Lecture 2 | Quantum Entanglements, Part 1 (Stanford)

Lecture 2 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded October 2, 2006 at Stanford University.
This Stanford Continuing Studies course is the first of a three-quarter sequence of classes exploring the "quantum entanglements" in modern theoretical physics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.
Complete playlist for the course:
kzbin.info_play_list...
Stanford Continuing Studies: continuingstudies.stanford.edu/
About Leonard Susskind: www.stanford.edu/dept/physics/....
Stanford University channel on KZbin:
/ stanford

[@MrFerdinandOo]

32:02 Bra-ket notation 33:44 - 38:05 complex numbers 43:30 complex conjugate (44:20 bra) 47:47 braket rule 49:16 50:43 quantum bit 1:14:25 observables

[@petergreen5337]

❤thank you

[@bear630]

Thank you for this. I cannot express to you how much I am enjoying this series of lectures. That they are available here is wonderful for someone like me.

[@xi7837]

this lecture series is probably a distant memory of yours but im the same I really like because as a child I can get the information for free

[@NazriB]

Lies again? Quaker Oats

[@cyb3rgh0stv7]

Eedsffz

[@xi7837]

@@Temmie_Dola Yes

[@lugia8888]

@@xi7837 he has such stupid complex variable notation lol

[@aadeagle]

same here friend, Iam in field of biology and business but also interested in physics, very exciting. it was unfortunate that when i was in high school no one inspire me, but since then i'm fascinated with nature but confuse in physics. susskind is one of my hero along with feynman

[@garydranch2014]

So greatful to Stanford U & Leonard Susskind for making available to the general public these lectures on multiple subjects of quantum mechanics and relativity theories, the nature of the very large and very small components of our universe. Now well into my seventh decade I come for the first time as a supplicant to the discipline of the study of physics. It is most gratifying to me that I am able to partake in the privilege of self-learning, inner-reflection and mental discipline that many others who hungered for such access to a community of knowledge have previously been denied. It is for this reason that I offer my utmost thanks and gratitude to the series producers, to Stanford University and to Professor Susskind for these priceless gifts that I am the beneficiary of. Bene facia!

[@elifonkonsolakis2521]

I cannot explain with written words how much I enjoy these lectures, Ty prof Leonardo, (with Italian pronunciation ;-)

[@mauriciobarda]

Thanks a lot Standford and professor Susskind for these courses on modern physics. I'm really enjoying them a lot

[@LukeTaylor101]

Seriously, i have no idea, how i landed no this video lecture. I was asleep when half way I heard the professor and woke up and realised i was studying Quantum Entanglements in my sleep. And..... this is Lecture 2. So you can imagine

[@hififlipper]

You woke up too early.

[@ZoeTheCat]

The Bra-Ket notation makes QM appear foreboding -- especially as you transition to Schrodinger eqtn examples. Susskind does a better job than most texts to demystify this notation. The same is true for Tensor notation within GR. I'm convinced that some Profs/Books just want to appear complicated rather than explain things in simple terms. Susskind follows Feynmans good example: "If you cannot explain things simply - you probably don't understand it very well."

[@cicafittipaldi]

This professor is the brain rewiring master i have never in my life learned so much in so little time.

[@jessestrehlow1979]

Thank you Stanford and Leonard for sharing such amazing knowlege.

[@mihastampek2018]

Oooooóóoooooooooooooooooooooooooooooooooooooóóóooóo

[@mihastampek2018]

Oooooóóoooooooooooooooooooooooooooooooooooooóóóooóo ooooóo

[@glennmain]

Thank you for the great vids Stanford and Leonard Susskind!

[@parthoroychoudhury860]

I don't generally comment in lecture videos, but this explanation is just so cool! The way he explains this complicated subject, this is the best explanation quantum mechanics video I have found so far.

[@itis4peace]

Great pleasure! Excellent teacher. If any of you guys at Stanford are not taking advantage of Leonardo's lectures, you're against physics in time reverse, and going through physics dynamics in inertia stage. Greetings, Professor Susskind.

[@axelkolm3894]

These lectures are really helpful. Thanks for uploading them

[@sdsa007]

I've been learning quantum subjects piece-meal from various incomplete sources...... but now I am beginning to understand how the abstract math is fitting together in terms of a physical observation!... I am so happy for this!

