Answering a question from the audience on uncertainty principle; Complex numbers 6:00; Vector spaces 17:40; Dimension and basis of a vector space 26:30; The vector space of functions 34:00; Dual vector space 36:30; Inner product 45:00; Basis vectors 1:00:00; Physical Significance of vector spaces 1:15:00; Linear Operators 1:43:00

Leonard teaches these topic so anyone can understand it. He is such a brilliant professor that it is like learning kindergarten.

finished my physics degree a year ago this is second and third year material, hes a good explainer a recomend this to anyone revising.

Lecture 1 notes Quantum mechanics Notes( now you have no excuse so learn ) 1 Double split experiment Interference Pattern Momentum Energy of momentum E(energy) = P2 / 2m Otherwise P2 / 2m P2 . 2m To put that in to simple terms that’s Energy = Momentum Squared Divided by Mass times 2 Energy = Momentum Times momentum divided by Mass times 2 It also = ½ p times P/m = ½Pv Symbols and there meanings .d= distance that it moves D= Delta X= Position ( in simple terms ) d(Delta X ) = Vit + ½at to the second power . I = imaginary number such as square root of negative 1 . A= acceleration W= Angular Frequency (W=2pieF) P= Momentum M = mass V = Velocity p/m C = speed of light C = ^F F= frequency of a wave measured in seconds T= time ( 1/F) ^= Landa = 1 wave length per cycle . in other words ( the distance of a frequency . Velocity is Landa devided by T In Conventional E= CP There’s a connection between the frequency and the wave length Momentum divided by mass is velocity Energy is velocity times Momentum Momentum is energy divided by the speed of light For Complex numbers www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut12_complexnum.htm Energy = Hf = hw P = ĥf/ c E/c=p Reverse E=CP . tell you what H and h and ĥ mean later . just refer to it as Constant H = h with loup h= h with bar And ĥ is just h . ^= Landa = Wave length of light , wave length of anything . The time it takes is Inverse of frequency T=1/frequency The distance is ^ Landa Whats the velocity of the wave C=^f C/^= f Remember that last equation with the H’s E=Hf = hw P=hf/c P=hf/c=h The h’s= Plasnck’s constant ( there’s more then 1 )if you need help go to www . answers.com/topic/planck-s-constant 1. Definition of Planck’s constant (n) Bing Dictionary o Planck's con•stant 2. basic physical constant: a basic physical constant that is equal to the energy of a photon divided by its frequency, with an approximate value of 6.6261 x 10-34 joule-seconds. scienceworld. wolfram.com/physics/h-Bar.html H= basic physical constant: a basic physical constant that is equal to the energy of a photon divided by its frequency (F) h (H bar)= In physics, Planck’s constant is the proportionality constant between energy and particle frequency: E = hν. When working with angular frequency ω = 2πν, it is convenient to introduce a new constant ħ equal to h/2π so that E = ħω. The symbol ħ is simply pronounced “h bar” and is sometimes called the reduced Planck constant. www.johndcook.com/symbols/2014/02/plancks-constant/ H= E/f h(H-bar) = H / 2 pie (Side Note:The frequency for light waves is 10 to the 15th power for ordinary light ) Calculators are needed in quantum mechanics ( Science Calculators ) other wise you find yourself writing a equation that may take up a whole Board or even 2 …… that’s with approximate numbers not even exact . Now lets get back to that Equation Einstein had told them E= HF =hw( that F is the frequency of the light describing the Photon ) (E/f= H or E/H=f) (Remember with the correction from the theory of relativity E= P squared / 2m = ½p times p/m = ½PV E= ½PV ) So on one side you have E=CP ( C being the velocity (and the speed of light ) P ( momentum ) On the other you have E = ½PV (and if we were using the speed of light it would be E= ½PC ) See the difference E=CP ( without relativity ) E= ½PC (with relativity) The only difference is the momentum is half . P=E/C ( Momentum is energy divided by the speed of light) Now here’s were things come together P(Momentum) = H(Remember H=/EF) F( Frequency) Divided by c(the speed of light) Or in math terms P=HF/c |P=HF/c| Remember C=^F F=C/^ That is the speed of light = Landa( the distance of a wave) times The Frequency . and to get the Frequency That is Frequency = The speed of light Divided by Landa( the distance of a wave( Wave length ) So when we plug it in other words V= ^/t {T=1/f } |P=HF/c| P=HF/c=Hc/c^ The C then cancels Hc/c^ = H/^ Basically P=H/^ Planck’s constant divided by Landa( the wave length ) The smaller the wave length (^)Landa) the larger the momentum(P) Thus Momentum and wave length are inverse to each other ^ < (Delta X ) Definition : Delta X ( Is Triangle X symbol ) But to make it easy for you I will just put D ) : D( Delta X ) is the position "delta," a Greek letter, typically stands for a change in (whatever the variable is). One of the more useful uses of delta x and delta t is to calculate velocity in the x direction. For example: If you start on a footpath (x) at your house (x = 0) and walk to the outhouse a hundred feet away (x = 100 ft), then delta x = (100 - 0) = 100 ft. www. chacha.com/question/how- do-you-calculate-delta-x-in-physics ( D= Delta X ) d= ViT + ½ at 2nd_power Often, when delta x is used in this fashion, you will also see the time period written as a delta quantity: That is D(Delta X ) = Vit + ½at to the second power That is in words Delta X = Velocity(Imaginary number) Times Time + Half of acceleration times time squared . That’s a mouthful now you see how a equation could literally Take up a whole board if it wasn’t simplified . Delta X here is the Position though D= Delta X= Position ( in simple terms ) Back to the equations P=HF/c| P=HF/c=Hc/c^ The C then cancels Hc/c^ = H/^ Basically P=H/^ ^ ket vector You can multiply any vector by a complex number and get another vector A|a> = |b> If you have 2 vectors any 2 vectors you can add them and get another vector |a> + |b>= |c> You can also go to Video 1:32:17 to understand this A|a> + B|b>= |c*> Si the symbol in 1:34:30simply means scienceworld.wolfram.com/physics/SI.html System international Funtions of 1 varible [si](x) = [si]r(x) + i[si]i(x) (r meaning real values) complex numbers in a nut shell )

