@user8718165 Qmechanic's edit did not inlcude your improvements. And now that I look at the post history, it seems they indeed reverted your edit but for some reason the system logs this as a rejection of the original suggested edit
@danielunderwood I have actually read a lot about classical field theories but haven't read much about quantization techniques. But I read in a book which claims these fields have been counted as quantum fields while a lot of books only call the fields under second quantization of them as quantum fields, which the book deems as improper.
@ACuriousMind Okay...that's fine. Anyways, thanks for enlightening me:-)
12:57 PM
I have only known the quantization techniques which have been taught in nonrelativistic quantum mechanics, I don't know much about the quantization techniques in quantum field theories.
1:17 PM
Well canonical quantization is pretty simple. I think QFT often goes the path integral approach, which seems to be quite difficult to pick up (at least for me). There's also geometric quantization, which...exists?
@user400188 quantum field theory is relativistic quantum mechanics in the occupation number formalism where one works with the quantum field operators that arise naturally out of the occupation number formalism. There is a more direct way to get to quantum field operators without going through the occupation number formalism so it might not be so obvious how it's linked to QM, but you can also go through normal QM to get to it
@bolbteppa One should caution that the fields arising from the particle mechanics is "the Weinberg viewpoint", while it is equally valid to view "quantization of classical field theory" as the starting point and then get the particles from the mode expansions of the field. The formalism doesn't really care whether we take fields or particles as fundamental.
The more direct way being quantizing classical fields by replacing Poisson brackets with commutators, but one has to ask where does this idea come from, it comes from the same place you can start from to get to the occupation number formalism, the uncertainty principle...
@user8718165 : I have now reinstated your edits.
Are you afraid of technology? ^
1:23 PM
@Qmechanic Wow!Thanks a lot
'The next step in the construction of LQG is to decide what the dynamics are. Technically, this is done either (A) by choosing a "Hamiltonian constraint" in parallel with the Hamiltonian formulation of GR, or (B) in the spin-foam formalism, by postulating some sort of sum over histories assigning an action to each spin foam. It is here which we encounter the major problem: There is no agreement over how to implement the dynamics! There are many ideas, but no consensus on what to do. ...
I would say that LQG really doesn't exist yet as a well-defined theory. Unless you consider dynamics to be an unimportant part of a theory. And finding sensible dynamics is a really hard problem, perhaps impossible. ...
Yet, despite the lack of dynamics, there's no end of papers where people do specific applications, like count black hole entropy, or even attempt to do quantum cosmology (basically by truncating the theory to a finite number of degrees of freedom, and then quantizing those degrees of freedom in a way which is "loopy" in spirit). But all of these things are totally provisional until one can embed them in an actual theory with dynamics. '
'One can put too much emphasis on quantizing gravity---really that's backwards, we need the classical theory to emerge from the quantum theory, not vice versa. When people calculate discrete area and volume spectra for spin network edges and vertices, they've got things backwards.'
I wonder how their approach relates to e.g. what Feynman tried to do
2:03 PM
@knzhou what did the exceptional high school students think about that mechanics question of a rope being pulled?
also i too think that conservation of energy applies in these kind of cases.For example, have a look at the 3rd question over here:
this is the answer pdf:
2:23 PM
@TheEastWind They're still doing it! It's buried inside a really long homework assignment. Give them a week.
It depends on how you choose to model the chain.
I've seen the exact same problem but assuming energy isn't conserved.
@TheEastWind I'm also interested in using InPhO questions for training the students. Is there a place I can get all of them, translated into English?
@user400188 QFT is a special case of QM. The axioms of QM are all assumed in QFT.
2:39 PM
Is this video really an example for HUP? I think this phenomenon happens because of diffraction. youtube.com/watch?v=a8FTr2qMutA
2:51 PM
@knzhou There is a bit of a linguistic confusion where people use "QM" both for the axiomatic system encompassing QFT and "0-dimensional QFT" aka "ordinary QM" and for "ordinary QM" excluding any true QFT
@ACuriousMind True!
I'm just being semi-prescriptive here. I think the word QM should be used so that QFT is a special case of it, because the first jump to QFT is one of the most jarring things when learning physics.
Anything that makes it feel more continuous would be better.
I agree
3:13 PM
@ACuriousMind IS HUP a good explanation for this experiment by Derek youtube.com/watch?v=a8FTr2qMutA
not going to watch a video, sorry
3:34 PM
@user8718165 sort of. What the video shows is classical diffraction of light waves rather than an effect due to quantum mechanics. However the maths behind the broadening of the spot is basically the same as in quantum mechanics.
@JohnRennie Oh so Derek just said about that because the math is similar...got it
@user8718165 you really need to understand Fourier transformations to understand the HUP.
These things should be very intimidating...right?
@user8718165 most things seem simple once you've learned them. To me Fourier transforms seem easy peasy because I learned them about forty years ago and have used them regularly ever since. I probably found them scary at first, but that was so long ago I've forgotten what it was like.
Even things like General Relativity seem easy once you've learned them. Though I'm unconvinced that anyone finds QFT easy even after they've learned it :-)
It will take me a long time just to get these QM stuff well along with the math behind it. Mastering anything requires dedication, intuition and practice :-)
@JohnRennie What does one study in QFT? Is it totally different from QM?
3:44 PM
@user8718165 yes
Wow I got a question
Q: Interesting relationship between diffraction and Heisenberg's Uncertainty Principle?