[@ddorman365]

I completely agree with you Lenny on what is necessary in thinking o understand Quantum Theory and your description of the juxtaposition of states is right on the mark, great job on describing 1/2 mono dimension, please hurry on to entanglement, peace and love, Doug.

[@eschuber8]

this video looks like the pixels are following the uncertainty principle, wow, the quality is not so hot

[@dragonjake6968]

really now the pixel of a video doesn't guarantee quality learning its what you can take from it and besides if you listen to the video its all well done

[@ximecreature]

Susskind doesn't use the board so much. So long I can still have access to content of this quality, I'm not going to complain about the pixel density

[@NoActuallyGo-KCUF-Yourself]

But it looks like the vectors are moving, man!

[@ApteraEV2024]

L😆L!!! 👍

[@jcbmack]

These lectures accomplish in a gentle way teaching the fundamentals.

[@Pidxr]

It's too bad these videos are either so highly compressed or just filmed at low resolution as to be distracting.

[@jojolafrite90]

Yeah, sure it's not the point. But at this point it's actually too bad and disappointing.

[@deathtotruthers1]

Agreed. This is the single best explanation of quantum mechanics (for idiots like me) I've ever found. And I've been looking for a long time. When I retire, I plan to go back to school and get a BS in physics. But until then, this will do.

[@jannisalexakis6436]

He is perfect in communicating knowledge!

[@belkyhernandez8281]

Watching this in 2022. We have been in an anti intellectual period for the last 7 yrs, maybe more. So making this and other subjects available to everyone is more important than ever.

[@avitalmorgenstern]

NOW I GET IT! Thank you prof. Leonardo

[@ahmedessam1426]

27:00 what if you put an AC current on the vertical magnet that makes electron N up and S down that they move in a very fast sequence that when electron moves down it found itself pointing North tries to move up it will find itself pointing North again : what state will it be? my opinion that it will jump off away from the magnetic field

[@ImGoingToMakeShowsWithOprah]

This is what I wish was talked about in elementary school. I would have been interested.

[@ThinkTank255]

Hey, don't blame him for your own mental shortcomings. It is a very clear presentation. This is EXTREMELY basic stuff. It's intended for people that have at least a BASIC understanding of quantum mechanics and a BASIC understanding of mathematics. He has another video series on quantum mechanics. You should probably watch that first.

[@pharmahuascasoul]

ThinkTank255 you so smart

[@Holobrine]

I love how a complex number times its conjugate gets you a form identical to the Pythagorean theorem.

[@jessestrehlow1979]

Complex numbers are just vectors that can be written in polar form using sin and cos. For example. 3+4i has a magnitude of 3^2+4^2)^(.5)=5 and the direction is 5cos(¥)+5sin(¥)= arctan(img/real)=arctan (y/x)=tan-1[4÷3]=36.8°

[@PetraCarExpo]

Amazing series!

[@aadeagle]

i'd heard the feynman lectures, its easily understood in layman's term, but i think prof. susskind explanation is much better, he's not talking like we physicist... but makes you feel like he is one of you, emphatically and patiently explaining physics concisely. a true modern day LEONARDO... I WISH HE COULD PLAY BONGO

[@SomeGuy-nr9id]

5:30 the mathematical term is a normalized vector or a unit length vector. Were all the components multiplied by themselves then added together sum to 1 which is valid across any number of dimensions. This part i know from doing so much graphical programming you work with tons of matrices and sometimes have to dive pretty deep into understanding them. Pretty much all the properties of the ideas have been named so they can be identified to explain or talk about harder concepts for manipulations. Cause you know video games.

[@Gregy624]

Vector entanglement can only occur as the result of the interaction of transluminal wavicles with a parallel polarization to the orbital Z axis relative to the direction of travel?

[@FlashJockey1]

so if you prepare an electron for say, an angle of 90°, you measure it's spin... one half of the time you get the same amount of energy that radiates if it was prepared for 180°, where does that radiated energy come from?

[@Oners82]

+Flash Jockey He answers that at about 27.20. The energy comes from the energy inputted into the system to create the magnetic field.

[@philipolson4842]

The first lecture mentioned reviewing calculus would be a plus. This took an extra week.

[@Holobrine]

17:20 What if you measured how long the electron takes to release a photon? Maybe it has to build up the energy first somehow.

