* @1:45 Linear operators * @4:00 Basis of vectors * @11:20 Components of a vector (representation of vectors in terms of basis vectors) * @14:20 Multiplication of linear operators * @19:36 Hermitian operators definition * @26:00 Eigenvalues and eigenvectors review * @30:38 Theorems on Hermitian operators and their eigenvalues/vectors properties * @40:10 Postulates of quantum mechanics * @54:24 Probabilities of measurements by a given observable * @56:00 Motion of a particle on a line * @59:23 Observable corresponding to the particle location * @1:07:34 Dirac delta functions as location eigenvectors * @1:13:15 Location wave function and probabilities * @1:20:20 Observable corresponding to the particle momentum * @1:28:33 Integral by parts formula * @1:34:00 Eigenvalues/eigenvectors of the momentum observable * @1:43:23 Momentum interpretation * @1:47:40 Incompatible quantities
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!!
Quantum mechanics Lecture 2 notes The distance of separation is L plus or minus Delta If you have 2 delta’s measuring the length between them . remember delta is position L+-D And when you want to find how fast it was moving you divide it by the time L+-D ____ T Or L/t +- D/t ( velocity ) Uncertain princible h( H-bar ) over delta h/D Review of complex numbers 8:15 Z= x + iy Z*= x - iy |z> Ket vector Okay what if (Z1*,Z2) and I take its complex conjugate 14:28 (Z1*,Z2) *= Z2*,Z1=> Z1*, Z2 interchanging 16:45 Complex vectors = * |A> . you can multiply them by complex number @ being complex number @|A>=| C > 19:20 Were x is real and @ is Complex Remember i is imaginary Square root of negative 1 @(X)=@r(X)+@i(X) These form a vector space Because you can multiply them by other complex numbers and get complex vector space And there are Colom vectors 23:30 Again the numbers are not variables they just represent that complex number ( a1 , a2, a3, a4 ) And you can indeed add colom vectors (a1, a2, a3, a4 ) + ( b1 , b2 , b3 , b4) = ([a1+b1] , [a2+b2], [a3 + b3] , [a4 + b4]) You can also multiply vectors by constants lets say @ is a constant @( a1, a2, a3, a4 ) = ([@a1], [@a2],[@a3],[@a4]) 25:00 Duel vectors are complex conjugate’s The inner product of a vector with itself is real = * and its always positive Z*Z (X+ iy)(X-iy) ì cancels. So the imaginary part cancels (X+ iy)(X-iy)= X squared + Y squared = Real and its always positive Square of the length of a unit vector is 1
خقق على الاقل ولا خلص حدًا خههغص انا ولا انتي اختي غلط ولا لا ولا اقول اخ اخ خلص انا اصلا انا هههههه خصخصة حفه والله انا اصلا انتي اول باول انا انا اصلا انتي اختي انتي ولا انتي لازم تعرفي اني انا اصلا انا ☝️ خخ انت ولا انتي ولا لا ولا خلص انا خ انا انا ولا انتي ولا لا انا هيك انتي اختي انتي ولا انتي خالتي انا انا اصلا انا ولا لا انا اصلا انتي اول باول هههههه لحهع انت الي لازم خولا انتي فحهخهخ انت ولا انتي هس ع ان لا لا انا ولا اختي انتي لازم تعرفي اني احب بنت اختي انت ه انت اصلا ص انا اخوي انت اصلا انا ح انت الي ه انت اصلا انتي اختي انتي ولا لازم اكل ولا لا انا خذي عحالك ولا لا خ انا اصلا انتي اختي انتي ولا انتي غالي انت اصلا انتي اختي انتي ولا لا انا ☝️ ولا ها لا انا اصلا هفه انا انا اصلا عهع انا خ انا اصلا انا ولا انا انا ه ولا عخ انا اصلا انتي بنت اختي انت اصلا انا انا اصلا اصلا انتي اختي انتي ولا انتي لازم احكي انتي انتي اختي ع هه ليش ح لا لا انا اصلا انا خخ ن انت عنجد انا نخل هلا غير صح انا اصلا انا ولا هيك هيك خ ولا لا انا اصلا انتي اختي انتي ولا حغ ولا انتي لازم هلا انا عع هلا هلا عنجد هلا هلا عادي ولا لا لا انا اصلا انتي ح غ انت الي خرر. لرد. زراصلا انا