hb20007I recently came across an interesting explanation of diffraction through an aperture which does not use Huygens' Construction but instead relies on Heisenberg's Uncertainty Principle: The Uncertainty Principle states that trying to pin a particle down to a definite position will create uncert...

4:09 PM
@knzhou all of them already are in english thus there is no need for translation.
here they are:
these are till past year. this year's paper i already attached.
@knzhou Where did you see this question(one end of the rope falling under gravity)? I used to think that all Inpho questions are original.
4:32 PM
@TheEastWind These are useful, thanks!
you're welcome
Are these the whole thing? I see in some of the earlier ones there is only multiple choice. Are there any rounds of competition after this?
As for the rope, it's actually a standard type of question. I've seen it in several contexts.
Well you see there are 3 levels actually.
Morin's book Classical Mechanics has an entire section devoted to questions of this type.
First is the nsep. That is too easy to bother with
Then comes inpho, which i linked
4:33 PM
He discusses the falling rope in detail in section 5.8.
then like every country we have a camp. I'm afraid the practice assignments of the camp are not publically available
Got it, thanks!
What about all those other things I hear about?
Does IIT JEE or whatever it is have good physics questions?
@knzhou yup, though not as tough as inpho
these 7 questions are meant to be done in 3 hrs
BTW, back when I helped coach the US team for the IPhO the year it was in India, I was told to give them very long calculations with a very short time limit -- apparently that was how INPhO-style questions were supposed to be.
this year the topper scored 74 out of 75 lol
4:35 PM
It looks about equal in difficulty to the USAPhO, but with more emphasis on speed.
@knzhou Oh. I never seen the USAPho to be honest.
Btw, these students you teach, is this a special kind of coaching or something?
The details about that are on my website.
It's special coaching, I do it so I have enough money to eat.
Basically I just give people a list of 30 hard (IPhO level) problems per week and tell them to solve them.
@knzhou Oh that seems interesting.
That's an impressive CV
They're taking the USAPhO next week, so we'll see how well this "learning by problem solving" turned out...
MIT Cambridge Oxford Stanford thats awesome
4:42 PM
Oh yeah, the grand tour and all...
I'm sure you'll accumulate a CV too, once you also get to the ripe old age of 23.
You haven't even started college yet, right?
I will this year though
Also curious, what's the most common book for people aiming to be training camp level in India?
Is it Irodov or is that wrong?
Good luck then!
Almost everyone does Irodov. After that they practice Krotov and David Morin
@knzhou Thanks
@knzhou Though in all honesty i really am not the best person to ask about this type of stuff. I am focusing more on IITJEE than olympiad physics
Cool! I have a copy of Krotov somewhere.
Was just wondering if there were books I hadn't heard about.
Oh man, good luck on that!
All of them go to one or the other coaching, and coaching questions are sometimes tougher than IITJEE.
4:46 PM
I went to grad school with a string theorist who did great on it.
I don't think I personally could have enjoyed such a system though.
Anyway, I have get back to work, but it was nice talking!
No one enjoys it though
except a rare few
Like one time we had a test of 100 marks only on kinematics. MAx marks were 30 or something
All right then. Catch you later!
@TheEastWind I don't know QFT. I know only just enough about it to have an idea how it's constructed. But compared to GR QFT seems a mass of details and special cases. I know lots of people love GR for it's simplicity and elegance. I'm just not sure QFT is loved even by it's most skilled practitioners :-)
(The HNQ is to blame :-)
5:27 PM
In reference to physics.stackexchange.com/questions/468998/… I bet it'd be hard getting the necessary experiments past an ethics committee... so we may never know how much gravity can a grizzly bear. ;)
How much gravity can a bear bear if only barely?