[@DoaaElshahat--]

Yes, it has to absorb some energy from the EM field.

[@physicsjagat]

You are great teacher,all the series are excellent

[@stewartcalder8941]

So what happens to the conservation of energy in the milliseconds between the field being switched on and the elctron flipping to its stable UP state? (I still have an hour left to watch)

[@roxiefaye144]

I'm curious what you doing now

[@apostolismoschopoulos1876]

@@roxiefaye144 I thought of the same thing lol. There are times as these when I encounter comments from 9 years old that I'm wondering what do these people do today?

[@ernestorodriguez6445]

When you measure the direction of the electron, how long do you have to wait till it emits a photon? Is it always the same time? Are we sure that if we leave the electron exposed to the field it will never emit the photon if it didn't after some time?

[@Oners82]

+Ernesto Rodriguez So many questions asked in the comments section that are already answered in the lecture (28.10)! No it is not always the same time, the time is proportional to the strength of the magnetic field. And yes the electron will always emit a photon eventually assuming that it is in a position with a non zero probability of doing so. But the experiment assumes a strong magnetic field so the emission happens very fast.

[@pete2331]

I'm currently working my way through the lectures of part 1. Does anyone know if there is a part 2 around somewhere or does it just move to part 3? I just don't want to miss out on any of it if I can help it. Thanks

[@boomfist]

I love how at 1:13:10 the wrinkles on his wonderful forehead actually crank the encryption algorithm into rendering interference.

[@skillabr1832]

It's video compression. Because there is complex movement, the delta in the picture does not get adequately captured when passed through the compression algorithm. Thus, you get a blurry image.

[@AntKneeLeafEllipse]

This is wild: a little more than half way, you say "Let's take a break," and it happened just after I took off my glasses to rub my eyes. As if you were talking directly to me. What's the probability of that?? This is pretty incredible, and I'm very grateful for this resource.

[@Jodoi1]

@pete2331 From Prof. Susskind's intro in this video, this appears to be the second part - I think it has been labelled incorrectly.

[@cochisewolf]

I've had a few classes where there were people, usually one or two, that asked so many questions that we couldn't get through the subject. So those of us who actually studied and met the prereqs suffered. I agree there should be a Q&A period after class for those not resourceful enough to find the answers on their own.

[@cochisewolf]

Professor Susskind is excellent. I would have to agree with another comment, one sure way to improve the presentation would be to not allow questions to be asked until it was over. I had an instructor once who told everyone to hold off any questions till the end of the lecture at which time he would devote time to them.

[@zphuo]

@8:45, why little bar magnet in magnetic field will process around the magnetic field? Why this action will loss the energy??

[@poopnakedyeah]

any processing object is accelerating and any acceleration causes radiation

[@majiddehghani4974]

Mr suskid is very patient in his teaching method

[@phycogsmithomniscience309]

it makes sense to me , the electron radiates the same energy no mater what to make a adjustment or it is not required to make that adjustment.

[@fpgaguy]

thank you so much for providing this

[@TheOvermaster]

@HyperBorealOperator Correct me if I'm wrong - but I believe it was a statistical mechanics course - so there is no loss of continuity by skipping (in this case, we can't watch it) part 2.

[@NoActuallyGo-KCUF-Yourself]

If it's like most science classes I've taken, it doesn't start to get hard until over half-way through the course.

[@hossboll]

1. duas direções possiveis p/ apontamento do vetor do e: cima ou baixo. no in between: quantum bit. 2. dependendo da config. fisica - disposição do campo/do e - , ha maior ou menor probabilidade de emissão do foton na transicao. quanto mais o e estiver alinhado com o campo (n->s), menor a prob de emissão do foton. (problema: qual o "referencial" do e? campo grav. terra? para haver dif. probs faz-se necessario haver diferentes percepcoes do espaço (?).)

[@karlxu1548]

21:00 This is confusing If electron can only be observed pointing up or down, how do you even get a probability distribution beyond up and down?

[@dogdowling]

To get a probability distribution, you need to repeat the experiment many times, re-preparing the electron in the exact same way each time. The point is that when you measure the electron, you get up with some probability p, and down with probability (1-p). To (approximately) determine p you need to repeat the experiment.

[@XXsardarXX]

Hey, I think I understood this stuff....amazing!

[@kemalaziz9696]

Are there lecture notes for the series?

