Lecture 3 | Modern Physics: Quantum Mechanics (Stanford)

[paddy110287]

haha First Video 330,000 views, second 177,000 third 77,000 its going down and down...pity cause they are great videos

[VellianoRosso]

k = 1/length = ħ^(-1/2) G^(-1/2) c^(3/2)

[VellianoRosso]

2pi=Tau

[simplelife1021]

Can someone explain how at 13:00 the basis vectors m and n become summation indices. I understand both notations (I think) but I just don't understand how he derived Knm from (nI K Im)

[fustilarian1]

What does it mean to be orthogonal in complex space?

[LiberalsUtopian]

Lets all purchase goods and services from Standford shop.

[n1a1s1i1m]

At 1:15:45, why is the integral of dirac-delta-function psi(lambda) and not x*psi(lambda)?

[Evan2718281828]

@n1a1s1i1m

If you mean why was the integral of the whole thing this, then the answer is that it's because the area under the dirac delta function is always 1 by definition, not x which i think you assumed.

[PezNicolas1]

@n1a1s1i1m because of the definition of dirac-delta function... just takes the values 1 and 0 so 1*=1 and 0*=0 they are of course real numbers....

[TheChadSitze]

FINALLY! A way to learn in depth QM material while in the nude!

[Bucketheadhimself]

@TheChadSitze Check out a book from the library?

[jackp22]

1:22:22 tumbleweed rolls through lecture theatre

[TheLiberalSoup]

I looked it up, "eigen" does mean proper. Leonard Susskind was right. No surprise there haha

[mlzg4]

Does anybody know if there are any mathematically rigorous QM courses on youtube? I am talking about something that starts with spectral theorem and describes operators in terms of generalized eigenvalues.

[TheChadSitze]

@mlzg4 I'd be interested in knowing this as well. Susskind is going into it more than I have found anywhere else but I assume there is an even more in depth discussion to be had concerning this material.

[jamma246]

I don't understand 35:00.

How can all Eigenvectors always be orthogonal? Surely if you take the identity matrix, which is clearly Hermitian, everything is an Eigenvector, and these are not all orthogonal.

[JWentu]

@jamma246

In case of the Identity matrix all lambdas are the same so you lose the condition lambda1 different from lambda2

[amadevs89]

Isn't the result of integration by parts as done at (1:30) correct only if the psis are normalized?

[cesarjom]

@amadevs89 No, never in that mathematical method is there any need for the eigenstates (psi function) to be normalized. its straight forward integr't by parts, which btw is a common method to show operators are hermitian in QM.

[mdinka]

is there a deeper sense in why just the features of Eigenvectors and Eigenvalues turn out to be so important for QM?

[csikjarudi]

@mdinka As I understand, in general the eigenvectors and eigenvalues are important mathematical objects due to their invariance under (proper) coordinate change. In other words, no matter how you represent a linear operator, it will have the same eigenvectors and eigenvalues, so they are the "essence" of the operator.

You can make an analogy of this to QM, and I think that shows the importance, at least to me. :)

[cesarjom]

@mdinka eigenfunctions have long been in use to solve many physical problems in nature (not just in QM). they turn out to be fundamenal in linear systems (ie vector spaces) where linear diff equations explain or model the natural phenomana. Asking if there is a deeper meaning to eigen-equations is like asking if there is deeper meaning to why the field of calculus if so important to newtonian mech.

[dalcde]

When we are finding P(x), shouldn't we be finding the probability for the particle to be in the interval (a, b) and take the integral from a to b since the function is continuous?

[dalcde]

1:09:01 Obviously Dirac cheated.

[supertrunksz]

In 1.29.00 shouldn't we have also another additive, to be precise the iΨ*Ψ ? Because when integrating by parts you have S(f.g')dx = f.g - S(f'g)dx

[cesarjom]

@supertrunksz YES but remember that the integral is over -∞ to +∞... so the term iΨ*Ψ is evaluated over the same interval (after its implicit integration). This term thus goes to 0 b/c the complete set of eigenstates Ψ form a Cauchy sequence (a req of belonging to a Hilbert space) which implies that at +or- ∞ the eigenstates are 0.

[11markus04]

good lecture, but the sound goes out around 1:23:00

[LennyLeonard85]

Free lectures on the internet is the best that happened to me in a long time. Thank you Stanford, MIT, Oxford and all the others! Really appreciate it!!