23:33 For anyone's who's interested... Property of a Hermitian matrix: transpose(L) = conjugate(L) Notation: T(A) => transpose(A) HT(A) => hermitian transpose (conjugate transpose) . => matrix multiplication * => complex conjugate L => hermitian matrix (H would have been confusing with the HT() hermitian transpose notation used) Starting with the identity... HT(A).L.B = HT(A).L.B Because L is a hermitian matrix... HT(A).L.B = HT(A).HT(L).B, Because both sides are a row-vector times a square-matrix times a column vector (1xn).(nxn).(nx1), each side of the equation evaluates to single value (1x1 matrix). A single value is unchanged by the transpose operation, so you can transpose just the right side of the equation... HT(A).L.B = T(HT(A).HT(L).B)) distribute the transpose and reverse the order of multiplication... HT(A).L.B = T(B).L*.A* pull out the conjugate... HT(A).L.B = (HT(B).L.A)* and rewrite in Dirac notation... = *
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....
Quantum mechanics Notes 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 other words V= ^/t {T=1/f } 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 |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 )
These lectures are brillliant! So much more interesting the the lectures I had at uni. Much more funidimentally based then the stuff we did. Although haveing said that, I probably would have struggled with the abstractness of these lectures without having two years of quantum mechanics already stored in the vaults of may brain; and also topological spaces!
at 6:31 he says the matrix elements form an mXn matrix, but he meant a DxD matrix ( m is an index ranging from 1 up to D, when D is finite ). although the expression “mXn” matrix is frequently heard when discussing linear operators from R^n to R^m.
1:32:27 This may seem trivial to you, but this idea of “Eigenvalues” of an operator is mind blowing. I am used to simple eigenvalues for simple matrices.
To a physicist this is the same thing as with matrices. To a mathematician this is a lot more complicated, but physicists are usually glossing over the complications.
An eigenvector is the "proper vector" corresponding to the "proper value" (eigenvalue) of an obserable. Once you have identified your eigenvalues, you then calculate your eigenvectors associated with those eigenvalues. However, remember that the eigenvector must be normalized as soon as you calculate it, so that it is a unit vector in the basis space of the operator. In other words, the eigenvectors of both the position and the momentum operators are all of magnitude 1....
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.
The inner product (that's your integral S(f.g')dx) is taken across the whole line from -infinity to infinity. He just assumes that the function Phi is going rapidly to the zero so "it's values on the border in the infinities are zero". And that's the missing term in the per-partes formula.
@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.
@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.
@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.
@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
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
@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.
Took me a while to find the whole in the Stern-Gerlach Experiment. Not a spooky as it seems, still have to go through all the experiments however the results make absolute sense once one assumption is removed. Assumption, the electron is causing this effect. What about all the up and down quarks? Do you think that maybe they have something to do with this? Here is how it would hypothetically work; Every Atom has an electronic field in it, not unlike the earth but probably more complex with quite a few polls depending on how many neutrons and protons there are. In a beam the atoms become entangled up and down, once they hit the magnetic field the up oriented will move one direction and the down oriented will move the other; however once out of the field they will reorientate themselves. Then when they hit the new magnetic field the process starts over again. That would also explain why Neutrons have a "Spin," even though they are neutral. They still have a field, however the total sum is zero. The same with atoms, they have fields and unless they are ions, their total magnetic field is zero.
