0:00:00 - Review of mesuremnts on a qubit 0:25:33 - Logic of Quantum Mechanics 0:26:17 -- Space of states of the system 0:30:19 -- Calculus of propositions 0:44:02 - Review of vector spaces 0:53:30 - Space of states of a Quantum system 0:58:10 -- General linear combinations of 'up' and 'down' 1:05:53 -- 'right' and 'left' as linear combinations of 'up' and 'down' 1:13:30 -- 'in' and 'out' as linear combinations of 'up' and 'down' and of 'right' and 'left' 1:41:42 - up next 1:43:17 - Q&A
First i thought 'naah he's going way too slow...boring.but after some time I realized how good this guy actually is at putting pieces together. wonderful lecture thanks a lot
frank zack You’re absolutely right. When you try to teach this lecture myself (only to test my comprehension ) I find myself having to revisit it many times. R Feynman lectures are similar in that regard.
Orthogonality means that measuring one does not give you information about the other. Measured X coordinate, I have no idea what Y will be. But, when you measured spin up = i, then you know for sure that spin down is -i exactly. This is opposite to what orthogonality means.
No. When the quantum measurement got you spin up state =1, then the spin down state =0. That's the geometric orthogonality sense that if you have a vector oriented in x-axis direction then its projection on an orthogonal y-axis is zero.
I think you have to accept that "up" and "down" here are simply labels for two completely distinct states. And in mathematical terms , distinct and different states -- just like vectors in normal state -- are a property of orthoganility. And frankly , I think it a little smug to say that any of the people in this class are simple /thick etc -- I just wish the sound was better to hear them .
I do think that the way he explains uncertainty is a little hard to grasp compared to presenting the dual slit experiment (a much more gentle transition into the WTF of quantum mechanics), but it gives you a much more solid and abstract understanding of how to work with these systems
@@samuelallan7452 the double slit contains there concepts though unless over over simplified. We end up with wave functions and still discuss collapse yet instead of simply being about observation and probability along simple axes its about waves position and interference etc. The same maths applied to that is a lot more complex surely
Yes! Prof Susskind's spin machines are just Prof. Adam's color/hardness boxes. Both are great explanations; I personally prefer Prof. Adams' narrative, but Prof Susskind uses dirac notation, which paves a smoother road for future maths.
@@ONS0403 Adams went out of his way to make it clear to his students that non-relativistic quantum mechanics is a systems theory (similar to thermodynamics or theory of linear systems). To ask "what is in the boxes" is not useful because the non-relativistic case is not self-consistent physics, and the self-consistent relativistic theory will overwhelm most students, so it can't be taught to beginners. Susskind is probably a little more interested in those students who will, eventually, graduate to the relativistic theory and he is therefor a bit more conventional.
am so appreciative that you always show the photo you are using. This has helped me tremendously in seeing colors more definitely even if I can't replicate it the way you do. Still trying, Thanks again and have a Happy Easter, stay safe and healthly.
A or B. If we measure along z axis we’ll sure get +1, so A is true, that means (A or B) is true. But when he talks about (B or A). If we measure B first we might get +1 or -1, so B might be true or might be not true. But after that we measure the component along z axis, so we orient the detector along z axis, and if the spin is also along z axis, we will certainly get +1, and A is true, so (B or A) will also be true 100%. I didn’t understand why A might be false if we measure B first
Let me try, after measuring it for 'B' the outcome has 50%-50% chances of being +1 or -1 and right after this, if we were to measure it for 'A' along z-axis we'd get +1 or -1 because right after one measurement (B) we have disturbed the Quantum State of the system as Lenny sir said, unlike Classical Experiments, we can't measure Spin without disturbing the Intial Quantum State of a System.
Note that he talks about the "state" of the coin, not it's position; he established in Lecture 1 that the state he's talking about is "heads" or "tails". Thus, all he's saying here is that bouncing visible light off the top surface of a coin -- to read heads or tails -- doesn't make the coin randomly flip over.
Since he asked. Apparatus is plural. apparatum would be the singular. It refers to “preparation” which is also how Dr Suskand referred to the observed Qbit
Take home lesson: there is always a coordinate system in which the particle is in a pure state. It's just that our apparatus may not be aligned with it. Question: Can this be generalized? Are quantum systems always in a pure state in some coordinate system we may not know about?
1:32:55 but that is because we have as human beings extended the space with a complex number. Why can’t we extend the space with different complex numbers? Like maybe have |f> and |s> defined with a different complex number that follows similar rules to i.
I had this question too, but it seems they are orthogonal in state, but not in space. Someone asked your question toward the end of the video. You may want to check that out.
To distinguish between the (x) and (y) directions. He has chosen his base states with which to describe the electron spin to be (up) and (down) along the (z ) axis, and the only way to find unambiguous descriptor coefficients for an electron spin being along either the (x) or (y) directions (of normal 3 dimensional space), is to use 1/root2 in the (x) directions and (i/root2 ) in the (y) directions
buenfunkshun I even don't understand why we need a third dimension. I imagine the spin vector like an arrow, or why not, my pencil. I have it before me on the table pointing at the ceiling, that's the z axis, than skip it so that it lays horizontally on the table. And only this angle between the z and x axis should matter. If I rotate the pencil while laying on the table - corresponds to different y values - this shouldn't matter. (Or if I skip the pencil by 30° from the z axis, shouldn't matter if I skip it towards me or away from me.) Think it has something to do with being a "space of states, not to confond with our "ordinary" 3D space"? Maybe that's also why the "in" and "out" vectors are complex? Also the cosine is the same for a pair of angles (within 360°) but the probability not - think we'll understand it further on, and hopefully see that matematically it makes sense. But for the rest I like the mathematics very much : ) Edit: I see now that it's basically the same question of elouv 3 years ago: «Why the "in" and "out" vectors have different coeficients (including imaginary i) from 'left" and "right" when written in terms of "up" and "down"?»