[Sakartvelo69]

brilliant lecture but it still isn't clear to me how the dirac delta function relates to the psi wavefunction...how does it satisfy the requirement that it is an eigenvector of X? dunno...maybe i'm stupid...i'll give it some more reading but i think he could have been a bit clearer there...

[reticulum78]

@Sakartvelo69 The dirac delta function operates in a continuous space in much the same way as Kronecker delta does in a discrete space. Any function can be visualized as a continuous sum (integral) of Dirac delta functions where the coefficients are the function values at specific points. X measures the position of a particle so the eigenfunction of this must be localized at one point, and delta(x) is the function that describes this localization.

[Sakartvelo69]

@reticulum78 thanks a lot...I've been reading about it and well, yes, that's what I've learned...in a nutshell:) cheers

[TheLiberalSoup]

@Sakartvelo69

could you help me with this? It seems from your comment you would understand what I need to. How is it possible for a function (the dirac delta function) to be a vector at the same time? I am just so lost when he is talking about the dirac delta function being an eigenvector. I don't see any vectors at all in his diagram, so I can't make sense of it. Could you help?

[1o618033988749894848]

@TheLiberalSoup he's talking about vectors in a more mathematical sense. we usually think of vectors as an array of three numbers to describe a position in space, but all you need to have a vector space is that the vector axioms are satisfied (he wrote them in lecture 2) . So the dirac delta function is a vector in the vector space of complex functions on a single real variable, just like phi(x). Its an eigenvector just because it satisfies H(phi)=lambda*phi. Its constructed so that it does that

[Evan2718281828]

@1o618033988749894848

ah, so like an example would be 3x+i5x or something? If that's what it is, then how is it possible to multiply nd operator, if it is actually a matrix, by that? It wouldn't be normal vector-matrix multiplication, so what would it be? Thanks for answering by the way.

[1o618033988749894848]

@Evan2718281828 I think the best way to think of it is that we're working in a basis of eigenfunctions. So if f1(x) and f2(x) were the first two eigenfunctions of a linear operator, the function f1(x)+f2(x) could be thought of as (1,1,0,0,0,...). Then if the matrix is diagonal, it is easy to see how the matrix multiplication will give you a new function which looks like a*f1(x)+b*f2(x), if a and b are the eigenvalues for f1 and f2.

[1o618033988749894848]

@Evan2718281828 A hermitian operator can always be diagonalized with real numbers on the diagonal (the eigenvalues). These real numbers correspond to observables. So for example, if the momentum operator acting on the first eigenfunction gave you 3 times the first eigenfunction so that P(1,0,0....)=(3,0,0...), the momentum of the particle in that eigenstate is 3. We usually make the assumption that the space is COMPLETE so that we can express, for example, 3x+i5x as a sum of multiples...

[Evan2718281828]

@1o618033988749894848

ha! I just realize that both of our usernames are named after the decimal digits of mathematical constants

[1o618033988749894848]

@1o618033988749894848 of the eigenfunctions. 3x+i5x might look horrible in the basis of eigenvectors, but in principle it would be an array of complex numbers which you'd multiply by a diagonal matrix. (To diagonalize a matrix is the same thing as to write it in the basis of its eigenvectors). Sorry for the horribly long "explanation" =P

[Evan2718281828]

@1o618033988749894848 haha dude! don't be 'sorry' for helping me understand something! hahahaha i loled when i read that cause im sitting here with my pencil taking notes of what your comments are saying and you're saying sorry! So but i have one more question which is, you said that in principle, ax+ibx type stuff would be an array of complex numbers, which i dont get that well so could you give an example of a complex function 'translated' (idk) from ax+ibx to an array?