At the beginning, he is talking about complex numbers as vectors spaces, it would be interesting to note the following: Complex numbers form a 1-dim vector space over a complex field C. But it can also be seem as a 2-dim vector space over the real field R. When you show a complex number in a x,y graph you are kind of representing it in the second interpretation. That is why, I think, he said that he wasn't drawing the vector space, just a XY plane. So the representation of vector fields over complex numbers have this problem of needing 2*n dimentions to represente. Which means that we can't visualize vector spaces of more than 1-dim over C.

im a physics student taking QM and this was immensely useful for explaining the physical meaning of dirac notation !!!

this lecture series is perrrrfect. it's comprehensive but to the point. i want to go beyond the silly pictures and analogies of popular physics but i dont want to spend 4 years doing it. this is great!

No better way to spend my free Saturday nights :)

4:00 on uncertanty 4:18 question 7:13 complex numbers review 13:30 complex conjugate of product 17:42 complex vector space 26:28 dimension of a vector space 32:56 on functions 34:34 conclusion 35:42 dual vector space 43:24 question 44:17 inner product 44:30 dot procuct 46:30 inner product 51:40 54:30 question 55:40 for functions and vectors 59:55 basis vectors 1:08:25 coeffeicents of a vector 1:14:43 significance 1:23:14 rules for 2 states system 1:25:12 6 state system - die 1:26:40 meaning of adding state vectors 1:32:20 -1:33:57 2 postultes of QM (revision) 3 questions till 1:40:55 1:41:58 operators 1:45:44 linear operators 1:47:35 examples and hometask

Bless you fellow traveller, well met, and good hunting to you.

Grandpa Susskind, we thank you. I mean that with all the respect in the world. Not an insult.

love the introduction of kronecker delta!

Susskind is one of the best.. Clear,,,,,, but deep, undo so clear. Thaks!

To nitpick something he said, basis vectors need not be orthogonal. In two dimensions, all you require is they don't span the same line.

Great lecture.

Thanks for the lesson

Thank you very much proffesor: Leonard Susskind's

Definition: a *vector space* is a set that is closed under finite vector addition (+) and scalar multiplication (.). Definition: a *complex vector space* is a vector space whose field of scalars is the complex numbers.