The probability of getting the same answer is the cosine of the angle of perturbation. For a small enough angle, we can treat the cosine of the angle as being equal to 1. If the angle discrepancy is 4/5 of a degree, the chance of getting the other answer is about one in 10,000. That's a big angular discrepancy, and a tiny degree of randomness. If the angle discrepancy is a tenth of that, the chance of a 'wrong' outcome goes down to a hundredth of that - 1 in a million. Does that help?
Im confused, In the book he relates the different logic that arises from experiments, A and B being meaningless in QM, to the Uncertainty Principle, but: "Historically, the uncertainty principle has been confused[5][6] with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems.....It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems,[8] and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology."
It's not accurate to say that in quantum mechanics, you only change the properties of the system that are different from the one that you measured. If the system starts in a mixed state, as far as the measurable property in question is concerned, the measurement makes it collapse into one of the pure states that it was a mixture of. Only if the system was in a pure state at the time of measurement does the measurement not affect the property of the system in question.
@elouv in and out need to be different to left and right. if they had same coefficientss then you could say left = in and right = out (or other way round). from what i understand, it doesn't matter which has the i in it, but only one of them can have it.
Leif, if I could understand how did he chosen the 2ns basis, I could tell you about the 3rd. The 1st, 2nd and 3rd bases seem to be conventiaonally orthogonal, as "up" and "left", as opposed to the new kind of orthogonality "within" the basis. I asked to explanation at 'physics.SE, question 67506'. It covers both issues: what is common and relationship between up-left and up-down orthogonality.
I have the same question as noobyfromhell. Suppose we measure the spin of the n axis to be +1. Then we measure an axis which is, say a degree off the n axis, and we measure -1. There are many axis which are close to both of the above axis, yet they cannot all have an average of the dot product spin because the predicted averages will be totally different. does that make sense? Im new to quantum and it gets stranger and stranger the more i think about it.
Don't understand the question but your new measurement collapses the state or prepares the particle in a new axis, so the only axes being considered at once are the last known spin and the one of new measurement
... ie - no reason in mathematics to assume components of vectors can't be known independently. Again, would understand that a measurement would be an operator, which would change things, that is be non-commutative, but that isn't a difficulty in math of vector spaces and probability but a pragmatic practical difficulty.
I have a doubt.... So if a spin is measured... It can be 1 or -1 and once it's either of the 2, we can conclude that the direction is the same as the apparatus if it's 1 or opposite of -1 from a single experiment alone?
The state changes after the mesurement. Whatever state it was before destroys after the measurement, and the new state is the one related to the particular value.
Thanks for the lecture. Question: The way Prof measured for A OR B boolean logic doesn't stand simultaneity. What was actually measured was B given that A is 1 or B given that A is 0. Would having 2 Apparatus simultaneously measure the same particle offer any new insights or would report just random arbitraty measurements with respect to each other. But within the measurements of a single apparatus they would be consistent with what Prof has explained.
Physically speaking can two detectors exist occupying the same space and time? I mean I viol not so it can't be done simply put but then that relies on what actual instruments are like stern Gerlach device or polariser. Presuming so then the detectors must interact with eachother and form some total detector. Like the stern Gerlach would combine magnetic fields and you'd just have some combination of both and not two
Question. Lets say you find spin up z. Now you have the aparatus find the x spin, but instead of looking at the data completely destroy the data. Now measure z spin. Is the z spin still probabalistic or is it spin up z every time?
When you measure the x-spin, you invariably effect the original quantum state i.e. z-spin. There is no way that you measure the x-spin without effecting the z-spin. So lets say you make a z-spin measurement of +1 and then do a x-spin measurement (it does not matter whether you look at the result or not) and then again do a z-spin measurement, you will find that z-spin measurement result will be completely random, it would be +1 or -1.
How do you destroy data? A fundamental of physics is that you can't due to conservation of energy so why would that have an effect and is the question due to misunderstanding 'quantum eraser'. Obviously the detection does the collapse but additionally there is no way of lowing those intermediate States he you don't know.... And despite popular proliferation of quantum erasers they collapse the state whether erasing or not, its just that they arrange results different when they are later sorted, he that's where you got the idea from
If I start with spin up and then turn aparatus fo 90 degrees i.e.along x axes I'l get 50%left and 50 right%.But if I turn back apparatus along z axes I'll get sometimes up sometimes down.Wouldn't it have to be always up as we came back to starting possition?
Measuring doesn't just measure it changes the spin. Measurement causes the 'preparation'. Particle has to spin in the direction you measure it. There are no values in between 1 and -1 possible, only statistical probability of those States over repetition. So your first state is erased after the next measurement and so treat the latest measurement direction as the new prepared state. Which is why we have to do this across many different particles, not just the same one over and over . It kind of makes more sense considering real apparatus to me, like use of magnetic devices or polarisers, there is is going to be some interaction with those devices that is orientation specific and that spin causes charge, surah that an electron can't suddenly become uncharged if you like, it must conform to the direction involved with the device.
Question - Why the hell are |u> and |d> vectors orthogonal? I don't want an answer "they are distinguishable by an experiment" - I want geometric interpretation... Proof, that =0. Intuition tells me, that =-1, what am I doing wrong?...
+Tomasz Dzieduszynski I would understand if this was the orthogonality in the imaginary space extension, but this is not the case since we didn't use that to define the spin along z axis...
+Tomasz Dzieduszynski All the possible states in a quantum mechanical system are mathematically orthogonal in an abstract object called Hilbert space. It has nothing to do with their orientation in "real space". If you have a system with two different states, |u> and |d>, they can be represented i Hilbert space as |u>=(1,0) and |d>=(0,1) which gives =0. If you have a system with three states (red, green and blue), you can represent them as red = (1,0,0), green=(0,1,0) and blue=(0,0,1). They will all be orthogonal to each other in Hilbert space. In "real space" we don't even know the meaning of orthogonal colors, haha. Not really a big expert on this but i think it works like this...