[1o618033988749894848]

@Evan2718281828 It depends on what the basis of eigenfunctions looks like. For example: suppose we had a system which had a Hamiltonian (a hermitian operator which basically gives the energy) which had the Legendre polynomials as eigenfunctions (you can wikipedia them... they're just a bunch of polynomials). We would also need some other constraints, like the particle cannot exist outside of the interval (-1,1)... then the Legendre polynomials can constitute a complete orthogonal basis

[1o618033988749894848]

@Evan2718281828 (I'm not really so sure about the completeness, but whatever). So this is a pretty artifical system, but it will illustrate the point. Since the second Legendre polynomial is just x, your wavefunction ax+ibx=(a+bi)x would just look like (0,a+bi,0,0....). If it were something like ax+b, it would be (b,a,0,0...). Now that you have your wavefunction expressed in terms of the eigenstates of the Hamiltonian, you can do Quantum Mechanics to it. In most systems the eigenfunctions

[1o618033988749894848]

@Evan2718281828 are things like gaussians or plane waves. For your linear wavefunction, it's not normalizable, hence the need to invent a weird system. For a wavefunction to make physical sense, we usually require that it decays to zero as x goes to infinity and minus infinity. There's also other technicalities... you'd probably be better off learning this stuff from a book. The David Griffiths book is the one I use, and it's excellent. Also I hadn't noticed the e in your name =P nice

[Evan2718281828]

@1o618033988749894848

ahhhhh i see. Thank you for explaining!

[bhigr]

This lecture gives you the gist of quantum mechanics. This lecture could very well be a regular course in QM.

[bhigr]

Eigen means something like belonging to something in a general sense. Eigentum is property. So proper is correct, but proper as in property not proper as in correct. In this context Eigenvector means the vector characteristic of the respective operator.

[IferMasterofFire]

On a side note: that's the last I'm going to say about other people beneath this jewel of internet freedom. Thanks again for uploading these lectures, Stanford University. They've been a great help.

[IferMasterofFire]

wow, that bluegrass guy is a creepy fanatic..... Should we tell him that nationalism has lead to the Nazi movement in recent history? And that there have been far more infulential civilisations? If I'd meet someone who referred to my country as The Great Nation, I'd slap him in the face and call the mental institution. Saying such things is something the Chinese are indoctrinated with, not something that anyone in a western nation should say.

[bluegrassaficionado]

@IferMasterofFire #1) Nazism is fanatical socialism. I'm against socialism, communism, and anything that's not total, unrestricted capitalism. The liberals in this country are not, so if anybody is close to Nazism it's them. #2) I welcome you to try to slap me in the face; I promise you will not like the result. #3) If you don't think America is great and you're not proud of this country, GET OUT NOW!!! WE DON'T WANT OR NEED YOU HERE!!! Poisonous, treasonous vermin.

[IferMasterofFire]

@bluegrassaficionado #1: Nazism, or rather facism, is a politically right movement based upon nationalism, with some socialist elements for their own people (which is far from socialsm). #2: meh #3: I'm proud to live in the old world. Even with Geert Wilders, our muslim hating populist squeaker of a politician, we still do better than you on the newsweek list of best countries to live in...

[bluegrassaficionado]

@IferMasterofFire No sense arguing with those too dense to understand the fundamental concepts.

[harshcarbon27valvi]

omg the way of his teaching is awsomeeeeee.

my quantam teacher suxxx.

he is far more better then him

[Mr4thamigouk]

Yes, these lectures are fantastic... I already did a Bsc some years ago and it is clear that looking back now, I didnt 'really' understand anything I learned! Susskind breaks the maths down to the simplest level that it is hard not to understand things...

Having said that I follow this lecture series better than I did General Relativity...

[repzic]

He keeps saying refer back to Quantum Entanglements etc, what order are the different Topics in?

[nickharvey7]

Could Heisenberg Uncertainty Principle be the same uncertainty that the observer will have with any future event? Could time and the geometry of spacetime be continuously formed by the momentum of EMR or light form atom to atom? There is no understanding of time in modern physics or why we have a future and a past. Could this be why we have the paradoxes of QM?

[TheYashpalsingh]

Thank a lot sir

[madambaf]

Politics becomes a subject relevant to science when the political admin dictates scientific progress. Bush admin highly restricted stem cell research. The current admin lifted the restrictions. Simple example but illustrates the effect of politics on science. Those "intellectuals" or "academics" associate themselves with the political philosophies that ensure they are freely able to conduct their research which propels our Nation to great heights. Others would rather keep us in the stone age.

[bluegrassaficionado]

It makes me so angry when there is political commentary in the classroom. Politics is NOT a subject which is relevant to science. Folks are paying to learn interesting scientific concepts, not to be subjected to competely subjective remarks. However, since the topic WAS introduced, I am always puzzled as to why so many people who label themselves as "intellectuals" or "academics" associate with the idiotic and obviously absurd political philosophies that are currently eroding our great Nation.

