So. So yesterday I gave a very speedy of I apologise, a too speedy introduction to the problem of compound systems. But we have a very tight budget of lectures. And I wanted not only to talk about explicitly about these composite systems and how you handle them, but also I think it's it's very valuable to discuss this classical Einstein, but also give Rose an experiment because it goes to the core of whether quantum mechanics is correct and what it really has to say about the universe.
And that wasn't going to be time. So I couldn't spend two lectures doing what I did yesterday. But it's been compressed in order that we can talk about this experiment, which is an important, crucial application of this apparatus, of how we apply quantum mechanics to compound systems. So Einstein is famous for saying that God is sophisticated, but he does not play dice.
He dislikes the the probabilistic aspect of quantum mechanics, not that he disliked or disapproved of the use of probability in physics. His his thesis work had been on kinetic theory and statistical physics. So he was quite comfortable with the idea that in classical physics you use statistical methods, probabilistic methods to do things like kinetic theory.
But he understood that the reason you were doing you, in that case, you were doing probability theory because you had incomplete information. So when you lack information, it's obvious that you have to you you have to assign probabilities.
The thing was worrying for him was working for him about quantum mechanics was that it asserted that even when you had complete information, which we know is in embodied in a set of a complete set of amplitudes, still the outcome of experiments is probabilistic and uncertain. And he felt that this was this was wrong because and the relation to God there, I guess, is that the omniscient God would not have the future uncertain, would know what the future was.
So God must know something that we don't know. We have we have an uncertain future because we are short of information. But the information must be there. We don't know.
We just don't know the information. So there must be some variables, some some variables which encodes the information about what's going to happen, which if you knew them and at some future time in physics, perhaps you would know what these variables were, and then you would be able to predict exactly what was what was going to happen.
And in 1935, Einstein with Podolsky and Rosen proposed this describe his thought experiment, which they argued demonstrated that there must, in fact, be these sorts of hidden variables. In in 1964, I guess it was John Bell analysed a similar experiment and showed that the predictions of quantum mechanics are actually incompatible with the existence of these hidden variables. And then in 1972, 20 years after Einstein's death, an experiment of this type was actually conducted.
And the the measure and many, many since have been conducted. And these measurements vindicate the predictions of quantum mechanics and therefore prove that these hidden variables cannot exist. So that's what that's what the agenda is today to describe this. So what's the the experiment that Bel describes is this, which is there are various versions of this, but you think that the key idea is the same.
Suppose you have some nucleus which is going to be unstable and it's going to commit a positron in this direction when it does emit and an electron in this direction. And Alice sits over here and measures the spin. She measures the spin of the components of spin of this electron as it comes by in a direction of her choice. We will call it a four Alice and Bob sits over here and he measures the spin in the direction of his choice.
The component of spin in the direction of his choice, which of course we will call B. So let's imagine Alice acts first. Sorry. And the idea is that the we know from nuclear physics that both before and after the decay, the spin on this nucleus is zero. We know, of course, that electrons and positrons are gyros they carry. They are spinning particles. They carry angular momentum because the angular momentum change of the nucleus is zero.
It must be that by conservation van momentum. The spin of this is oppositely directed to the. The spin of that so that the angular inventor of the electron plus the positron is zero. So supposing Alice measures the spin first and Alice gets that a dot, as it turns out to be, plus a half.
So she finds that the component is spin along. Her chosen vector A is s. What she then says to herself is this If Bob measures along a heap, if Bob chooses to put B equal two way, then he's guaranteed to find the answer minus half. Right, because if he measures with B along B, the components of the spin along b a vector B which is close to the vector A he's not very likely to get it, plus the half he's most likely to get, minus a half, but I can't guarantee [INAUDIBLE] get that.
[INAUDIBLE] get minus a half. Right. So the point of that is that that that kind of thought, that Alex's thought process makes it absolutely clear that Bob's measurements are going to be correlated with Alex's measurements. And what we want to do now is put that on a quantitative basis and ask, what does quantum mechanics have to say about this? Okay, so what we want to do is talk about the correlations between the measurements that beam the bob makes and the measurements that Alex makes.