Nice experience of leareng QM from father of string theory

@mdinka ... braA | ketB is just notation (Dirac notation) for the inner product, which is DEFINED as sum (over n) of (An*)(Bn) where An is nth component of vecotr A and note An is a complex number.

@roboflop314159 In this very simple theory creation is a continuous process and the flow of time is formed by the continuous inward absorption and outward emission of light or EMR. The process that forms Heisenberg’s Uncertainty Principle also forms the Arrow of Time!

Thanks for this

actually, this formulation of quantum mechanics is fairly self-contained. You don't need to do any classical mechanics first. You can do them in either direction. Two different ways of thinking

I came for Modern Physics and left with a better understanding of Probability and Statistics..

Beautifully clear.

... or equivalently ketB is analogous to a column vector A. And braA is the transpose conjugate (*) of the column vector ketA. So in this matrix representation, braA | ketB is the same as our original definition: sum (over n) of (An*)(Bn). It does NOT make sense to write ketA | ketB... that would be like trying to multiply 2 column vectors (dim Nx1). Inner product is a number (scalar), so you must use Dirac notation that is consistent with that result.

Linear operators are closely related to linear transformations in linear algebra (a matrix)... I suggest you watch 3Blue1Brown essence of linear algebra youtube series. As a side note, squaring is not a linear transformation because there is no transformation matrix that you can find that could possibly give you the desired output (each vector in the space transforming so that its length gets squared.... not possible!)

Thanks Suss

@aaronasterling you are right, I don't think QM can get clearer than this.

Thank you very much for these lectures......

It's just the definition of the dot product in that space. As the guy is saying it satisfies the necessary requirements of a dot product. Dot products can be defined differently in all different kinds of spaces.

@jamma246 - my only point (and I think I was not very clear about it) is that for higher dimensional complex numbers (eg quaternions having 2 additional imaginary terms), it is not true that (ab)*=a*b* if a,b are quaternions. For same, it is true that (ab)*=b*a*. This result is consistent with the fact that quaternions (eg a = 1w+ix+jy+kz where w,x,y,z are reals) do not commute, b/c in the basis (1,i,j,k) of a quaternion, ij = -ji while ii = -1.

@cesarjom He doesn't say that they don't commute. (ab)*=b*a* means that (ab)*=a*b* by commutativity of the complex numbers. Writing it that way doesn't make anything wrong, but it works for later on.

This man can teach me quantum stuff . Loved these ❤️ videos

Well, delta can mean different things depending on what you are talking about. Obscurusvita told you that delta meant change. In classical physics, some times change is represented by an arrow over the variable and not delta and some times by delta. In the case of this lecture, Delta stands for the undertainty priciple. Go to google or wikipedia and type Uncertainty_principle to get an idea of what delta stands for in this case.

@hackulous Yes! but when the superposition collapses it will only have one position in space and time. Could this not be a new moment in time as part of the time continuum? Because classical physics is an approximation of quantum physics is it not possible that Heisenberg Uncertainty Principle is the same uncertainty that the observer will have with any future event?

@nementerc... note that a great theoretician does not neccesarily mean you have a great communicator (ie lecturer) particularly to the layperson. Such is the case of Dr Susskind, his insights into his areas of physics are priceless to graduate-level and follow physicists where he can provide much more detailed mathematical arguements... for the layperson there is not much advantage as Susskind lacks the eloquence and succinctness of say a Richard Feynmann.

I'd love an educated response. When electrons are spayed at a target and show equal portions of spin up and spin down, does the detector respond by demonstrating the opposite spin? I have no idea what is used to detect spin, but I imagine it is a particle, photon or another electron. So a spin up electron hits the target, and the target thing is coupled in the opposite direction? I ask this because my impression of an electron is it is both spin up and spin down until detected. And that would suggest an opposite state on the target end.

thank u sir

@jamma246 they need to be orthogonal due to the practicality of not needing to calculate several integrals. He just didn't mention it at this point.