+Tomasz Dzieduszynski I think it's one of the fundamental rules i quantum mechanics that the possible states of a system forms an orthogonal basis in Hilbert space. :)
So if we ever measure a quantum asteroid to be headed straight to earth we just have to measure the component of its velocity in another direction to avert a collision with earth :) Or better yet, we should preemptively measure the component of the velocity of every quantum space rock in the direction of the sun so we can all have a good night of sleep without the fear of the prospect of quantum armageddon.
It seems he does 2 different types of inner products: one with bra kets as complex conjugates, and the other where he makes a row vector and a column vector. This confused me. I don't remember him discussing 2 types of inner product.
He talked about in the last lecture but I think that it is good to think about it as the same operation. The bra kets notation is abstract and applies to all quantum states even though different systems can require different representations. In contrast, the row and column vectors are a specific representation of some bra and kets which is useful for doing actual calculations
Jm Jones I'm a bit confused... If by the act of measuring the spin you set it, why subsequent experiments changing the apparatus orientation don't set the spin to a new orientation as well? What's special about that first measurement? Is the apparatus in some sort of "preparation mode" that you turn off after first measurement? Thanks for your help!
Ferran Núñez Martínez Its nice to help. No, the first mesurement is not special. QM allows you to calculate the probability of some outcome given the present state. for example you set the appartus in the y axis and the outcome is A. if you measure again in y the result is 100% A and 0% B, but if you calculate the probability for the outcome in the X axis you will obtain 50% C and 50% D. and when you make the measurement the state will change and you will set it in C or D.
Thanks, that was helpful!I think I've got it. So you measure Y, you get either -1 or +1, but from now on, if you keep measuring Y, you'll always get +1. Then you turn it, measure X and you have 50% of getting +1 or -1 but, from now on, as long as you keep measuring x, you will get +1.
The ppl attending these lectures are not stanford students, they are just ordinary people who are interested in learning some physics. And this series of lectures is intended for such ppl who also know some pre calc and math
00:25:30 - Susskind here tries to explain why QM logic is different to classical - he does so be pragmatic considerations of the non-commutability of QM operations - a measurement changes the forthcoming tests ! He then states the reason for the differences in logic is attributable to the fact that the space of states is a 'vector space' - but his explanation does not lead to this generality - any thoughts anybody on how this can be done ? ie - no reason to assume in mathematics ...cont
I don't think he stated that here, he mainly wanted to say that the behaviour in a vector space when talking in the context of classical physics is something you cannot expect in quantum physics. The measurement or the way you make the measurement has a very huge effect here
One "explanation" is that this is *the ways things are* so no explanation is necessary (or sufficient). All a physicist has to show is that a mathematical model is explanatory of experiments. However, historically non-commutativity was known to the founders of QM (Born) to be a mathematical feature of matrices and matrices represent QM-ical measurements and matrices go with vector spaces (i.e. they are structure-preserving maps on them).
For any given system how to find the number of dimensions.? How did he conclude that spin of state is a 2 dimensional vector space.? Has it anything to do with the 2 posible outcomes..?
Can someone explain why he used 'i' to write in and out relative to up and down, when it wasn't included in the left and right vectors? I saw another comment here mentioning something about directionality... does the use of 'i' indicate directionality somehow?
i, confusingly, is also the square root of -1. One of the students was confused by this as well, so he relabeled the vector as capital I to disambiguate it with the i=sqrt(-1)
@@mihirgupta3824 I'm confused too. He introduced i into the formulae without explaining why and none of the students asked him why. So either the reason is obvious to everyone but us, or an explanation is needed. His book doesn't explain this either.
He could have chosen the 'in' and 'out' direction to have real components. In the end this is just a rotation about the z axis by 90°. It's like a choice of coordinate system. And note that you can multiply by a phase... |r> can also be written as |r> = 1/sqrt(2) i (|u> + |d>) the overall factor of i doesn't change the state. But with the choice he made for what |r> is, the phase relation between the coefficients of |I> is fixed.
It seems odd that there should only be three pairs of orthogonal basis, if I wanted to define a universe in which there could be a different amount of pairs of orthogonal basis what would I have to do?
Then you would have to define a space where a number has multiple dimensions. In normal complex space, a number has 2 dimensions. In quaternion space, they have 4. In octonion, they have 8. You can use these dimensions to write different kinds and numbers of orthogonal basis, but it is pretty hard as you may have imagined. In this context, 2 dimensional numbers are enough to describe the vector space of quantum mechanics.
Peter ThomasG1971 bit late, but you'd probably be screwed. If you made it more, then laws of gravitation would be wonky and nothing would stay together, too little and you can't have life (digestive tract splits you in two)
No, they’re are an infinity of orthogonal basis. What’s limited to 3 is the choices of pairs of alphas that exhibit the symmetry of expression as linear combination of the pairs of orthogonal vectors
@@joefagan9335 can you elaborate more on what you mean by "symmetry of expression"? what I understand is that any basis can be expressed as linear combination of any other basis.
Logic, a simple innate knowledge?,..in a defined context of connectivity such that a meaning is predetermined, as in a coordinated system, because the absolute limit of "minimum" is a point of connection - certainty in the infinite uncertainty of nothing. It's only a "minimum" of uncertainty to describe the process of possible change in an existential certainty, the description equivalent to the "sum of histories" is recirculating, as in the one-electron concept and phase state, of QFT, in oscillation +/- eternity-now. And thus begins the descriptive process of building a proportionate language, coordinated system of logic, and devices to apply that logic with some quantized degree of certainty. (?) ----- Minimumuncertainty because connected unity of Euler's Intuitions, e-Pi-i sync-duration connectivity function, roots 1-0 probability. All derived in concept from Professor Susskind's Black Hole Singularity positioning, the Universal Apature of Reciproction, time-timing cause-effect of/by Professor Disney's Observation of WYSIWYG Logarithmic QM-TIME Completeness.., Unity, and Professor Hartnoll's Super properties including zero-infinity connection axis in Universal Black-body Singularity Superspin.
Its too long to explain in a 500 character responce post limit, but I was actually making an elusion to the delayed choice quantum eraser experiment of Yoon-Ho Kim, R. Yu, S.P. Kulik, Y.H. Shih and Marlan O. Scully if you want to search it. Not knowing about it and making a classical physics responce is how I knew Chewyfield wasn't what he claimed to be.