Given that Bob is going to choose vector's be at his discretion. So let's talk about the quantum mechanical predictions. So what we do is we choose which we're free to do. The z-axis has to be a long atlas's vector. We can do that without loss of generality. And we will find this is I'm about to write down a result that will emerge in the next couple of weeks from our work in angular momentum, maybe emerge next week from our work in angular momentum.
But for now, I have to ask you to take on trust that because the electron plus the positron together have no angular momentum. It must be that that wave function can be written like this as E plus p minus minus, e minus, p plus. So that is to say that the states we know that we know that a spin, a half particle, again, this this all needs to be justified properly when we do angular momentum.
But we anticipated these results once before, early in the course that a spin half particle has the complete set of states, which are the state in which we're guaranteed to get plus along some direction for the spin and minus along the direction for some spin. There are a complete set of states. So here are the complete set of states for the electron. Here is the complete set of states of the positron. And this says that there is a probability of a half.
Right. This this one of root two is the amplitude. To find that the electron is plus in the Z direction and the and the positron is minus in that direction. And this is the this is and there's a similar amplitude with a minus sign for the opposite possibilities. So the origin of this expression will emerge shortly. I must ask you to take this on. Trust. So this is the state of the system, the composite system of the electron and the positron taken together.
So we talked about the collapse of the wave function in these circumstances yesterday. Alice makes her measurement, she finds plus, which means that she collapses the wave function into this. So this is. This is before Alice. Makes a measurement. After Alice has made her a measurement and found. Plus we have that ASI is simply E plus p minus, which is to say that there is unit amplitude that Bob will find minus if he measures along the z axis.
In other words, if he chooses B to be the Z axis, which we've established is which we've chosen to be the same direction that Alyce chose. So that's consistent with what Alice said. What happens if he takes to be some other direction? Well, what we need to do is express some of the directions, but is so we need to write the cat. So we would like to calculate the amplitude. Well, the probability that if that when B uses some other direction, he finds it to be positive.
He finds that S of his positron is along that direction, has plus a half along that direction. So in order to do that, I have to ask you to take something on trust that we will derive. But we've seen before, which is that this thing is equal to sine theta upon to e to the i phi upon to of positron. Down plus costs. These are upon to each of the minus I find on to for the positron plus.
So what does that say? That says the the the state of having of of being certain to give you a half along the vector B for the positron is given by this amplitude that's just some complex number. Times the amplitude times the state where you will definitely get minus a half on the z axis, plus this amplitude times the state in which you're guaranteed to get plus a half on the z axis for the positron and Theta and Phi are the polar angles of B,
right? B is the unit vector. So it is defined by a couple of angles. And the feature in PHI they give you the orientation of B with respect to the Z axis and the and here we using the complete set of states along the z-axis. So that's a result we will derive. But I'm asking you to take it on trust for now. So what is the probability that Bob measures place on B? The answer is that we it's according to the dogma of the theory.
It's this because the state of the system, the state of the positron after Alice is made, his measurement, her measurement is definitely this. So this is that's that's how it works. The apparatus. So basically we flip this around, we take the emission that joint of this thing, bang it into minus. And guess what? We get the complex conjugate of this coming out.
Oops, that science is wrong to e to the minus i fi and I've written on to and I've written the probability, which means I need to do a mod square to a mod square. This factor goes away and we're looking at sine squared. These are on to the it follows straight away. You could also calculated that the probability that Bob finds minus on B is one minus. The probability he finds plus on b is equal to cos squared theta upon to. So that's that puts precisely on a quantitative basis what Alice said.
Alice said that if, if, if Bob chooses a vector B which is very similar to my AA, which is the case when theatre equals nought, then he's guaranteed. Well, if he's identical, he's guaranteed to find minus because this becomes this becomes this becomes one and that becomes nought. And if he chooses a vector B, which is similar to my vector array, he's not it's not guaranteed that [INAUDIBLE] get minus, but he has only a small probability of getting plus.