Could we have an objective understanding of quantum mechanics if we explained it as an emergent interactive process unfolding photon by photon? This idea is based on: (E=ˠM˳C²)∞ with energy ∆E equals mass ∆M linked to the Lorentz contraction ˠ of space and time. The Lorentz contraction ˠ represents the time dilation of Einstein’s Theory of Relativity. We have energy ∆E slowing the rate that time ∆t flows as a universal process of energy exchange or continuous creation. Mass will increase relative to this process with gravity being a secondary force to the electromagnetic force. The c² represents the speed of light c radiating out in a sphere 4π of EMR from its radius forming a square c² of probability. We have to square the probability of the wave-function Ψ because the area of the sphere is equal to the square of the radius of the sphere multiplied by 4π. This simple geometrical process forms the probability and uncertainty of everyday life and at the smallest scale of the process is represented mathematically by Heisenberg’s Uncertainty Principle ∆×∆pᵪ≥h/4π. In such a theory we have an emergent future unfolding photon by photon with the movement of charge and flow of EM fields. This gives us a geometrical reason for positive and negative charge with a concaved inner surface for negative charge and a convexed outer surface for positive charge. The brackets in the equation (E=ˠM˳C²)∞ represent a dynamic boundary condition of an individual reference frame with an Arrow of Time or time line for each frame of reference. The infinity ∞ symbol represents an infinite number of dynamic interactive reference frames that are continuously coming in and out of existence.
@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.
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?
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.
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.
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.
When he talks about the probability of measuring an eigenvalue P(λ) = where λ is a complex number how is it possible to get a complex number as a result of an experiment?
@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
Thanks for precision. However, focusing on technicality kills intuitive meaning of kets... I like Feynman's way that relates quantum laws to ordinary experience.
I'm not sure you'll like this answer :o)... Although it is possible for the numerical values "lambda" to be equivalent, the actual eigenvalues of the operators are going to be in the units of that operator. Consider a Harmonic Oscillator, for example; the lambda roots of the characteristic equation for a 3x3 position operator are {0, +SQRT3, -SQRT3} , but the actual eigenvalues are {0, +SQRT (3hbar/2mw), -SQRT (3hbar/2mw)}....
@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...
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.
I should have phrased my question more adequately. Since eigenvectors are normalized by definition, do momentum and position eigenvalues all every exactly coincide. Like lambda equaling {3,1,2} for both the position and the momentum.
Probably a silly question, but is the delta function orthoganal? If you take lambda + epsilon as the value you are looking for wouldn't this clash with lamda? I've probably missed something important!
It seems to me that in order to describe the momentum of a particle, you need at least two dimensions. i.e. one is not enough since an x-ray and a radio wave both travel at the speed of light in a vacuum. Usually when we apply energy, it is in one direction. even if it is a wave of energy, it seems as though it is just energy escaping into different dimensions, even if it is a planar one trapped in 3-dimension space. Similarly, quantum entanglement might just be energy escaping into a different dimension, this one with dimension >= 5 as it is invariant to both time and space. (3+1 = 4). If this is true, then it might take zero energy to rotate the plane of a wave around the axis of direction. In other words it might do it spontaneously i.e. waves could spontaneously change their perpendicular axis relative to direction. Does this have implications for higher dimensions? Does direction for instance have many meanings?
can the eigenvector describing position be of the same length as an eigenvector describing the momentum? (this would mean they are just in different directions, and that the eigenvectors of the position and momentum of a particle aren't necessarily orthogonal). This may be a stupid question...
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.
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.
The p^ operator gives roots of {0, +(Im)SQRT3, -(Im)SQRT3}, and its eigenvals are {0, +SQRT(3mw*hbar/2), -SQRT(3mw*hbar/2)}. My point is to make sure you are distinguishing the roots of the characteristic equation (calculated from the determinant) and the actual eigenvalues. Another glance: Det|A-V*1| = 0.... V=lambda. V must have the same units at A, but we tend to drop the coeff's (h-bar, w, etc) to work the determinant. So, we really work Det|a-v*1|=0, where A=(const)*a, V=(const)*v...
@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
@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
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)
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.
When I took linear algebra I didn't understand what the hell we were doing with all this. It doesn't make sense until you study it in diff eq and quantum....
@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.