At 11:13, he states that it would be strange to see this behavior from a classical vector. But he's wrong. I don't say this lightly, I say this because I want an explanation for a mistake I've seen both here and in other places that try to explain quantum mechanics. The detector has already been defined to only give a binary result. That it gives a binary result, then, says nothing about the nature of the object being measured. You cannot conclude from a detector that only gives two possible outputs that the thing being measured has only two possible states. You would need a detector that could theoretically output fractional results in order to make that conclusion. 20:26: He finally addresses this, but his answer is basically just a shrug. SIgh.
These are lectures and questions should be written down and answered at the end of the lecture. The lecturer should not be interrupted over and over again!
As I comprehend it, the normalisation constraint helps us determine the fourth variable/parameter if we know at least three of them. So, we can do away with one parameter (This elimination procedure will play a role again below). Let's now look at the phase factor idea. If we multiply the state vector by a phase factor such that one of the components of the state vector ends up with an imaginary part becoming zero, the physical nature (as per Lenny, as he says in the video) of the state vector does not change. So, with one of the components fully specified by a real number, say x, and the other as say a + ib we can use the normalisation constraint again, x^2+a^2+b^2=1. Now, using the elimination procedure, if we did not know either x, or a, or b, we could recover it, if we knew the other two, e.g. if x were unknown, we could recover it by x=sqrt(1-a^2-b^2). So, we only need TWO parameters to fully characterise the state of the system.
Pavan Choudar he has to find 2 complex numbers that satisfy their magnitude squared add to one, and also that I and O can be expressed as linear combinations of both U and D and R and L in a symmetric sort of way. There’s no way of doing this with purely real numbers and so i creeps in
@@darklord69420 ppl viewing a lecture are an audience; they also happen to be students. But that's not even the thing to be pulled from my comment. Call them whatever you want; they still have poor lecture etiquette.
@@busterdancy1857 This is not kindergarten or primary school. This is university with adults who have the right to come and go as they please. If you can't stand that as a lecturer, then the solution is simple: don't be a university lecturer because your ego is in the way of teaching adult material to adults.
|u> and |d> represent axes in an abstract vector space. Compare this with a real 2-d space with orthogonal unit vectors y and x (these should have the ^ on to show they are unit vectors). Then any vector in our 2D real space could be written as: A = ax + by Since |u> and |d> are the orthogonal unit vectors in the abstract space, then any vector in that space may be written in the form: |A>=a_u |u> + a_d |d>
This is some expert-level bullshit: Susskind had several classes with the same name, and you couldn't collect to a playlist of something? The are dozens of "The Theoretical Minimum" in this chanell, recordings from different years, just dumped without order.
He is not wrong, he says in classical physics its possible to determine it to infinite precision which it is, he says nothing about it actually being possible to do.
0:00:00 - Review of mesuremnts on a qubit
0:25:33 - Logic of Quantum Mechanics
0:26:17 -- Space of states of the system
0:30:19 -- Calculus of propositions
0:44:02 - Review of vector spaces
0:53:30 - Space of states of a Quantum system
0:58:10 -- General linear combinations of 'up' and 'down'
1:05:53 -- 'right' and 'left' as linear combinations of 'up' and 'down'
1:13:30 -- 'in' and 'out' as linear combinations of 'up' and 'down' and of 'right' and 'left'
1:41:42 - up next
1:43:17 - Q&A
Thank you so much Luca for these comments, these are so helpful to me.
Did you also write notes for these lectures?
1:23:49 in and out in terms of left and right
1:33:02 degrees of freedom in specifying state
❤thank you
Always like these lectures from Prof. Susskind. They provide good intuition before diving into a bunch of math.
First i thought 'naah he's going way too slow...boring.but after some time I realized how good this guy actually is at putting pieces together. wonderful lecture thanks a lot
frank zack, you called Leonard Susskind, "this guy". :)
frank zack You’re absolutely right. When you try to teach this lecture myself (only to test my comprehension ) I find myself having to revisit it many times. R Feynman lectures are similar in that regard.
Thank you Prof. Susskind. You are helping me understand Dirac, which I have wanted to do for years.
After 9 years Mr Dixon
Have you understood dirac?
@@meowwwww6350 Nobody has
Brilliant explanation!!! His lectures make me want to study quantum physics. It is so interesting under his explanation!
“Nothing left to say… (Sigh) … say it anyway.” 😂 @24:56
Orthogonality means that measuring one does not give you information about the other. Measured X coordinate, I have no idea what Y will be. But, when you measured spin up = i, then you know for sure that spin down is -i exactly. This is opposite to what orthogonality means.
No. When the quantum measurement got you spin up state =1, then the spin down state =0. That's the geometric orthogonality sense that if you have a vector oriented in x-axis direction then its projection on an orthogonal y-axis is zero.
"lots of subtlety but nothing really difficulty .. mathematics of quantum mechanics is easier than classical physics .. "
I think you have to accept that "up" and "down" here are simply labels for two completely distinct states.
And in mathematical terms , distinct and different states -- just like vectors in normal state -- are a property of orthoganility.
And frankly , I think it a little smug to say that any of the people in this class are simple /thick etc -- I just wish the sound was better to hear them .
I do think that the way he explains uncertainty is a little hard to grasp compared to presenting the dual slit experiment (a much more gentle transition into the WTF of quantum mechanics), but it gives you a much more solid and abstract understanding of how to work with these systems
@@samuelallan7452 the double slit contains there concepts though unless over over simplified. We end up with wave functions and still discuss collapse yet instead of simply being about observation and probability along simple axes its about waves position and interference etc. The same maths applied to that is a lot more complex surely
@@jorgepeterbarton Yeah I was a different person 3 years ago, I have since become more educated
"Logic makes some sense"
+
someone print it for him on a black sweatshirt
L j oh n in n mmm mmm mmm
Me
Combine these lectures with the lectures of Prof. Allan Adams you will complete interrelating this subject theoretically & mathematically
Yes! Prof Susskind's spin machines are just Prof. Adam's color/hardness boxes. Both are great explanations; I personally prefer Prof. Adams' narrative, but Prof Susskind uses dirac notation, which paves a smoother road for future maths.