What's the velocity of an unladen grizzly bear?
If you shaved the bear then the question would be how gravity can a bare bear bear?
@ACuriousMind African or European bear?
5:52 PM
will someone tell me the name of the proof about levitation of stationary magnets?
also, does it have any constraint about small or big magnets?
and, what if we put very large plate magnets on top of each other, facing same face
will it flip? i feel like if tries to, when it bends a little up, it escapes magnetic field and gravity dominates
maybe we make them bowl like
6:17 PM
plz just tell me the name
@PranshuKhandal Earnshaw's theorem?
@knzhou Yeah, Irodov is the big one. Gnädig's text was also popular during my time.
@Blue thank you sir
6:40 PM
@Blue That book humbled me more than anything else, back in high school.
And now Gnadig and co. came out with a sequel which is even harder.
7:05 PM
@PranshuKhandal Nope. It's not going to happen. Many people have tried, both before & since Earnshaw's theorem, but all have failed. You may enjoy this site coolmagnetman.com/magindex.htm which has tons of amateur-level info on magnets, & heaps of magnetic gizmos you can make or buy. It even has a few sensor & control circuits for electromagnetic levitation.
@PM2Ring that's true all have failed, but curiosity remains :)
also i have asked this question out there
i didn't knew the theorem name
@PranshuKhandal Yes, I noticed. And I've suggested a duplicate. ;)
You can do it with a spinning magnet, like the Levitron, but it's not easy to achieve stable levitation, so the Levitron wasn't much of a commercial success. Modern maglev gizmos have control circuitry that modulates the strength of the electromagnets thousands of times per second. And even those things can be fiddly to set up correctly. If you want truly stable maglev, you need a diamagnetic material, or a superconductor.
8:01 PM
what is meant by the exchange interaction?
our professor mentioned is as a motivation for Hund's rules, but I can't find it in my book (I use Griffiths)
oh never mind, it seems that we get a $S_1\cdot S_2$ term in the Hamiltonian when we have two particles, so I guess that makes sense then
I'm guessing this effect is tiny, as I've never seen it before, though we've worked with sping-orbit coupling, which is the same idea i guess
In physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical force, it is not a true force as it lacks a force carrier. The effect is due to the wave function of indistinguishable particles being subject to exchange symmetry, that is, either remaining unchanged (symmetric) or changing sign (antisymmetric) when two particles are exchanged. Both bosons and fermions can experience the exchange interaction. For fermions, thi...
' In Eq. (3), C is the Coulomb integral, B is the overlap integral, and Jex is the exchange integral. These integrals are given by:'
right, I've seen exchange in spatial coordinates, but I haven't seen it where we include spin, I think
I'm guessing the idea is the same
oh, it makes sense that all up-states yields the highest (spatial) exchange interaction, because they are symmetric, and thus yield an anti-symmetric spatial part, which we know has lower exchange interaction
There's a way to represent those terms as eigenvalues of a 'spin exchange operator'
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries. They are σ 1 = σ x ...
I haven't seen a way to make this obvious yet :(
aw too bad:p but it seems cool
1 hour later…
9:36 PM
@EmilioPisanty how's life?
I've been chatting in Spanish and Russian with a new colleague who is of a Russian family but used to work with Spanish speakers on a daily basis.
It turns out that I have much to learn in the way of vocabulary.
So much of what he says uses slang... I can barely keep up.
Q: Where should I post a question about logic circuit minimisation?

AustereTigerI have a question about some logic circuits (I have the Boolean functions and truth tables) that I would like to minimise. Where would I post a question about this? I was thinking Physics firstly but possibly somewhere else?

1 hour later…

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