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
+Alin Catargeanu Ψ (and thus Ψ*) are defined such that Ψ*Ψ converges. This came up in the first example, where int(xΨ*Ψ)dx was real and bounded. Hence the uv term after integration by parts amounts to taking functions at values (+/- infinity) where they will be zero.
bfyguy I actually don't know. Quantum entanglements really should be after it. I think its better to watch this first becuz you can understand it even without entanglements and then you can watch entanglements and you will gain a deeper understanding.
@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
i'm still curious is studying too much physics can cause baldness for men? i watched several lectures and different physics lecturer and almost all of them are bald..
Are you being offensive, disrespectful, or just making a really bad joke? I would have thought that if you are here watching this lecture you would be more at owe with this incredible teacher!
i just want a scientific answer and it's serious as part of satisfying my curiosity..I admire those people that teaches science in youtube for me to have a privilege to discover more because i have no money to pursue school so i rely on watching lectures especially to this man and those other professor's in physics that i had watched..I am sorry if it's too offensive but that's part of my observation, if nobody could answer it clearly then i'll have to do the job to discover it..
LoL i am studying physics for discovery and to satisfiy my passion..and i'm not aiming for that shitt Nobel Prizes because i will not carry that after i die..
@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.
@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 :)
@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.
You dont know me but you dont know nothing until you know this blog thats being started to describe QM, quantum theory is wide ranging, Quantumtheorypageone dot blog spot dot com , quantum mechanics would be all inclusive most physicists dont know what QM was means to us regarding how to predict to know what we are doing all together, regards what Relativity etc Micro Mechanical Occurences condition the means, Economy is much deeper than most can know its takes guts to know this'
* @1:45 Linear operators
* @4:00 Basis of vectors
* @11:20 Components of a vector (representation of vectors in terms of basis vectors)
* @14:20 Multiplication of linear operators
* @19:36 Hermitian operators definition
* @26:00 Eigenvalues and eigenvectors review
* @30:38 Theorems on Hermitian operators and their eigenvalues/vectors properties
* @40:10 Postulates of quantum mechanics
* @54:24 Probabilities of measurements by a given observable
* @56:00 Motion of a particle on a line
* @59:23 Observable corresponding to the particle location
* @1:07:34 Dirac delta functions as location eigenvectors
* @1:13:15 Location wave function and probabilities
* @1:20:20 Observable corresponding to the particle momentum
* @1:28:33 Integral by parts formula
* @1:34:00 Eigenvalues/eigenvectors of the momentum observable
* @1:43:23 Momentum interpretation
* @1:47:40 Incompatible quantities
Thank u ,dude
I fucking luv u my man
❤thank you very much
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!!
Quantum mechanics Lecture 2 notes
The distance of separation is L plus or minus Delta
If you have 2 delta’s measuring the length between them . remember delta is position
L+-D
And when you want to find how fast it was moving you divide it by the time
L+-D
____
T
Or L/t +- D/t ( velocity )
Uncertain princible h( H-bar ) over delta
h/D
Review of complex numbers 8:15
Z= x + iy
Z*= x - iy
|z> Ket vector
Okay what if
(Z1*,Z2) and I take its complex conjugate 14:28
(Z1*,Z2) *= Z2*,Z1=> Z1*, Z2 interchanging 16:45
Complex vectors
= *
|A> . you can multiply them by complex number @ being complex number
@|A>=| C >
19:20
Were x is real and @ is Complex Remember i is imaginary Square root of negative 1
@(X)=@r(X)+@i(X)
These form a vector space
Because you can multiply them by other complex numbers and get complex vector space
And there are Colom vectors 23:30
Again the numbers are not variables they just represent that complex number
( a1 , a2, a3, a4 )
And you can indeed add colom vectors
(a1, a2, a3, a4 ) + ( b1 , b2 , b3 , b4) = ([a1+b1] , [a2+b2], [a3 + b3] , [a4 + b4])
You can also multiply vectors by constants lets say @ is a constant
@( a1, a2, a3, a4 ) = ([@a1], [@a2],[@a3],[@a4])
25:00
Duel vectors are complex conjugate’s
The inner product of a vector with itself is real
= * and its always positive
Z*Z
(X+ iy)(X-iy) ì cancels. So the imaginary part cancels
(X+ iy)(X-iy)= X squared + Y squared
= Real and its always positive
Square of the length of a unit vector is 1
It's dual vectors, not duel vectors. Duel vectors is what musketeers used to fight with each other, aka. swords.