I fell asleep listening to a Mr Ballen video and woke up with this on

1:21:30 When he was describing vector spaces of Heads and Tales . You know how 1 vector plus another makes a 3rd vector. I just figured out the 3rd vector of heads and tales ( logically speaking ) the rim of the coin when the coin lands on its rim .. its probability would be low for it to just land and stay on its rim ,but in addition it would be High if the rim just touched the floor and it falls back to heads or tales . That's a interesting idea it almost always lands on the edge of it then falls .

... more general any Z vector with coefficients in Zn is a vector space ... taking the sum and product as modulus n operations

@Mattprole Eucledian geometry is no less difficult, it's just more familiar.

I would be nice if questions asked were written across the video, since audio of them is unheard. Anyway, goooo Leonard! Quantum the hell out of those Mechanics!

@unimoggie Yes, I think it does. We know that alpha h star times alpha h is a positive real. He just stated that probabilities should also be positive reals, so I think it would logically follow that he means the complex conjugate. Also, he hasn't given any other definition of star other than complex conjugation. So why should we assume it has a new meaning.

What if we were to replace the surface of the Coin or Die, with a viscous fluid? That it becomes non discrete. Would the use of set still be preferred or could vectors be used?

58:35 the inner product of the two function vector spaces looks exactly the same as for ordinary vector spaces. Integral over dx of the functions sums all components over the interval. Instead of φ*(0), φ*(1) .... you have φ*(0), φ*(0+dx) as the next component and so on.

it would be great if we could hear the questions being asked

Could Heisenberg Uncertainty Principle be the same uncertainty that the observer will have with any future event? If time and the geometry of spacetime is continuously being formed by the momentum of EMR or light. 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?

@fantar132 could you elaborate because susskind is saying that indeed a single particle is affected by the uncertainty principle

"When waves from the universe enfolds down the spiral vortex of an atom, it will recoil, expanding into three-dimensional spacetime." "The magnetic fields are always enfolding+/-unfolding spacetime at right-angle's." "The symmetry of spacetime is broken where the two waves meet, creating the wobbly particle effect. As these like charged wave-center's repel becoming equally spaced around the spiral vortex, these polarized wave-center's share the same expanding moments of time until acted unpon."

Nice lecture leading gently into the subject. I miss two things that help to have some intuitive representation of abstract concepts (maybe it will be handled in a later lecture): 1. in the inner product, the ket represents the initial state and the bra the state in which the system ends up (or is projected). Feynman explained that nicely in his introduction to QM. 2. the Argand representation of the complex plane, which helps to visualize the changes of phase in quantum systems.

Actually the space of boolean vectors is in fact a vector space if it is defined over the boolean field ... it means that the coefficients are also booleans. The only difference is that now product is "and" and sum is "or" ....

I think this man is only in this dimension part of the time.

run this at 2.0x speed .. and u have a overview of quantum mechanics in just 10hrs .

Thanks for sharing these great classes. Could someone explain how/why we can mathematically consider the dual vector space as the conjugate space ? The way he describes it makes it sounds like it's easily deduced from the definition of a dual space, but for someone like me who has not done mathematics for a while it is not that obvious. (He does not just say that there is a bijection between the conjugate and the dual, but that they are equal)

1:13:10 so this is sum over instead of sum over i? Is this because vsubi was replaced with the braket?

in 26:30 to satisfie the facts that he tells after , should'nt V1 and V2 be perpendicular to each other?

unable to download these videos i have low bandwidth net connection so i used dap but halfway through it started asking for authentication cant download through itunes either because it doesnt support pause/resume i have slow internet but i want these videos plz help

What is the difference between these lectures, and his lectures in Theoretical Minimum: Quantum Mechanics course?

@floopsie666 if you know how to solve diff eq, (an undergrad physics requirement) then you can solve many problems of the schrodinger eq.

Don't you know how hard you whacked the electron with the photon? Can't you figure out the electron's velocity before the whack by conservation of momentum?

ah sry 1 more thing what you said before means that, a non-independent couple of vectors can create a system. but an independent can create a cartesian?

Susskind: "It's just Linear Algebra" Me: "Hum, I clearly remember the linear algebra text book saying that It was talking about just finite dimensional spaces"

at the risk of sounding slow. if i follow this correctly; it would seem that it would truly be mathematically impossible to violate the uncertainty principle since there would always be a value associated with you looking at the system.

hey guys how can i find more information about the course, like transcripts, textbook used, lecture notes, assignments, course web page....etc

Vector, this is our director.