@@ONS0403 Adams went out of his way to make it clear to his students that non-relativistic quantum mechanics is a systems theory (similar to thermodynamics or theory of linear systems). To ask "what is in the boxes" is not useful because the non-relativistic case is not self-consistent physics, and the self-consistent relativistic theory will overwhelm most students, so it can't be taught to beginners. Susskind is probably a little more interested in those students who will, eventually, graduate to the relativistic theory and he is therefor a bit more conventional.
The students need to stick with the topic and not try to prove how smart they are!
Sorry you didn't understand their questions
best quantum lecturer for explaining
am so appreciative that you always show the photo you are using. This has helped me tremendously in seeing colors more definitely even if I can't replicate it the way you do. Still trying, Thanks again and have a Happy Easter, stay safe and healthly.
A or B. If we measure along z axis we’ll sure get +1, so A is true, that means (A or B) is true. But when he talks about (B or A). If we measure B first we might get +1 or -1, so B might be true or might be not true. But after that we measure the component along z axis, so we orient the detector along z axis, and if the spin is also along z axis, we will certainly get +1, and A is true, so (B or A) will also be true 100%. I didn’t understand why A might be false if we measure B first
Let me try, after measuring it for 'B' the outcome has 50%-50% chances of being +1 or -1 and right after this, if we were to measure it for 'A' along z-axis we'd get +1 or -1 because right after one measurement (B) we have disturbed the Quantum State of the system as Lenny sir said, unlike Classical Experiments, we can't measure Spin without disturbing the Intial Quantum State of a System.
Note that he talks about the "state" of the coin, not it's position; he established in Lecture 1 that the state he's talking about is "heads" or "tails". Thus, all he's saying here is that bouncing visible light off the top surface of a coin -- to read heads or tails -- doesn't make the coin randomly flip over.
Since he asked. Apparatus is plural. apparatum would be the singular. It refers to “preparation” which is also how Dr Suskand referred to the observed Qbit
- Prof. Susskind is a teacher of the highest order, thank you Prof.
He cracks me up. Even if I only understand a small portion of his lectures, his demeanor is priceless :P
successfully solved the equations in terms of | i > and | o > using matrix approach
UP and DOWN are orthogonal! I need to get my head around that. Back to the beginning - watch it again. There's good learning material in here.
Take home lesson: there is always a coordinate system in which the particle is in a pure state. It's just that our apparatus may not be aligned with it.
Question: Can this be generalized? Are quantum systems always in a pure state in some coordinate system we may not know about?
1:32:55 but that is because we have as human beings extended the space with a complex number. Why can’t we extend the space with different complex numbers? Like maybe have |f> and |s> defined with a different complex number that follows similar rules to i.
If |u> is +z direction and |d> is -z direction, how are they orthogonal? Their inner product is not 0 but -1.
I had this question too, but it seems they are orthogonal in state, but not in space. Someone asked your question toward the end of the video. You may want to check that out.
I'm still confused, why did he use i for relating in and out to up and down? Why isn't that pair of equations exactly the same as for right and left?
To distinguish between the (x) and (y) directions.
He has chosen his base states with which to describe the electron spin to be (up) and (down) along the (z ) axis, and the only way to find unambiguous descriptor coefficients for an electron spin being along either the (x) or (y) directions (of normal 3 dimensional space), is to use 1/root2 in the (x) directions and (i/root2 ) in the (y) directions
buenfunkshun I even don't understand why we need a third dimension. I imagine the spin vector like an arrow, or why not, my pencil. I have it before me on the table pointing at the ceiling, that's the z axis, than skip it so that it lays horizontally on the table. And only this angle between the z and x axis should matter. If I rotate the pencil while laying on the table - corresponds to different y values - this shouldn't matter. (Or if I skip the pencil by 30° from the z axis, shouldn't matter if I skip it towards me or away from me.)
Think it has something to do with being a "space of states, not to confond with our "ordinary" 3D space"? Maybe that's also why the "in" and "out" vectors are complex? Also the cosine is the same for a pair of angles (within 360°) but the probability not - think we'll understand it further on, and hopefully see that matematically it makes sense. But for the rest I like the mathematics very much : )
Edit: I see now that it's basically the same question of elouv 3 years ago:
«Why the "in" and "out" vectors have different coeficients (including imaginary i) from 'left" and "right" when written in terms of "up" and "down"?»
Noun
apparatus (plural apparatuses or apparatus)
1:32:56 3 dimensions of space can be represented in two dimensional vector space terms
This guy basically checkmates every single one who tries to challenge science
A version of this video with the question cut off would make the understanding of the topic a lot smoother
The probability of getting the same answer is the cosine of the angle of perturbation. For a small enough angle, we can treat the cosine of the angle as being equal to 1.
If the angle discrepancy is 4/5 of a degree, the chance of getting the other answer is about one in 10,000. That's a big angular discrepancy, and a tiny degree of randomness. If the angle discrepancy is a tenth of that, the chance of a 'wrong' outcome goes down to a hundredth of that - 1 in a million.
Does that help?
Im confused, In the book he relates the different logic that arises from experiments, A and B being meaningless in QM, to the Uncertainty Principle, but: "Historically, the uncertainty principle has been confused[5][6] with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems.....It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems,[8] and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology."
''Tha blackboard''
It's not accurate to say that in quantum mechanics, you only change the properties of the system that are different from the one that you measured. If the system starts in a mixed state, as far as the measurable property in question is concerned, the measurement makes it collapse into one of the pure states that it was a mixture of. Only if the system was in a pure state at the time of measurement does the measurement not affect the property of the system in question.
@elouv in and out need to be different to left and right. if they had same coefficientss then you could say left = in and right = out (or other way round).
from what i understand, it doesn't matter which has the i in it, but only one of them can have it.