@@Alkis05 sometimes vectors shoot each other. We persevere.
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خقق على الاقل ولا خلص حدًا خههغص انا ولا انتي اختي غلط ولا لا ولا اقول اخ اخ خلص انا اصلا انا هههههه خصخصة حفه والله انا اصلا انتي اول باول انا انا اصلا انتي اختي انتي ولا انتي لازم تعرفي اني انا اصلا انا ☝️ خخ انت ولا انتي ولا لا ولا خلص انا خ انا انا ولا انتي ولا لا انا هيك انتي اختي انتي ولا انتي خالتي انا انا اصلا انا ولا لا انا اصلا انتي اول باول هههههه لحهع انت الي لازم خولا انتي فحهخهخ انت ولا انتي هس ع ان لا لا انا ولا اختي انتي لازم تعرفي اني احب بنت اختي انت ه انت اصلا ص انا اخوي انت اصلا انا ح انت الي ه انت اصلا انتي اختي انتي ولا لازم اكل ولا لا انا خذي عحالك ولا لا خ انا اصلا انتي اختي انتي ولا انتي غالي انت اصلا انتي اختي انتي ولا لا انا ☝️ ولا ها لا انا اصلا هفه انا انا اصلا عهع انا خ انا اصلا انا ولا انا انا ه ولا عخ انا اصلا انتي بنت اختي انت اصلا انا انا اصلا اصلا انتي اختي انتي ولا انتي لازم احكي انتي انتي اختي ع هه ليش ح لا لا انا اصلا انا خخ ن انت عنجد انا نخل هلا غير صح انا اصلا انا ولا هيك هيك خ ولا لا انا اصلا انتي اختي انتي ولا حغ ولا انتي لازم هلا انا عع هلا هلا عنجد هلا هلا عادي ولا لا لا انا اصلا انتي ح غ انت الي خرر. لرد. زراصلا انا
23:33 For anyone's who's interested...
Property of a Hermitian matrix: transpose(L) = conjugate(L)
Notation:
T(A) => transpose(A)
HT(A) => hermitian transpose (conjugate transpose)
. => matrix multiplication
* => complex conjugate
L => hermitian matrix (H would have been confusing with the HT() hermitian transpose notation used)
Starting with the identity...
HT(A).L.B = HT(A).L.B
Because L is a hermitian matrix...
HT(A).L.B = HT(A).HT(L).B,
Because both sides are a row-vector times a square-matrix times a column vector (1xn).(nxn).(nx1), each side of the equation evaluates to single value (1x1 matrix). A single value is unchanged by the transpose operation, so you can transpose just the right side of the equation...
HT(A).L.B = T(HT(A).HT(L).B))
distribute the transpose and reverse the order of multiplication...
HT(A).L.B = T(B).L*.A*
pull out the conjugate...
HT(A).L.B = (HT(B).L.A)*
and rewrite in Dirac notation...
= *
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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....
These lectures are very easy and amazing for a deeper understanding.
Quantum mechanics Notes
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 other words V= ^/t {T=1/f }
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
|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 )
Thanks
These lectures are brillliant! So much more interesting the the lectures I had at uni. Much more funidimentally based then the stuff we did. Although haveing said that, I probably would have struggled with the abstractness of these lectures without having two years of quantum mechanics already stored in the vaults of may brain; and also topological spaces!
True that
A great gift the information age gives us where we can listen to the top notch lectures like this 👍👍👍
at 6:31 he says the matrix elements form an mXn matrix, but he meant a DxD matrix ( m is an index ranging from 1 up to D, when D is finite ). although the expression “mXn” matrix is frequently heard when discussing linear operators from R^n to R^m.