[TheGrayChannel]

@bluegrassaficionado I agree. Politics shouldn't come from the podium or the pulpit.

[TheBobathon]

@bluegrassaficionado there was no political commentary, just a single remark, that clearly wasn't serious, in 18 hours of careful and thoughtful exposition of quantum mechanics. And you haven't paid for anything, you're getting all these excellent lectures on 'interesting scientific concepts' entirely for free. Watch again, see if you can spot them :)

[bluegrassaficionado]

@TheBobathon Serious or not the point I was addressing in my post is still valid. And by the way, claiming he was clearly not serious is just a lame way of attempting to excuse inappropriate remarks. Political jabs are never just lighthearted comical relief; you can always read between the lines.

[TheBobathon]

@bluegrassaficionado why in heaven would you waste your time reading between the lines? Most reading between the lines is just projection and fantasy, saying more about the contents of your mind than the intention of the speaker.

Perhaps you see Susskind irresponsibly indoctrinating vulnerable citizens and you see yourself bringing clarity and independent thought, whereas I see a brilliant physics lecturer and an abusive and opinionated twat. We both have a right to our views, I guess.

[bluegrassaficionado]

@TheBobathon The intricacies of conversations in the English language and basic psychology must escape you. Reading between the lines is done in every conversation you hold with another human being, unless you take what everyone says to you at face value which would make you a gullible fool. Since you asked, I represent a voice of reason crying out to those too greedy and ignorant to realize how they are contributing to the erosion of the greatest civilization in the history of the world.

[TheBobathon]

@bluegrassaficionado Hmmm. Right.

[mrbeatdoctor]

Well if you are going to waste discussion time for politics then I would like to thank you for the wonderful mess that your guy left us with. Your political philosophy worked out just great. Things were so great after your republican ideas. lol

[bluegrassaficionado]

@mrbeatdoctor #1) Get out of America. #2) If you choose to stay, I hope you enjoy paying off trillions of dollars worth of debt foolishly incurred by the current, unconstitutionally reigning regime. Unless of course you're one of those lazy, liberal, facist-beast freeloaders who the democrats constantly let thrive by giving them the money of hard-working folks so they can pay you off to vote for them. Too selfish to look beyond your own greed, your ignorance will (and has) hurt us all.

[ccsitaround]

great lecture on a difficult subject--just wondering if the students are using a textbook(name?) for this course? thanks

[anacryon]

You may want to try "Introduction to Quantum Mechanics" by Griffiths, it has a similar entry level to these lectures

[ccsitaround]

thank you--just wondering/I'm using the textbook-Quantum Physics for Dummies about $5.00-- an excellent research source.

[nejtilsvampe]

@ccsitaround ironic name for it :P obviously quantum physics is NOT for dummies :P

[Entropy3ko]

The same textbook I used almost 10 yrs ago at university (now of course there will be newer editions of it :P).

A great book, very clear

[mrelectron111]

thanks for your answer : i hope for you to search about Dr. Jose Delgado - experiments on humans and animals, also there is Video about how he stoped a bull going to attack him _ check it Out!

[D33veeoss]

Elves? I'm not sure the exact attributes of Elves nor which types are reputed to exist. I would need to study the subject. We could start with point of origin and the practicality thereof.

At best we find something interesting.

At worst the issue remains a non sequitur as the facts do not follow the claims.

You study that, and I'll get back to what the good doc and myself are concerned with.

[D33veeoss]

Actually, before you worry about nanomites that embed themselves into the water supply, you'd first have to become alien proof, believe in Jesus, and send me $10 000. I'll pay you back soon, I promise. You'd have to stop taking vehicle transportation, stop eating food you did not drink yourself, not drink water you didnt filter or get from a well. No TV or anthing electronic as that could be what triggers the nanomintes, etc etc.

So let it go pending further information and insight. :) :) :)

[mrelectron111]

not Elves - Dr.Jose Delgado

I have read about Dr. Patric Flanagan neurophone, But Dr.Flanagan was able to make a deff to hear, But dr.jose delgado was able to alter behaviours

[D33veeoss]

If professors teaching about gravity start every new idea with how we've gotten from the idea that Newton had when his ceaseless mind pondered, the apple falling, every idea can be grasped naturally.

Teaching information and then showing the information's roots with expectation of understanding is completely backwards.

The excitement Newton felt is contagious even to lay-people. That's all I'm trying to say.