[@robertdavenport7950]

does the same electron emit only one photon or does it do it over and over again?

[@schmetterling4477]

The electron doesn't emit a photon, at all. The atom as a whole emits a photon. While we talk about "the states of the electron", that's a misnomer. In quantum mechanics we are describing the physics of the entire system of nucleus and electrons. So, yes, the atom can be in a highly exited state with very large angular momentum and then it can emit a cascade of photons.

[@apollion888]

I fucking love this guy, Susskind rocks!

[@PelczarTomasz]

One future Nobel prize will be someday about elementary body well known as Flerow...

[@MrWorshipMe]

I would change it to 60% :) - and only if we talk about experimental physics PhDs - most theoretical physicists are one notch above the rest. And after going through all of Feynman's and Susskind's lectures - I would recommend trying to tackle Penrose's "The complete guide to the laws of the universe", which takes you even more into the depths of theory and math of QM and GR.

[@An_Idiot_in_the_Wild]

Penrose's books are absolutely brilliant.

[@AnilKumar-mz8dj]

suppose i am applying alternate electromagnetic magnetic field by changing direction of current, for an electron in a same energy level.then how long electron will emit photons..??will electron disappear or something else?

[@nmarbletoe8210]

Anil chauhan an alternating magnetic field would make the electrons flip back and forth repeatedly, for as long as you kept the field turned on. each flip you'd get photons. you'd have a photon factory

[@Gwunderi25]

+Anil chauhan I'm not sure whether you mean that the electron, when alternating the magnetic field, somehow "evaporates" by emitting photons? But the electron will always have the same mass/energy (don't know what's correct). The photon is not somehow part of the electron; I think the photon is radiated because the state "spin up" (in the direction of the field) is the energetically most "convenient" state for the electron? (think it's like a kind of potential energy that the electron gives up by becoming spin up?) In any case it's not to compare with water evaporating or something like this, so no worry, the electron will not disappear : )

[@phewpow]

Is it possible that there are 1000 or x amount different kind of electrons, and that it's not probility, but that you just can't say which kind of electron you are handling at the moment?

[@musicalfringe]

That's known as a hidden-variable theory, and John Bell established mathematically that, although hidden variables aren't disallowed, explaining quantum behaviour through them changes the other assumptions allowed in your theory, e.g. locality.

[@MEpianist]

Big THANKS to KZbin, Stanford & Prof. Suskind! Although I disagree with your decision to edit out the questions...

[@grunder20]

Excellent work if I may say.

[@bean8287]

Around 20:00 in, when talking about the state of the electron as either spin up or down, how do you actually know that the electron is spin up or down per se. If you are determining this by whether a photon is radiated or not, and they are all radiated with the same energy, then how do you know that the electron has actually completely re-aligned itself with the magnetic field, and not simply made a partial alignment, corresponding with the amount of emitted energy. It seems like the only way to know for sure is to wait and see whether more radiation is emitted, implying the electron made another incremental rotation. But such an experiment would be tricky, because just because you don't detect radiation yet, could mean there might be some in the future. Perhaps my answer lies ahead in this video/series, I'll have to see, hopefully in the meantime someone could fill in the gap for me. Edit: Ahh looks like I only needed to wait another minute, as he explains that you would not detect further photons after the first one, I will have to do some research after watching this to see the specifics of the experiment!

[@musicalfringe]

This is to the best of my understanding - someone please correct me if I blunder. The electron never partially re-aligns. It either remains in its original (indeterminate) state or emits and aligns completely. Particle spin axis seems to be an example of something that's fundamentally discrete in a very counterintuitive way in QM. Partial measurements or even ambiguous post-measurement states don't seem to be allowed. If my thinking is right, indeterminacy of spin only appears when you (a) change the measurement axis relative to the particle's last-measured spin axis, (b) allow enough time between measurements for the particle's interaction with the environment to push its state away from the last-measured state. Ignoring case (b), the indeterminacy only ever appears in the form of the next measurement being probabilistic to the degree that its axis is perpendicular to the previous measurement axis. After that, though, the spin axis is once again completely known.

[@petevenuti7355]

Is the probability curve the sine of the angle?