1:32:27 This may seem trivial to you, but this idea of “Eigenvalues” of an operator is mind blowing. I am used to simple eigenvalues for simple matrices.
To a physicist this is the same thing as with matrices. To a mathematician this is a lot more complicated, but physicists are usually glossing over the complications.
Yes you heard eu gen ics
An eigenvector is the "proper vector" corresponding to the "proper value" (eigenvalue) of an obserable. Once you have identified your eigenvalues, you then calculate your eigenvectors associated with those eigenvalues. However, remember that the eigenvector must be normalized as soon as you calculate it, so that it is a unit vector in the basis space of the operator. In other words, the eigenvectors of both the position and the momentum operators are all of magnitude 1....
He is going straight through the first chapters of Shankar... which i really appreciate.
Oh I thought it was Zettili. It must take a similar approach.
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.
Ннееее ее ненене Ен на её ееннн
The inner product (that's your integral S(f.g')dx) is taken across the whole line from -infinity to infinity. He just assumes that the function Phi is going rapidly to the zero so "it's values on the border in the infinities are zero". And that's the missing term in the per-partes formula.
Don't worry, figured it out when fully expained in Lecture 5. :)
I didn't know that moving a camera right and left during 2 hours could be so helpful.
Thanks
@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.
@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.
Yes, and F.T. of delta function is sine function that's why it is spread in the p
@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.
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
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@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
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
@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.
1:28:55 I think he is missing the term F.G on the right hand side.
Took me a while to find the whole in the Stern-Gerlach Experiment.
Not a spooky as it seems, still have to go through all the experiments however the results make absolute sense once one assumption is removed.
Assumption, the electron is causing this effect.
What about all the up and down quarks? Do you think that maybe they have something to do with this?
Here is how it would hypothetically work; Every Atom has an electronic field in it, not unlike the earth but probably more complex with quite a few polls depending on how many neutrons and protons there are. In a beam the atoms become entangled up and down, once they hit the magnetic field the up oriented will move one direction and the down oriented will move the other; however once out of the field they will reorientate themselves. Then when they hit the new magnetic field the process starts over again.
That would also explain why Neutrons have a "Spin," even though they are neutral. They still have a field, however the total sum is zero. The same with atoms, they have fields and unless they are ions, their total magnetic field is zero.
great lecture on a difficult subject--just wondering if the students are using a textbook(name?) for this course? thanks
Could we have an objective understanding of quantum mechanics if we explained it as an emergent interactive process unfolding photon by photon? This idea is based on: (E=ˠM˳C²)∞ with energy ∆E equals mass ∆M linked to the Lorentz contraction ˠ of space and time. The Lorentz contraction ˠ represents the time dilation of Einstein’s Theory of Relativity. We have energy ∆E slowing the rate that time ∆t flows as a universal process of energy exchange or continuous creation. Mass will increase relative to this process with gravity being a secondary force to the electromagnetic force. The c² represents the speed of light c radiating out in a sphere 4π of EMR from its radius forming a square c² of probability. We have to square the probability of the wave-function Ψ because the area of the sphere is equal to the square of the radius of the sphere multiplied by 4π. This simple geometrical process forms the probability and uncertainty of everyday life and at the smallest scale of the process is represented mathematically by Heisenberg’s Uncertainty Principle ∆×∆pᵪ≥h/4π. In such a theory we have an emergent future unfolding photon by photon with the movement of charge and flow of EM fields. This gives us a geometrical reason for positive and negative charge with a concaved inner surface for negative charge and a convexed outer surface for positive charge. The brackets in the equation (E=ˠM˳C²)∞ represent a dynamic boundary condition of an individual reference frame with an Arrow of Time or time line for each frame of reference. The infinity ∞ symbol represents an infinite number of dynamic interactive reference frames that are continuously coming in and out of existence.
Wtf is that , your theory is blowing my mind ,noice
@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.
@Alex Gabel it's also German and has the same meaning
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?
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.
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.
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.
... It's most certainly not a stupid question; it is quite critical to understand what Eigenvectors are and mean.