And that's because theta will be small and in his probabilities looking like science with these two. So what Einstein, Podolsky and Rosen said. Well, the question is, why is the result that Bob gets somehow dependent on the measurements that Alice gets? And in particular, it looks like the result of Bob's measurement depends on which direction Alice chose, because this this angle feature is the angle between Bob's vector and Alice's vector.
And we can imagine that, Alice. Let's imagine that Alice goes first and chooses a direction. Apparently, Bob's the probability of Bob's out of outcomes depends on thesis, therefore depends on analysis choices. But supposing these this positron an electron is sent out at relativistic speed. Perfectly plausible that they are then Alice and Bob. Well, Alice makes a measurement and Bob can make a measurement in the rest frame of the nucleus is essentially the same time.
And if Bob acts that quickly, then there is no time for a light signal from Alice to rip to reach Bob. Setting out after Alice has made her measurement. So Bob definitely makes his measurement incomplete, and it has to make his measurement in complete ignorance of what choices Alice may or may not have made. And indeed, if in this relativistic case, it's easy to to see that who acts first? Different observers. Observers moving at different speeds with respect to the to the nucleus.
And Alice and Bob will disagree about who acts first. The whole question of who acts first is neither here nor it clearly can't affect the physics because it's an observer dependent statement according to relativity. So. How so? So. So. So. How is it that the result of Bob's measurements depend on Alice's choices? When it's not logically possible for a signal to go from here to there in order to effect it.
Well, Einstein, Podolsky and Rosen said what it must be is that actually the result of Alex's measurement is preordained. We don't know what the what the result is going to be, but that's because we're pig ignorant. But God knows it's it's foreordained because the the result is encoded somehow in the state of the electron not written on the board because we're using this clapped out quantum mechanical rubbish.
And similarly, inside the positron, there's also this magic information, this DNA, this whatever, which which foreordained, the result of Bob's measurements. And then everything is okay. That was their interpretation of this of this problem. So now let's talk about Bell's inequality. So that was the state of affairs for, I guess, 30 years, right? 1935 until 1964. So John Bell said, okay, so let's let's calculate something. Let Sigma Ray be the result of Alex's measurement.
Oops. And it's obviously going to be plus or minus a half. Right. What whatever number she comes up with is going to be plus either plus a half or minus a half. And simply and similarly, sigma B being plus or minus a half is a result of Bob's measurement. And let's calculate the expectation value of Sigma A, Sigma B. So there are four cases to consider because they can both measure plus a half. They can both measure minus a half one can measure plus one half. My mother minus half, minus a half.
And that is two different ways. So so this thing is going to be there are four possible values that sorry, Sigma Theta Times Sigma B can be either could be either plus or minus a quarter.
And the possibilities to consider are the probability that Alice gets plus at times the probability that Bob gets plus, given that Alice gets plus plus the probability that Alice gets a plus and Bob gets sorry, A's gets minus and Bob gets minus the probability that Bob gets minus given that Alice has got minus, right? So in both these cases in both these cases, that product is going to be plus, right? Because in this case, this is going to be plus a half and that's going to be plus a half.
In this case, Alice Sigma is going to be minus a half and Sigma basically minus half. The part is going to be plus a half. And then we have some minus cases, which is the probability that Alice gets plus say and Bob gets minus given the delta got plus and then we have minus the probability that Alice gets minus and Bob gets plus given that Alice got minus. Okay, we have to make Bob's probabilities, his conditional analysis, because we've seen that they're correlated.
We can argue that the probability that for ALS get plus is the probability for Alice to get minus, namely it's a half. We don't. When Alice makes her measurement, we don't know a blind thing, so both possibilities are equally likely. So that must be the that must be what these probabilities are for Alice. And we've just worked out what the probability we just worked out what the probability for Bob to get plus was.
You know, we've worked out these probabilities so we know that the probability for for Bob to get. Plus given that Alice got plus we found that that was sine squared thetr on to right. So that's science with these are on two by symmetry you could work it out but by symmetry this will be science with these are upon to right.