Leif, if I could understand how did he chosen the 2ns basis, I could tell you about the 3rd. The 1st, 2nd and 3rd bases seem to be conventiaonally orthogonal, as "up" and "left", as opposed to the new kind of orthogonality "within" the basis. I asked to explanation at 'physics.SE, question 67506'. It covers both issues: what is common and relationship between up-left and up-down orthogonality.
Bra and ket..which is British for bracket....just love his sense of humour !
I have the same question as noobyfromhell. Suppose we measure the spin of the n axis to be +1. Then we measure an axis which is, say a degree off the n axis, and we measure -1. There are many axis which are close to both of the above axis, yet they cannot all have an average of the dot product spin because the predicted averages will be totally different. does that make sense? Im new to quantum and it gets stranger and stranger the more i think about it.
Don't understand the question but your new measurement collapses the state or prepares the particle in a new axis, so the only axes being considered at once are the last known spin and the one of new measurement
@@jorgepeterbarton thank you, that answers my question!
Plural of apparatus is apparatuses
Thank you prof susskind, so helpful!
little Experiment/ logic of quantum mechanics: 5:15
Merriam-Webster says : "plural apparatuses or apparatus".
what a great thing,,, thank you stanford for putting this on youtube thank uuu tytytyty
From the course website:
January 16, 2012
Topics:
References
... ie - no reason in mathematics to assume components of vectors can't be known independently. Again, would understand that a measurement would be an operator, which would change things, that is be non-commutative, but that isn't a difficulty in math of vector spaces and probability but a pragmatic practical difficulty.
I have a doubt....
So if a spin is measured... It can be 1 or -1 and once it's either of the 2, we can conclude that the direction is the same as the apparatus if it's 1 or opposite of -1 from a single experiment alone?
The state changes after the mesurement. Whatever state it was before destroys after the measurement, and the new state is the one related to the particular value.
If you place this alongside anything by Ken Ham and Answers in Genesis you find a greater meaning and explanation of life
What is the difference between these lectures, and his lectures on Modern Physics: Quantum Mechanics course?
Why the "in" and "out" vectors have different coeficients (including imaginary i)
from 'left" and "right" when written in terms of "up" and "down"?
Simply because they are different vectors.
Great explanation 🙏♥️
Thanks for the lecture.
Question: The way Prof measured for A OR B boolean logic doesn't stand simultaneity. What was actually measured was B given that A is 1 or B given that A is 0.
Would having 2 Apparatus simultaneously measure the same particle offer any new insights or would report just random arbitraty measurements with respect to each other. But within the measurements of a single apparatus they would be consistent with what Prof has explained.
+Bharadwaj OVS duuuuuude your name is crazy
Physically speaking can two detectors exist occupying the same space and time? I mean I viol not so it can't be done simply put but then that relies on what actual instruments are like stern Gerlach device or polariser. Presuming so then the detectors must interact with eachother and form some total detector. Like the stern Gerlach would combine magnetic fields and you'd just have some combination of both and not two
Question. Lets say you find spin up z. Now you have the aparatus find the x spin, but instead of looking at the data completely destroy the data. Now measure z spin. Is the z spin still probabalistic or is it spin up z every time?
When you measure the x-spin, you invariably effect the original quantum state i.e. z-spin. There is no way that you measure the x-spin without effecting the z-spin. So lets say you make a z-spin measurement of +1 and then do a x-spin measurement (it does not matter whether you look at the result or not) and then again do a z-spin measurement, you will find that z-spin measurement result will be completely random, it would be +1 or -1.
How do you destroy data? A fundamental of physics is that you can't due to conservation of energy so why would that have an effect and is the question due to misunderstanding 'quantum eraser'. Obviously the detection does the collapse but additionally there is no way of lowing those intermediate States he you don't know.... And despite popular proliferation of quantum erasers they collapse the state whether erasing or not, its just that they arrange results different when they are later sorted, he that's where you got the idea from
If I start with spin up and then turn aparatus fo 90 degrees i.e.along x axes I'l get 50%left and 50 right%.But if I turn back apparatus along z axes I'll get sometimes up sometimes down.Wouldn't it have to be always up as we came back to starting possition?
No bc measuring along the x axis basically resets the electron.
Measuring doesn't just measure it changes the spin. Measurement causes the 'preparation'. Particle has to spin in the direction you measure it. There are no values in between 1 and -1 possible, only statistical probability of those States over repetition. So your first state is erased after the next measurement and so treat the latest measurement direction as the new prepared state. Which is why we have to do this across many different particles, not just the same one over and over
. It kind of makes more sense considering real apparatus to me, like use of magnetic devices or polarisers, there is is going to be some interaction with those devices that is orientation specific and that spin causes charge, surah that an electron can't suddenly become uncharged if you like, it must conform to the direction involved with the device.
A little closer to the business end of this discussion, skipping explanations of classical AND - OR = 00:35:30
Bad advice. The mathematics being discussed in the first 35 mins. is essential to understanding QM.
prof Susskind, the plural of apparatus is apparatuses ;-)
Can one says that quantum physics experiment observations of is statistically determined?
how did the "i" get in there
Question - Why the hell are |u> and |d> vectors orthogonal? I don't want an answer "they are distinguishable by an experiment" - I want geometric interpretation... Proof, that =0. Intuition tells me, that =-1, what am I doing wrong?...
+Tomasz Dzieduszynski I would understand if this was the orthogonality in the imaginary space extension, but this is not the case since we didn't use that to define the spin along z axis...
+Tomasz Dzieduszynski All the possible states in a quantum mechanical system are mathematically orthogonal in an abstract object called Hilbert space. It has nothing to do with their orientation in "real space". If you have a system with two different states, |u> and |d>, they can be represented i Hilbert space as |u>=(1,0) and |d>=(0,1) which gives =0. If you have a system with three states (red, green and blue), you can represent them as red = (1,0,0), green=(0,1,0) and blue=(0,0,1). They will all be orthogonal to each other in Hilbert space. In "real space" we don't even know the meaning of orthogonal colors, haha. Not really a big expert on this but i think it works like this...