When he talks about the probability of measuring an eigenvalue P(λ) = where λ is a complex number how is it possible to get a complex number as a result of an experiment?
PublicCommerce you forgot that the eigenvalues of hermitian matrices can only be real, so lambda is real
awesome lectures. would be helpful if they were labeled based on topics discussed however
Woke up to being halfway through this video, i’m learning, somewhat
@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
Thanks for precision. However, focusing on technicality kills intuitive meaning of kets... I like Feynman's way that relates quantum laws to ordinary experience.
Where could I find some good lectures from Feynman? On these topics.
I'm not sure you'll like this answer :o)... Although it is possible for the numerical values "lambda" to be equivalent, the actual eigenvalues of the operators are going to be in the units of that operator. Consider a Harmonic Oscillator, for example; the lambda roots of the characteristic equation for a 3x3 position operator are {0, +SQRT3, -SQRT3} , but the actual eigenvalues are {0, +SQRT (3hbar/2mw), -SQRT (3hbar/2mw)}....
@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...
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.
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
I should have phrased my question more adequately. Since eigenvectors are normalized by definition, do momentum and position eigenvalues all every exactly coincide. Like lambda equaling {3,1,2} for both the position and the momentum.
@bluegrassaficionado I agree. Politics shouldn't come from the podium or the pulpit.
Probably a silly question, but is the delta function orthoganal? If you take lambda + epsilon as the value you are looking for wouldn't this clash with lamda? I've probably missed something important!
...putting back the (const) when we write the eigenvals of the operator. Hope this helps
It seems to me that in order to describe the momentum of a particle, you need at least two dimensions. i.e. one is not enough since an x-ray and a radio wave both travel at the speed of light in a vacuum. Usually when we apply energy, it is in one direction. even if it is a wave of energy, it seems as though it is just energy escaping into different dimensions, even if it is a planar one trapped in 3-dimension space. Similarly, quantum entanglement might just be energy escaping into a different dimension, this one with dimension >= 5 as it is invariant to both time and space. (3+1 = 4). If this is true, then it might take zero energy to rotate the plane of a wave around the axis of direction. In other words it might do it spontaneously i.e. waves could spontaneously change their perpendicular axis relative to direction. Does this have implications for higher dimensions? Does direction for instance have many meanings?
can the eigenvector describing position be of the same length as an eigenvector describing the momentum? (this would mean they are just in different directions, and that the eigenvectors of the position and momentum of a particle aren't necessarily orthogonal). This may be a stupid question...
thank you--just wondering/I'm using the textbook-Quantum Physics for Dummies about $5.00-- an excellent research source.
1:12:42 taking a break
@jamma246
In case of the Identity matrix all lambdas are the same so you lose the condition lambda1 different from lambda2
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.
The one is out and welcomes u all ☝️
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.
Isn't the result of integration by parts as done at (1:30) correct only if the psis are normalized?
The p^ operator gives roots of {0, +(Im)SQRT3, -(Im)SQRT3}, and its eigenvals are {0, +SQRT(3mw*hbar/2), -SQRT(3mw*hbar/2)}. My point is to make sure you are distinguishing the roots of the characteristic equation (calculated from the determinant) and the actual eigenvalues. Another glance: Det|A-V*1| = 0.... V=lambda. V must have the same units at A, but we tend to drop the coeff's (h-bar, w, etc) to work the determinant. So, we really work Det|a-v*1|=0, where A=(const)*a, V=(const)*v...
@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
@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
Priceless
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)
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.
Understood bra.ket.complex conjugate linear operator.hermitian operator and others
At 1:29:12, wasn't it missing an extra FG there?
Same question in my mind
Very satisfying erasing at 59:16 thank you
Awesome! Now we're getting into the meat and potatoes!
LOOL
The loud cougher that's in EVERY lecture cracks me up! He really get's that grunt going
😂😂😂 I was about to say that
When I took linear algebra I didn't understand what the hell we were doing with all this. It doesn't make sense until you study it in diff eq and quantum....