Because if Alice has got minus we know that Bob is jolly unlikely to get minus if he chooses if he chooses an angle, a vector which is close to a and we figure these that's this one here. We've already shown that the probability is for Bob to get for Bob to get minus given the date has got plus we've already shown is cos squared feature on two. So both of these are going to be cost squared. These are upon two and both of those are going to be sign squared.
These are onto which means that sigma sigma be expectation value is a quarter of sign squared thetr on Trump's theta onto. Twice over because we get two times like that minus twice cos squared three to respond to o times a half. Sorry, sorry, sorry. So here's a half and here's a half from the probabilities of A's. So those twos are not really there. And what is this cost squared minus science squared is cost twice the angle. So this is minus a quarter of costs theatre.
And what is cost theatre? Cost theatre is actually a B, it's the angle between A and B, so this is minus a quarter of a dot B, so that's what quantum mechanics predicts is the expectation value of the product of these two measurements. So now what Bell did was calculate what this would be in a hidden variable theory. So draw a line. Now we're into another conceptual framework. What we're going to say is that so there is some function. Sigma e which will depend on.
So what's this? This is this thing. Here is the value that you will find for the spin. The component is the spin of the electron along the vector a. We think this is a random variable because we don't know the values taken by the hidden. This is a set of hidden variables. This is a an end vector with components which are the hidden variables that we don't know. But Einstein, Podolsky and Rosen claim must exist to make the outcome of these experiments causal.
So. So this is not a probabilistic quantity. This is something this is either a half or it's minus a half, right? Depending on the values that these variables hidden from us about which we do not know. And of course, on the direction in which you measure the component of spin. All right. So this is equal to plus or minus a half in a causal way. And similarly, there must be sigma p.
This is the positrons. This is the positron spin that's also going to be plus or minus a half, depending causally on these things. We don't know what this function is. We don't know what these variables are. We don't know how many of these variables there are or anything but what we do.
But we will do know by conservation triangle momentum that is minus sigma electron at V and B, because we know that the the positron spin is oppositely directed to the electron spin by conservation of angle mentum. So if you get plus a half here, you are certain to get minus a half here. So this is this equality is conservation. Of angular momentum. So what we do now is evaluate the expectation value which quantum mechanics told us.
So we do sigma three depending on a times sigma p, depending on p expectation value. We write this out as in classical probability theory. Now, what's that going to be? Well, this expectation value means averaged over all possible values of the hidden variables, the things that we don't know. Right. So the reason that the outcome that this thing seems uncertain to us is because this thing is unknown to us and we therefore think of this as a random variable.
So what was this expectation going to be? It's going to be an integral over the components of V. We have to sum of all possible values of what we don't know. Times. Sum. Probability density. But we don't know.
Times Sigma e of V comma a. Times Sigma P of the comma b. So basically we just we just take an average of this product which is completely determined by V and then but we take an average with this appropriate way over all the possible values of V to get the experimental expectation value standard probability theory. The next thing that we do is we replace this by the corresponding sigma e using that neg that, that switch of sine business.
So we argue that this is minus the integral d to the envy row of sigma e three comma a sigma e v comma b So anything that's changed here is we've acquired a minus sign and that P has become an E. Now we say, okay, now let's imagine that we make this measurement with some other vector. Right? Supposing we now calculate the same expectation value between A and the vector C, just some other vector. And then we have the expectation value of sigma e a sigma p b minus.
That's an next. That's a complete expectation value minus the expectation value of sigma e a sigma p c some vector, some other vector C And what's that going to be according to this apparatus? It's going to be it's going to be minus the integral D to the end the row depending on v open a bracket. No. Sigma e of v comma a will be a common factor. And then we will have sigma e of v comma b minus sigma e of v, comma c right.
Because the right hand sides are both going to have this factor because we've taken the expectation value using sigma of a sigma, even in both cases. And what will differ in the two cases is that term in the back. So one time it'll be on, one time it'll be C. So that's what we get now build as something slightly nifty.