+Tomasz Dzieduszynski I think it's one of the fundamental rules i quantum mechanics that the possible states of a system forms an orthogonal basis in Hilbert space. :)
good explanation.
he says it around 1:00:30
So if we ever measure a quantum asteroid to be headed straight to earth we just have to measure the component of its velocity in another direction to avert a collision with earth :)
Or better yet, we should preemptively measure the component of the velocity of every quantum space rock in the direction of the sun so we can all have a good night of sleep without the fear of the prospect of quantum armageddon.
Providing the asteroid isn't bigger than a bucky ball :)
GREAT! THANK YOU SO MUCH...!
Only One can up
No foundation -No up and down
I like it when he gets annoyed at people asking him questions.
Sorry, I'm not sure what you're saying. Why will they be totally different?
can't find lecture 1 of this series :(
ua-cam.com/video/iJfw6lDlTuA/v-deo.html
thank you to make me understand
Electromagnetic waves do not exist Electromagnetic waves are real
The Indian dude got his ass whipped "Why have you jumped to two particles ??"
😂
@khajiit92 thank you for your answer!
It seems he does 2 different types of inner products: one with bra kets as complex conjugates, and the other where he makes a row vector and a column vector. This confused me. I don't remember him discussing 2 types of inner product.
He talked about in the last lecture but I think that it is good to think about it as the same operation. The bra kets notation is abstract and applies to all quantum states even though different systems can require different representations. In contrast, the row and column vectors are a specific representation of some bra and kets which is useful for doing actual calculations
How do they set the spin to a certain point before measuring it?
measuring it you set it, that's the whole point :D
Thanks
Jm Jones I'm a bit confused... If by the act of measuring the spin you set it, why subsequent experiments changing the apparatus orientation don't set the spin to a new orientation as well? What's special about that first measurement? Is the apparatus in some sort of "preparation mode" that you turn off after first measurement? Thanks for your help!
Ferran Núñez Martínez Its nice to help. No, the first mesurement is not special. QM allows you to calculate the probability of some outcome given the present state. for example you set the appartus in the y axis and the outcome is A. if you measure again in y the result is 100% A and 0% B, but if you calculate the probability for the outcome in the X axis you will obtain 50% C and 50% D. and when you make the measurement the state will change and you will set it in C or D.
Thanks, that was helpful!I think I've got it. So you measure Y, you get either -1 or +1, but from now on, if you keep measuring Y, you'll always get +1. Then you turn it, measure X and you have 50% of getting +1 or -1 but, from now on, as long as you keep measuring x, you will get +1.
amazing thanks!!
@HelloIAmDaniel You get what he says but cant find a video on the internet?
6:32
By the right-hand rule, shouldn't the z-axis point downwards?
duh... this is not an intro class don't get hung up on the little things
[Xi] watt's that suppose to men, ops mean, a-men. 'n' is a number n-ame. [Xi] front cover of collins internet linked dictionary of mathematics
Why isn't i times i i^2..... and she goes to Stanford....
The ppl attending these lectures are not stanford students, they are just ordinary people who are interested in learning some physics. And this series of lectures is intended for such ppl who also know some pre calc and math
00:25:30 - Susskind here tries to explain why QM logic is different to classical - he does so be pragmatic considerations of the non-commutability of QM operations - a measurement changes the forthcoming tests ! He then states the reason for the differences in logic is attributable to the fact that the space of states is a 'vector space' - but his explanation does not lead to this generality - any thoughts anybody on how this can be done ? ie - no reason to assume in mathematics ...cont
I don't think he stated that here, he mainly wanted to say that the behaviour in a vector space when talking in the context of classical physics is something you cannot expect in quantum physics. The measurement or the way you make the measurement has a very huge effect here
One "explanation" is that this is *the ways things are* so no explanation is necessary (or sufficient). All a physicist has to show is that a mathematical model is explanatory of experiments. However, historically non-commutativity was known to the founders of QM (Born) to be a mathematical feature of matrices and matrices represent QM-ical measurements and matrices go with vector spaces (i.e. they are structure-preserving maps on them).
@1:17:40 It would have been helpful if the Prof. had derived the solution for |i> and |o>.
He derived it . See 1:25:00
For any given system how to find the number of dimensions.? How did he conclude that spin of state is a 2 dimensional vector space.? Has it anything to do with the 2 posible outcomes..?
ashieshk he didn’t determine that it was 2 dimensional vector space - he stated that it is one of the postulates.
the dimensionality of a space is given by the maximal number of mutually orthogonal vectors
Can someone explain why he used 'i' to write in and out relative to up and down, when it wasn't included in the left and right vectors? I saw another comment here mentioning something about directionality... does the use of 'i' indicate directionality somehow?
Even I'm very confused about why he did that? Can somebody answer this please?
i, confusingly, is also the square root of -1. One of the students was confused by this as well, so he relabeled the vector as capital I to disambiguate it with the i=sqrt(-1)
@@mihirgupta3824 I'm confused too. He introduced i into the formulae without explaining why and none of the students asked him why. So either the reason is obvious to everyone but us, or an explanation is needed. His book doesn't explain this either.
He could have chosen the 'in' and 'out' direction to have real components. In the end this is just a rotation about the z axis by 90°. It's like a choice of coordinate system.
And note that you can multiply by a phase... |r> can also be written as
|r> = 1/sqrt(2) i (|u> + |d>)
the overall factor of i doesn't change the state. But with the choice he made for what |r> is, the phase relation between the coefficients of |I> is fixed.
It seems odd that there should only be three pairs of orthogonal basis, if I wanted to define a universe in which there could be a different amount of pairs of orthogonal basis what would I have to do?
Then you would have to define a space where a number has multiple dimensions. In normal complex space, a number has 2 dimensions. In quaternion space, they have 4. In octonion, they have 8. You can use these dimensions to write different kinds and numbers of orthogonal basis, but it is pretty hard as you may have imagined. In this context, 2 dimensional numbers are enough to describe the vector space of quantum mechanics.