@QuaterionEM I specifically say we have no understanding of time in physics because we cannot explain why we have a future and a past. We can look back in time through light years of space at the stars. But when we look down at the atom we only see probability. Could this be because Heisenberg Uncertainty Principle is the same uncertainty we have with any future event?

How much of these help address global warming and CO2 issue?

Susskind makes interesting generalizaion: (ab)*=b*a*... this implies non-commutative prop however for a 1-Dim complex a,b this is not true, they do indeed commute and (ab)*=a*b*. It is not until we generalize complex numbers a,b to a higher dim 3-Dim or greater that we run into non-commutative prop. Let a be a quaternion such that a=1w+ix+jy+kz, over a vector space spanned by basis (1,i,j,k) where i,j,k are complex such that ii=jj=kk=-1 and ijk=-1. Only then (ab)*=b*a*.

try going to there site and sarch this lecture.

you can learn any university course you want for free, if you put in time and effort, the DEGREE is what you won't have.

I would like better picture quality.....Pretty tough to understand.

How many lectures are there in total in QM?

@jamma246 Oh, he meant orthnormal basis.

thnx :)

How can something being so fundemental to the existence of things be so asymmetrical? How is there so many questions that cannot be answered through scientificly known theories? There is so much for human kind to learn.

Why some people insist Susskind should explain "clearly"? QM is not something you can find the analogy from our macroscopic experience. Just get used to it.

all of the forces are the universe abstracting from the 3D matter the 4-dimensional destruction of it. By holographing properties that we observe from particles, we observed that their activity requires 11 dimensions to describe, but it's all meaningless to us, because in reality it's just the completely incomprehensible barrier necessitated by 3 spacial dimensions.

You mean by a complex number with i part zero? Yes, that is a real number such as plain number 5 is " 5 + i0 " .the imaginary part is zero. The regular real integer numbers. Vectors are like complex numbers.

I understand that, but the Degree is important, that is if you want to be hired under your particular field. You will not be hired despite your intelect and research over someone who has a Degree.

I don't understand why he says that in classical mechanics we don't use vectors. We use position and momentum as vectors and we could add for example two position vectors and get just another position vector (the same with forces or fields or other vectors) and we can define inner product (dot product). We can also define a basis for three dimensional space (versors x y z). So I don't really understand what he means when he says that in classical mechanics we use points of a set and in quantum mechanics vectors.

this is a very interesting subject and id like to learn it. but ts just too long complicated and hard.

we never saw michael :(

yup

I just wanted to express my dumfounding thanks for the unbelievable generosity of Leonard Susskind and Stanford University for providing a free Physics education online from who in my opinion is quite genuinely the best physics teacher on the planet... No #Whiteprivilage necessary

How did I get hear, not mad that I am just confused

Anybody can watch Lady Gaga and "enjoy" it, but you need a undergraduate treatment in Math(s) for STEM majors to even understand this. He does a good job of giving a refresher, but you need Algebra, Calculus, Linear Algebra, and at least some Physics under your belt before you can even tackle the first two videos.

Bro, I fell asleep watching a chicken boiled in nyquil video, and ended up on this

Good luck, but I recommend to first try to understand some Linear Algebra and Classical Mechanics first. Especially the maths and terminology would make far more sense.

I don't know if at 17:12 B is the complex conjugate of A

nice vid

amazing lectures , but i have a complaint , science is valuable we should improve the quality of recording lectures and sharing notes and books for those who want to learn , why the useless videos on youtube is recorded with high tech equipments and animations while the important material that actually matters is not in the same quality . keep spreading science in the way it should .

Ugh, that's why I'm so lost, I was watching the entanglements one lol.

I'm getting a message stating that lecture 1 is for private audiences? Is anyone having the same problem?

Nice one. Put up some identity, then we'd be inclined to give you some cool links depending on what you're into. Welcome anyway, its pretty mad out here, but you can learn a lot if you want to. More than one ever dreamt.

Briliant!

I am currently majoring in particle physics actually. I'm starting up quantum mechanics this coming semester. I was just voicing my opinion. Sorry. =/

@ICarnag3I Linear Algebra