He makes the observation that well, but he knows that sigma sigma squared V comma B is a quarter because he knows that this number is either plus or minus a half, depending on the values taken by V and B, the square of this number is guaranteed to be a quarter. So we can we can say we can insert into here, we can insert a four sigma squared e a v comma B without any harm, right?
Because we just inserting a one. So he says that this expectation value this commodity, I'm going to write it out again. This expectation value on the left is minus the integral d to the end v rho sigma e of v comma a bit of write down for sigma squared e v comma b brackets sigma e v comma b minus sigma e v comma c very helpful. I'm sure what we now do is we take we break this sigma squared into sigma and sigma and we take one of the sigma is inside here.
When one of these signals comes in here, we get a sigma squared again, which is a quarter times four is one. So we get a one appearing here and then we then of course, this sigma, the sigma that I brought in appears there as well.
So the next line is this is equal to minus D, each of the N.V. rho sigma re of v comma a sigma e of v comma b brackets one minus four times sigma re of what we carried this one in V, comma B and we've already got one there, which is a sigma of V, comma C close brackets, close brackets. So this is what that expectation value what the top is, it's this.
So why spell done this. What we now argue is that this bracket so so this product of things here is going to be either plus or minus a quarter, right? Because all of these things, they're causal functions and that they are either equal to plus a half or they're equal to minus. This product is equal to either plus a half a quarter or minus a quarter. We don't know. But whatever happens, so this bracket is either equal to zero or something positive.
It has the brackets equal to two or nothing. Right. So what we really need is that this bracket is is greater than or equal to zero. It's not negative. This thing in the front here is is a fluctuating quantity. It's equal to plus or minus a quarter. So what we can argue now is let's take the modulus of both sides.
The modulus of the left side is whatever it is, the modulus of the right side just means we drop this and we can argue that this integral this integral is going to be smaller than the actual integral here is going to be smaller than what we would get if we replace this with plus a quarter, because sometimes that is minus a quarter and we'll be taking away from the integral given that this thing here is never this thing here is never negative,
there's no way that we can never get a positive result, a positive contribution to the integral when this is negative. So if we if we assume that this is always positive, we're going to overestimate this integral. So let me write that down. We will overestimate, overestimate, integral if we replace. Sigma e v com a sigma a v comma b by plus a quarter, because sometimes it's minus a quarter and that minus sign is never cancelled by any minus sign over here.
So then I can argue that the modulus of the left side, which I'm fortunate enough to write out again. That's p sorry. Sigma e of a sigma p of C the modulus. That's an expectation value. Now in your modulus sine is less than or equal to because I'm going to write down something which is which is too large. I've deliberately made it too big of the integral rho that is been. That factor has been replaced by a quarter. This quarter can be taken outside and then we're staring at one minus four sigma.
This is of the comma B sigma, e of the comma. See, now we make the observation of the integral this. So we break this integral into two parts. It's this stuff times one, but that integrates up to one because this is the probability density. And the probability density has to be structured so that if you if you integrate probability density time overall all parameter space, you get one. So this and this make a quarter.
So this is this, this thing I'm going to write down what is equal to which is it's equal to one from the court excuse me. It's equal to quarter brackets of one from here then. Now let's consider this onto this stuff here. This on to this stuff here is roughly speaking where we came into this that that this times this was the expectation value of of sigma on sigma and this minus sign we can soak up by changing that back into a P that's retracing logic that we did up there.
So this becomes one plus four times the expectation value of sigma e b sigma P of C expectation value where the V has disappeared from here because we've done an expectation value operation, we've we've averaged away all the V dependence in the proper way. So we have this is Bell's inequality that we have here.
Now it's a statement about expectation values associated with the two particles and three possible vectors A, B and C. So the next thing to do is to ask all the predictions we have perfectly. We've calculated the predictions of quantum mechanics for these expectation values. We've already done that. So the question to ask now is all the predictions of quantum mechanics consistent with this inequality. I guess we need to be able to see everything simultaneously. And I've I've not held that right.