@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.
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
Susskind is always easy on the ear.
He is like Bob Ross, except for physics.
@@adminnvbs9166
I have a question. I remember integration by parts reads :Integral u dv = u v - Integral v du. Am I missing something ?
Alin Catargeanu
I would think the u v would be ∫ Ψ*(x)Ψ(x) dx = 1, and your constant of integration would take care of the 1.
Rolf Frankland in fact, it would be (1/i)(Ψ*(x)Ψ(x)) @ inf minus the same @ neg inf, which is a probability to find a particle at +/- infinity =0.
Also ψ*ψ is always real, so it's complex conjugate is also real and it doesn't matter in this case.
+Alin Catargeanu Ψ (and thus Ψ*) are defined such that Ψ*Ψ converges. This came up in the first example, where int(xΨ*Ψ)dx was real and bounded. Hence the uv term after integration by parts amounts to taking functions at values (+/- infinity) where they will be zero.
Hey everyone if your stuck ill post my Notes Lecture 1 & 2
Joshua Blackmon they helped me alot!
i never so much as passed highschool algebra but i find these lectures fascinating
❤thank you very much
k = 1/length = ħ^(-1/2) G^(-1/2) c^(3/2)
Should we learn the quantum entanglement course before going into this one?
No need ill post my notes , its suggested .
quantum entanglements is after this course
bfyguy I actually don't know. Quantum entanglements really should be after it. I think its better to watch this first becuz you can understand it even without entanglements and then you can watch entanglements and you will gain a deeper understanding.
@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
i'm still curious is studying too much physics can cause baldness for men? i watched several lectures and different physics lecturer and almost all of them are bald..
Are you being offensive, disrespectful, or just making a really bad joke? I would have thought that if you are here watching this lecture you would be more at owe with this incredible teacher!
i just want a scientific answer and it's serious as part of satisfying my curiosity..I admire those people that teaches science in youtube for me to have a privilege to discover more because i have no money to pursue school so i rely on watching lectures especially to this man and those other professor's in physics that i had watched..I am sorry if it's too offensive but that's part of my observation, if nobody could answer it clearly then i'll have to do the job to discover it..
You have studied enough physics to deduce this? Maybe someone should give you a Nobel Prize. Congrats on your groundbreaking discovery.
LoL i am studying physics for discovery and to satisfiy my passion..and i'm not aiming for that shitt Nobel Prizes because i will not carry that after i die..
Fynx Factora
Neither will Nobel Prize winners,, it is just a recognition as long as you are alive.
What does it mean to be orthogonal in complex space?
Means inner product zero ,as we do in case of 3d point vectors (a^.b^ =0 )
@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.
@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 :)
Its all taught in Mathematics E in the final year of high school (at least it is here)
Susskind is so old-school: He keeps calling the white board "black board," and the markers "chalk!"
That's adorable. :-)
whos is the one that is coughin up blood in every lecture????
Thats thanos
Absolutely Brilliant..!
1h23min sound cuts out.. anyway to continue this?
nvm it continues
Go to 1:24:28
I understand 😎😎💯
Super! 👍🙏🙏
Anyone know what book he used for this class?
There is a book series related to the lectures, by Susskind himself! They're called 'The Theoretical Minimum', and are very good indeed.
@bluegrassaficionado Hmmm. Right.
I do not remember but I think its Ψ*Ψ is 0
@jc00178 yeah he is really amazing actually
@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.
You dont know me but you dont know nothing until you know this blog thats being started to describe QM, quantum theory is wide ranging,
Quantumtheorypageone dot blog spot dot com , quantum mechanics would be all inclusive most physicists dont know what QM was means to us regarding how to predict to know what we are doing all together, regards what Relativity etc Micro Mechanical Occurences condition the means, Economy is much deeper than most can know its takes guts to know this'
Math should not have any biases
If only there were some mechanism provided to skip ahead...
guy in the audience at 2:05 has one hell of a sneeze.
he was there last lecture as well, would hate to have someone like that sit behind me
The part of the video at 1:09