Convey the excitement of "Eureka!" and the audience will understand and innovate!!

[rmnbrw]

Wenn the problem in the QM was, that there were NO newton apple or something like this, from where they could derive something and exactily this was a hard part. They hat to GUESS things and than go back and check, why this COULD, not should be correct and other things not. At time QM was developed everyone was shocked because of this.

[D33veeoss]

That's exactly right. It's the ontologists of the day whose minds are designed to wonder.

"What do you do Mr Smith?"

"I operate on brains through advances in neurology."

What do you do Mrs. Smith?"

" I wonder about things like the exact practical application of each part of the brain. I test. I verify. I evolve theories that are conducive with reality. I then show the steady handed surgeon what to do."

I am the woman behind the curtain.

[orlanino]

I would dare to say that guys like Ponomarev and Fritjof Capra explained the matter better. And Ponomarev started from the apple.... he actualy enlarged the Planks' constant, and than described the "quantum" jungle and "quantum" billiard table. More clear than this

[D33veeoss]

I'm not one hundred percent sure of what those cats were all about, or their style, but I think you're talking about giving a defined picture/storyline/ visualization? (sorry to strawman you)

Anyway, that is tops, in my opinion, because as soon as a student is 'lost' , the teacher yells out, "Where Are You??" and the student can respond,"I'm at the spot where we all noticed the owl." "Oh OK, face S/E, do you see the mountain peak? OK head down that trail."

And all are united.

:)

[D33veeoss]

Practical implication.

Constant pondering, incessant battering, and natural born ceaseless wonder/curiosity of the practical implication of the surrounding facts (aka obvious reality) is what bore QMechanics, Theory of Gravity, Lightbulb filament and an understanding of relationship between matter and energy.

Do you feel the Newton apple tree story is trite?

That's EXACTLY what triggers great minds.

That's the precise model of how 'practicality meets imagination' will trigger these formulas.

[barry123www]

Eigen: adj. own, belonging to oneself or itself; typical, characteristic.

(from an online dictionary)

[AurumenK]

Wow this guy is a very good teacher. Thanks alot this has massively improved my understanding of qm.

[goddontplaydice13]

first of all, that was some good refresh on linear algebra, second of all, wow, i took basic qm before, but not in such a detail manner, i have to say; the professor did a very good job in term of lecturing. The good thing of video record is that i able to stop and evaluates the parts that i really dont get, and able to have complete notes 0.o, well for people a little bit on slow side, its a god sent lol. anyway i like physics, and thank you for the lecture, im looking forward to the next one.

[xdruguserx]

I appreciate this video a lot =D

I've been interested in all the quantam theories out there and this is starting to help me finally understand the math involved with all the theories =D

[rugbyanden]

Interesting... Then why have you not told us about this earlier? Muslims never came through during... say the middle ages... with advanced quantum research. Why, oh why, if you already knew all this?

Answer: Religious scholars write some text in general terms, and afterwards, anything that resembles that theory is used as "look we are right".

Its like me writing "one day the sun will go up" and then the next they when it does claim to be in contact with god. So please, speare the BS.

[4unitedMAN]

but thats precisely what they did.research your history and learn of the Islamic enlightment that lifted europe out of the dark ages dark ages, search for: Physics in medieval Islam.

please spare me your BS remarks.

[rugbyanden]

True, Europe has learnt quite a lot from the muslims during the dark ages, however, they never talked of this stuff, so please, spare us the fairytales... hrm, sry religion!

This is a video about quantum mechanics, not religion!

[redhat11234]

Ha, that is in NO WAY high school math, do not be a knob, he is being thorough to an audience that either has little Linear Algebra background, or has not seen it in many years.

[rugbyanden]

Its all taught in Mathematics E in the final year of high school (at least it is here)

[rugbyanden]

It bugs me that he spend so much time on the math. I mean seriously. Its a university course in physics and they are going through high school math.

[keysle]

what kind of highschool math did you take? i wish i knew what column vectors are...

maybe its just his vocab..

[tehinfidel]

If only there were some mechanism provided to skip ahead...

[chaztikov]

We really need to start teaching the chinese alphabet...I'd personally like to see more distinguishable symbols throughout math and physics...but who cares what I think

[cursorminer]

yeah, I think maths and physics notation in general is crap - possibly the main source of confusion and dismay in these subjects!