So let's, let's write the let's write down here let's find the predictions of quantum mechanics. Okay? This is the crucial thing. The prediction of quantum mechanics is that this product, which is the other calculation for reasons which if you stare hard at it, you'll realise that there's a notational issue, there's a reason for this. This in the in the hidden variable calculation is called sigma e sigma p because remember Bob is measuring the positron, Alice is measuring the electron.
So this is the this is the this is actually the same physical quantity that we've calculated down there and it's equal to minus a quarter of a dot B, so we can go straight back. So now we put in Sigma E a sigma e p the expectation value is minus a quarter, a dot B, which is from quantum mechanics. What does that do? Well, let's let's check out the left hand side. What does the left hand side look like? It's going to be the modulus of a quarter a dot c well. My minus eight dot B, right?
So the overall minus sign gets lost, but it's going to be the modulus of this thing here. That's the left hand side. What is the right hand side going to be? It's going to be a quarter of one minus a dot B. Sorry, I don't see. So now we need to ask ourselves, is it true that this right hand side is bigger than this left hand side? And in this matter, we can choose A, B and C exactly as we will.
Right. Because Bella shown the for any vectors, A, B and C, his inequality has to has to hold if there are hidden variables there. There's so far no restriction on A, B and C that any three vectors. So and if the quantum mechanical results violate bell's inequality for any vectors A, B and C, then quantum mechanics will be inconsistent with these hidden variables.
So at this point, we do a choice. We choose A dot, B equals nought, and we choose C is equal to is equal to a C cost of CI plus B sign up CI. So what are we doing? We're simply of some angle. We're just choosing A and B to B orthogonal vectors and we're choosing C to be a vector that lies between A and B, and we've got ourselves a parameter of CI which allows us to move C from pointing along A to pointing along B in a continuous way.
So with so just concretely, the picture is here is A we choosing age B this way we're choosing B to be that way and we're choosing C to be like that somewhere in the plane, stuff it in. And what do we get? We find that the left hand side is the modulus of a quarter a dot c is is a dot C is cost of CI a dot b is not and the right hand side is a quarter of one minus sign of CI. Plop these up. And what do you find? Sorry. Can we change these back to can we change that to sign up size.
Because my diagram will look better if I do cos of ci. Sign up ci. Cos of CI. Right. Okay. So obviously there's nothing in that. It's just a change in in the figure two unfortunately. Right. Then what do we get? We find that the right hand side looks like when upside is small, the right hand side is looking like CI squared on eight or something. Anyway, it's rising quadratic, Lee, and it goes to one. This is PI by two. Meanwhile, the left side is is basically a sign curve.
So we know what that looks like. It looks like this. So this is the left hand side. This is the right hand side. And Ballard has shown that the left hand side is smaller than the right hand side. So for four, for only two values, smaller than or equal to the. The quantum mechanical results are consistent with Bell's inequality for only upsized nought and upsized pi by two. The quantum mechanical results violate this inequality for all values. So basically so we conclude QM is inconsistent.
With these hidden variables. Once you've got a nice, clean statement of this sort that there's the quantum mechanics is inconsistent with something which EPR reasoned was should be the case was very you know, the indications were that it was the case. You clearly the right thing to do is to go out and make a measurement and allow nature to decide for you whether quantum mechanics is right or hidden variables.
Right. So in 1972, this was first done using not an electron and positron pair, but using pairs of photons. That's usually how this is done. The analysis is slightly more complicated if you use photons than if you use spin half particles. So we followed barely using spin off particles. But basically many of these experiments have now been conducted.
And the experiments vindicate the experiment, vindicate the quantum mechanical predictions with a level of precision that you you know, the it's clear that the the experimental results are inconsistent with with with hidden variables. So the experimental results and that's from 1972 onwards, there have been many always refined experiments are consistent. With QM and inconsistent.
With hidden variables. So that means that quantum mechanics is not going to be replaced by a hidden variable theory at some time in the future. Because you cannot construct a, you know, hidden variable theory along these lines is not going to be consistent with experiments are already conducted, so there's no point speculating about it. So to come back now to Einstein, Podolsky and Rosen, what is wrong with the arguments which indicate that somehow these measurements knew about a measurement?