Peter ThomasG1971 bit late, but you'd probably be screwed. If you made it more, then laws of gravitation would be wonky and nothing would stay together, too little and you can't have life (digestive tract splits you in two)
No, they’re are an infinity of orthogonal basis. What’s limited to 3 is the choices of pairs of alphas that exhibit the symmetry of expression as linear combination of the pairs of orthogonal vectors
@@joefagan9335 can you elaborate more on what you mean by "symmetry of expression"? what I understand is that any basis can be expressed as linear combination of any other basis.
Logic, a simple innate knowledge?,..in a defined context of connectivity such that a meaning is predetermined, as in a coordinated system, because the absolute limit of "minimum" is a point of connection - certainty in the infinite uncertainty of nothing.
It's only a "minimum" of uncertainty to describe the process of possible change in an existential certainty, the description equivalent to the "sum of histories" is recirculating, as in the one-electron concept and phase state, of QFT, in oscillation +/- eternity-now.
And thus begins the descriptive process of building a proportionate language, coordinated system of logic, and devices to apply that logic with some quantized degree of certainty. (?)
-----
Minimumuncertainty because connected unity of Euler's Intuitions, e-Pi-i sync-duration connectivity function, roots 1-0 probability.
All derived in concept from Professor Susskind's Black Hole Singularity positioning, the Universal Apature of Reciproction, time-timing cause-effect of/by Professor Disney's Observation of WYSIWYG Logarithmic QM-TIME Completeness.., Unity, and Professor Hartnoll's Super properties including zero-infinity connection axis in Universal Black-body Singularity Superspin.
Prof. Susskind's forehead is very interesting...
Its too long to explain in a 500 character responce post limit, but I was actually making an elusion to the delayed choice quantum eraser experiment of Yoon-Ho Kim, R. Yu, S.P. Kulik, Y.H. Shih and Marlan O. Scully if you want to search it. Not knowing about it and making a classical physics responce is how I knew Chewyfield wasn't what he claimed to be.
At 11:13, he states that it would be strange to see this behavior from a classical vector. But he's wrong. I don't say this lightly, I say this because I want an explanation for a mistake I've seen both here and in other places that try to explain quantum mechanics. The detector has already been defined to only give a binary result. That it gives a binary result, then, says nothing about the nature of the object being measured. You cannot conclude from a detector that only gives two possible outputs that the thing being measured has only two possible states. You would need a detector that could theoretically output fractional results in order to make that conclusion.
20:26: He finally addresses this, but his answer is basically just a shrug. SIgh.
Clark Mary Lee Kimberly Hernandez Jessica
These are lectures and questions should be written down and answered at the end of the lecture. The lecturer should not be interrupted over and over again!
I do not fully comprehend Susskind's argument at 1:36:00
As I comprehend it, the normalisation constraint helps us determine the fourth variable/parameter if we know at least three of them. So, we can do away with one parameter (This elimination procedure will play a role again below). Let's now look at the phase factor idea. If we multiply the state vector by a phase factor such that one of the components of the state vector ends up with an imaginary part becoming zero, the physical nature (as per Lenny, as he says in the video) of the state vector does not change. So, with one of the components fully specified by a real number, say x, and the other as say a + ib we can use the normalisation constraint again, x^2+a^2+b^2=1. Now, using the elimination procedure, if we did not know either x, or a, or b, we could recover it, if we knew the other two, e.g. if x were unknown, we could recover it by x=sqrt(1-a^2-b^2). So, we only need TWO parameters to fully characterise the state of the system.
@@arjunchandra THANK YOU
Smith Susan Miller Frank Garcia Joseph
Can somebody help me prove 1:03:37
Can anyone please tell me why complex number i is shared with ket vector i and o?
Pavan Choudar he has to find 2 complex numbers that satisfy their magnitude squared add to one, and also that I and O can be expressed as linear combinations of both U and D and R and L in a symmetric sort of way. There’s no way of doing this with purely real numbers and so i creeps in
I think it was a blunder on his part to choose to call in "i" instead of another letter, and later he realized that and changed it to capital I.
My goodness! The people in this audience need to learn proper lecture etiquette.
they are not audience they are students XD
@@darklord69420 ppl viewing a lecture are an audience; they also happen to be students. But that's not even the thing to be pulled from my comment. Call them whatever you want; they still have poor lecture etiquette.
@@busterdancy1857 This is not kindergarten or primary school. This is university with adults who have the right to come and go as they please. If you can't stand that as a lecturer, then the solution is simple: don't be a university lecturer because your ego is in the way of teaching adult material to adults.
How do we know that
|A> = a_u |u> + a_d |d>?
At 58:20
This is how a vector is defined by using unit vectors (|u> & |d>) and coefficients (α_u & α_d).
|u> and |d> represent axes in an abstract vector space.
Compare this with a real 2-d space with orthogonal unit vectors y and x (these should have the ^ on to show they are unit vectors).
Then any vector in our 2D real space could be written as:
A = ax + by
Since |u> and |d> are the orthogonal unit vectors in the abstract space, then any vector in that space may be written in the form:
|A>=a_u |u> + a_d |d>
What if we were in 100 dimensional space tho???
100 dimensions work simply, if orthonormal
Its interesting that I can follow along with some of these things better than a few of the kids in this class. High school education here.
kids? they're mostly adults in there.
I AM a highschool student
Basically logic works differently here...
That's because it's not logic but non-commutative algebras.
This is some expert-level bullshit: Susskind had several classes with the same name, and you couldn't collect to a playlist of something? The are dozens of "The Theoretical Minimum" in this chanell, recordings from different years, just dumped without order.
I think the plural of apparatus is apparati.
He is not wrong, he says in classical physics its possible to determine it to infinite precision which it is, he says nothing about it actually being possible to do.
I find no "Lecture 2 | The Theoretical Minimum"
cool, now I’m going to eat a sandwich
There is one student keep asking dumb questions...
I love students thinking with their mouth open