[jc00178]

susskind is just AWESOME... i am taking a quantum mechanics module in school now and can never understand wat the lecturer is teaching... ...but, with this , most of my doubts are cleared... now i have a better understanding of quantum physics. susskind wonderfully makes this rocket science understandable....

[senor47]

That's because he is very good in his field. That is why people want to go to these schools. You learn the hardest stuff in a simpler and more effectiive way.

[ThePaul1E]

@jc00178 yeah he is really amazing actually

[ArjenDijksman]

I'm not sure that what is said at 47:02 can ever be attained and verified: "In other words, if by one means or another, you created an electron in an eigenstate of some observable such as its position and you measure the position, the measurement will always yield every time the eigenvalue of the appropriate operator, the position operator."

There is always an intrinsic indeterminacy in the measurement of the position of an electron.

[demitrian21]

Suppose you measure the position of the particle you will get a value and that value can't change or be different everytime you measure it. With that said, the position of a particle is the eigenvalue. However, it is known that if you measure the position, you can't know the momentum of the particle. In this case, momentum is not the eigenvalue of the particle. Vise-versa, if u measure the momentum you cannot know the position of the particle. In this case, momentum is the eigenvalue.

[ArjenDijksman]

Well, I'm unable to verify this experimentally. Measuring many times independently the precise position of an electron yields different values under the same conditions.

[demitrian21]

Yes yes, one can't find the exact position but one can use the probability density function to find its position. When one get a value by using the probability density function that value is the eigenvalue.

[ArjenDijksman]

Great teaching of a difficult subject. I agree with davidw that this is more math than physics. It seems to me that we take QM the other way round. The state vectors should be seen as concrete representations of quantum systems while the complex components represent them abstractly, compare at 11:08: one way of describing vectors is just to describe them symbolically or abstractly as vectors but another way is just to give the coefficient A_m.

[ArjenDijksman]

I don't entirely understand the meaning of what is said at 1:01:00 : "You can't multiply a ket by x. You can only multiply it by the operator X." In its quantum course (chapter 20), Feynman multiplied the state vector by the position x, which surely is a real number.

[mrbluesky323]

I would guess feynman was just being sloppy... As I understand it, it's just a matter of technicality because it's meaningless to "operate" (multiplication is an operation) on a ket or bra with anything but an operator. It's just a matter of proper notation.

[ArjenDijksman]

Thanks for precision. However, focusing on technicality kills intuitive meaning of kets... I like Feynman's way that relates quantum laws to ordinary experience.

[mrbluesky323]

"focusing on technicality kills intuitive meaning of kets."

Yea, i pretty much agree... technicalities are a matter of tradition. The people who have been in the field longer use that notation merely because that's what people use. It's more important, i think, for communicating efficiently with experts in the field than it is for actually learning the material.

[barry123www]

The notation should make everything easier not harder. It is the product of years of refinement. Without the vector space notation, you are dealing with an even more complex integral formulation (Integrals involving the modulus of the complex valued wavefunctions) that obscures the simplicity of what's going on .

[D33veeoss]

Agreed. I can't learn with this method. The teacher needs to have a total circumspect understanding of the subject matter so that it can be presented in a method that is practically applicable at all times.

The ideas that he is teaching, he must not really understand.

Because the implications are mindblowing.

Inventors, in fact, start with possible mindblowing possibilities based on reality/fact.

(Turning fact into implication and testing)

That's how QM began.

It needs to be retro-taught.

[D33veeoss]

So that there can be complete understanding.

Let's take Dr. Phil or any PhD like him; they know the facts, but do not have any 'outside the box' or circumspect tact or natural ability.

This is because the minds that taught his teachers how to direct his thoughts, were not properly received. Understandably so. Who can teach someone to think like Einstein?

Constantly organizing information, to the point of OCD, so that their imaginations will feed off their conscious information.

Brilliant.

[blastbeating]

don't know if you're right or no, bus the notation is always a great deal in physics... some mathematicians joke about the bastard notation that we use in physics lol.

maybe the ket or bra are operators itself... please note the maybe and my very limited math knowledge

[NInails19]

Susskind Kicks ass!

[davidwizard2006]

we are more of studying math rather than physics

[Frostalzzz]

math and physics go hand in hand <3

[F00dTube]

Math is to physics as masturbation is to sex. :p