I think a lot of the. Well. Sorry. So the things that you should take away from this are first that when you measure something, you do two things. You disturb the system and you gain information about the system. So when Alice when Alice measured that electron and found it plus a half for the spin in her direction, a she disturbed the electron, but she didn't disturb the positron because the positron was somewhere else and there wasn't.
The positron couldn't possibly be disturbed by anything down to the electron until there had been time for light signal to go from her operations to wherever the positron was. So she definitely doesn't disturb the positron, but she does disturb the electron. Therefore, she disturbs. She changes the state, she physically changes the state of the electron positron system. And that's why she's changed. She's collapsed the wave function from that linear combination to this here.
But she has gained information about B because of the correlation that existed in the original in the original set up between her electron and the positron. By knowing by having discovered what was the state of affairs with the electron, she she was able to make some quite strong predictions about what B might find, what Bob might find on measuring the positron.
This is this experiment emphasises a theme is quite common and is quite recurrent in quantum mechanical calculations and it's very important to think holistically. To do this problem, you have to think about the electron positron system. It's no good thinking. Oh, I can do with electron all I can deal with a positron both together have to be considered because of these correlations in the system.
The a lot of the confusion that I think Einstein, Podolsky and Rosen had, and that is in many treatments of this, of this experiment arises from slipping into the error of thinking that because Alice has found a plus a half for the component of spin on her vector a that the spin is pointing a long way. As we shall see, a spin off particle has a always plus a half of spin in the directions of all three coordinate axes.
When you've made a measurement of the Z component, you can know that the answer. You can know that it's it has a positive value. Four Sigma Z. But you don't know this.
But but you don't you don't know what the values of sigma x and Sigma Y are, but you know that they have you don't know the values, but you know that they do have values which are comparable to that of Sigma Z. So what you should physically think of is that Alice is determined that the spin of her electron points in the Northern Hemisphere. Well, in the hemisphere that has her vector a for its pole.
She does not know. It's pointing that it's aligned with a she only knows it's in the northern hemisphere of that. So if when Bob makes his measurement and then she can say, aha. So I now and she then knows for certain that the positron has its spin in the southern hemisphere of her vector a. Right. Because but she does not know where it points there because she doesn't know where her electron points in her hemisphere. She doesn't know where the positron points in its hemisphere.
She only knows now all she's learned is which hemisphere that the positron is pointing in. So she can exclude, as quantum mechanics says, she can exclude only one result of B's measurement, namely, if B chooses to. If Bob chooses to measure along the vector A, then he will not find. Plus a half, because the top hemisphere has no point in common with the bottom hemisphere and at least knows that the positron is in the bottom hemisphere.
So the I think the bottom line is that there isn't a logical problem if we just keep it focussed on the, on the idea that what is preordained is that is which hemisphere. The electron or the positron is is pointing in a not the direction. It's an error to think of these spins as pointing in a particular direction. It's a it's it's it's difficult to escape from the idea that a vector points in some in some direction.
But then it's difficult when we do relativity to get used to the idea that time is relative and that two events that are simultaneously one event that happens before another event for in our frame of reference in somebody else's frame of reference reverses the order of the events. So the absoluteness of time is something it's very difficult to escape from. But we all grow up. We get used to it.
Time isn't absolute and in and quantum mechanics is telling us that no vectors don't point in particular directions. Uh, they, there's a, in the case of spin of particles, the best you can say is that they have particular hemispheres in which to point and we'll as we go on. So the next item on the agenda is angular momentum.
And that will enable us to to look at this a little bit more closely about under what circumstances it is the case that that that a a gyroscope or whatever seems to point pretty much in a definite direction. And we'll find in just the same way that things move only because they have ill defined energy. Things point in a definite direction, only because they have ill defined angular momentum and electrons do not have a well-defined Anglicanism.
They have well-defined tangle, mentum, and that stops them pointing in any particular direction. Okay, hold on.