[@frun]

Yes

[@am101171]

Vectors is just a way to represent things with several values. If you walk to your kitchen you do so at a certain speed but also with a certain direction, say an step a second with direction from the living room to the kitchen, that is a vector basically. If you multiply your vector by 2 (the scalar 2) you end up with 2 steps per second from the living room to the kitchen.addition: If your mom walks at same speed from the kitchen to the living room then bumps into you no one is going anywhere.

[@varunkryadav]

In the video it is mentioned that the new state of electron(turning on the magnetic field) depends upon the earlier state of electron. Can someone explain to me how and why?

[@AlecBrady]

Because you square the vector (i.e. form the inner product of the vector with itself) to get the probabilities. So total prob=1/rt(2)*1/rt(2)+1/rt(2)*1/rt(2) =1/2+1/2 =1

[@ericpaterson]

Quick question. Does the electron in the magnetic field release different amounts of radiation depending on the angle its situtated at? if so, then it seems there are many different states that it can be in. It either releases no radiation, or there are many different amounts of radiation it can release. That does not sound like two states. Any help would be much appreciated, thanks

[@dudleybrooks515]

The *strength* of the radiation (equivalently, the wavelength of the photon) which the electron can either release or not release is fixed by something *other* than the original angle of the electron. (I think it's fixed by the strength of the magnetic field, but I'm not sure.) In any case, the electron only has two "choices": release that particular energy (wavelength) or don't release any energy/photon at all. The *probability* that the electron releases it is what is determined by the original angle.

[@naimulhaq9626]

Every electron is a magnet with a magnetic moment. While watching Cosmology Lectures, by Leonardo, I can remember him explaining how a lots of magnetic monopoles that vanished with inflation, and consequently became difficult to find or detect. Is it possible that the magnetic moment of electrons are the result of some amalgamation of electrical and magnetic fields,due to the lost Leonardo's monopoles?

[@drumstruck751]

can you get the stars to make sense within a year of orbit acceleration with two long hot seasons and two short cold seasons. That way we can make sense why winter is shorter than summer. meaning summer is the long part of the oval when were closest to the sun, winter is in the quickly changing portion of the oval.

[@ArilandoArilando]

I don't understand how multiplying +bi with - bi "cancels out" the i. Can someone explain this to me?

[@mariapinilla3420]

(a+ib)(a-ib)= a^2+iab-iab-i^2*b^2, so the middle terms cancel, and since i^2=-1, then the answer is = a^2 - (-1)b^2 = a^2 + b^2

[@cesarsosa4617]

because by definition i = sqrt(-1) because of this, the square of i is -1 (a real number) as shown here: i^2 = sqrt(-1)^2 => i^2 = -1 (the square root of a squared number is the number itself) then: (b*i) * (- b*i) = - (b^2) * (i^2) = - (b^2)*(-1) = b^2 The result is positive because there are two minus (-) signs being multiplied. * implies multiplication

[@flawns]

it's important to watch the whole video before commenting. if you only stayed patient for a few more minutes.

[@jacksainthill8974]

Why haven't you responded to anyone who has been kind enough to help?

[@ArilandoArilando]

Are you talking about me?

[@Eay5paev]

At 1h02'55'' he says "we'll come to that", talking about -A being the same physical state as A but A+B not being the same state as B-A. Where does he explain this? This issue has always puzzled me!

[@zenicv]

Does a complex vector and its conjugate exist in the same vector space?

[@roelofvuurboom5939]

Yes, you can construct it for the vector itself. If a+ib is in the vector space, then so is a and hence also -2a. Add -2a to a+ib. That is -a+ib. Multiply by -1. That is a-ib.

[@lugia8888]

Linear combinations are permitted in vector spaces - that is the point of all the axioms. Basic Linear Algebra.

[@brad1368]

I've been watching a lot of Prof. Susskind's videos and he almost always has a coffee and some little bag of treats...does anyone know what he is eating? It appears to be the same thing each time and it looks pretty good.

[@Aryanpars]

Total Awesomeness!!! :P

[@SalimMaroun-fb5rd]

Thank you you are the man

[@pranav.dilsephysics]

It's amazing ✨

[@AlecBrady]

@varunkryadav Doesn't everything depend on what its earlier state was?

[@inox1ck]

The electron vector can still be explained classically. if you imagine the vector being elastic not rigid, it bends up to a point, then if you force it further it yields the energy then it straighten itself up. Another way is unless the electron is ready to emit the energy it continues the precession indefinitely. This is still quantum but it is imanginable

[@inox1ck]

Actually the first statement doesn't explain what happens.

[@Ymirheim]

@Lodine66 Actually I found most of the questions intelligent and insightful which is what you get when the students are not undergraduates.

[@Svenskivsk]

I think what he is saying is that there are only two states an electron can have (in this situation), and if the electron is prepared at a different angle it changes the probability that the electron will emit a photon of definite energy, correct?

[@abdullahnaeemamalik]

Why are the weights to the basis complex numbers?

[@F00dTube]

At around 1:27 on guy in the audience makes the remark that P(+) - P(-) isn't the expectancy value and that when both Probabilities are equal the total probability is 0. The formula should be 1P(+) + -1P(-).In this case it works because the absolute value of each possibility is 1.

[@manheer1000]

how does north pole of electron point towards north pole of magnet at 18:21

[@captaindapper5020]

Electrons have negative electric charge, and will move towards positive magnetic poles since opposites attract.

[@captaindapper5020]

Also, Lenny's human and makes mistakes sometimes :)

[@am101171]

The vector space is your house in this example. Hope it helps.

[@trex9966]

at what time does he say this?

[@shakagenghis]

conceptually this is very helpful. up/down could be yellow/purple- some states x and y that are connected observationally to a certain behavior. Very helpful. I really should have watched the whole lecture before I asked the question, but nevertheless I appreciate the help.

[@mnfchen]

This lecture series is a bit slow, but gotta be patient. I went through Linear Algebra in college, and these people obviously haven't =/ Great lecture btw!!

[@StephanLuik1]

A very nice explanation. However, I wonder if it could have been done in a quarter of the time if students actually knew complex numbers and matrix multiplication rules?

[@petevenuti7355]

Is magnetic moment intrinsically one dimensional?

[@fragmentsoferik]

I don't understand what he says around 1:21:00 that the probability for the observable (which is either + or -) is [P+] - [P-], which he calls the averageness of headsness or tailsness.. That should be 0,5 - 0,5 and I don't understand what he wants with that.. Sombody please explain..

[@jcovello]

Anyone know if: A: Putting the electron in magnetic field and watching for photon is hypothetical or really how the experiment if performed. B: If actual experiment we know there is a probability of photon being emitted but can we tell if electron gets reoriented in all situations or only when photon is emitted.

[@WPMcNeill]

If you find the idea of a single electron being held in place by a nail distractingly artificial, try reading about the Stern-Gerlach experiment, which is the actual way spin was first observed.

[@dweeper1020]

QUESTION: If we prepare the electron to be horizontal and put it in a vertical magentic field, there is a 50% probability that it will emit an electron. If it does, we know it is now aligned with the magentic field. If it does NOT, what alignment is the electron in (or can we ever know). Please only respond if you KNOW if the answer, not if you think you know it. I'd like to know.

[@dweeper1020]

Don't think of the electron pointing up or down (this is just a convenience he uses), think of it as the spin state - spinning around it's axis (either spinning in one direction or the other). All elementary particles have a spin quantum number. These spins create a magnetic dipole moment that "point" in a certain direction depending upon the direction of spin. You can find more details on spin state in wikipedia.

[@mc4ndr3]

z* z is non-negative, not necessarily positive. It's the square of the magnitude, not the magnitude itself.

[@AlecBrady]

@Eztoez I haven't got any maths beyond A-level, and it all seems quite simple to me.

[@ukvaishnav]

@av733 not really but if you have done little idea of linear algebra and ket-bra notation (which is just the vector space in different suit) then go ahead you'll hardly have problem. If you don't then first finish the Quantum Mechanics lectures.

[@johnesposito5508]

i just fell asleep watching this, and i had a dream that i drew a picture of Richard Simmons on his white board as an equation and he got really pissed..... Is Richard Simmons the key to quantum entanglement ?

[@agimasoschandir]

Maybe you were not exercising your hand by drawing enough - was there any music being played? Who was Richard Simmons being paired with - were you happy (he had spin pissyness, the you spin happiness)?

[@gricka31]

well questions are always helpful, often for multiple people, but they should be asked at the end of the lecture, because they are often resolved later on in the lecture

[@ComradeSpookfessor]

When lectures talk about crazy things I normally just Google them. Although I understood it perfectly, you mustn't have.