The Science of Everything Podcast - Episode 84: Advanced Quantum Mechanics Part II

Episode Date: July 7, 2017

Continuing on from the previous episode, here I discuss some more advanced topics in Quantum Mechanics, including Noether’s theorem, the particle statistics of Bosons and Fermions, perturbation theory, and the EPR paradox. Recommended Pre-listening is Episode 83: Advanced Quantum Mechanics Part I.

You're listening to The Science of Everything podcast episode 84. Advanced Quantum Mechanics Part 2. I'm your host, James Fodor. So this episode follows directly from the previous episode, Advanced Quantum Mechanics Part 1, which turned out to be longer than I'd expected, so I've split into two episodes. That's definitely recommended pre-listing for this episode.

In this show, what I want to talk about are some more advanced ideas in quantum mechanics, including Norfolk's theorem, particle statistics, perturbation theory and a little bit about the EPR paradox. So these are some of the more juicy concepts, which require a little bit of background to sort of understand. Hopefully you'll find this interesting. But do be warned that these are some relatively advanced topics,

so it may be heavygoing if you don't already have some background in QM, which is why I recommend the previous episode and also the principles of quantum mechanics episodes that I think episode 14 that I did some time back. Okay, so let's get into it. In the previous episode, I outlined as best as I could, without of any visual aids, the core ideas of Dirac's brachette formalism of quantum mechanics, and the basic idea of being able to represent an arbitrary quantum system

as a superposition of eigenstates, and each eigenstate corresponding to some measurable observable. Now, I'm not directly going to continue on from that, but I'm going to assume that as sort of background in the discussion that follows, and some of the things I talk about will require more than others. But basically now I'm going to introduce some key ideas that are pertinent to a more advanced study of quantum mechanics and that are just interesting in their own light.

So the first I'm going to talk about is Nertha's theorem. Now, Nertha was a mathematician who was, well, obviously responsible for Nertha's theorem and also had quite a large influence in a number of ways. he was a female mathematician at a time when there weren't very many female mathematicians, so it's particularly noteworthy for that reason as well. If you're wondering about the name, it's spelled N-O-E-T-H-E-R, and I've heard it pronounce NERTHER-S, so hopefully that's more or less correct. So what does NERTHES theorem actually say?

Basically, the theorem is about symmetries. So the theory says that for any physical system that has a continuous symmetry of some property, and we'll get to what that means in a moment, there is always an associated conserved quantity that does not change over time. So the basic idea of NIRTHIS theorem is to say that symmetries correspond to

conserved quantities, and conserved quantities correspond to symmetries. A symmetry essentially is something that looks the same or stays the same after you do something to a system. So, for example, if you stand in front of a mirror,

your face looks basically the same, even though the image has actually been flipped left to right. Your face is not exactly left-right symmetrical, but it's pretty close because you have an ear on each side, or most people have an ear on each side, an eye on each side of the face, and the mouth is symmetrical, more or less, and so on. So faces are essentially symmetrical. Humans are bilaterally symmetric animals. So your face looks essentially the same even after it's been flipped left to right in the mirror. That's not the case for writing. Writing is not symmetrical. That's why writing looks back to front in a mirror,

because it is back to front. It's not symmetrical. Some letters, in fact, are symmetrical, like the letter O or the letter U, but the letter E is not symmetrical. If you flip that back to front, it looks completely different. So that's just an example of a symmetry.

Something is said to be a symmetry if you do something to something, and that thing stays the same. So in the case of the letter O, it has a lot of symmetries, but it has a left-right symmetry such that when I flip it to the left or to the right, it looks the same. So that's one simple example of a symmetry. In terms of the physical systems that we're talking about, there are many different symmetries that are potentially of interest, and I'll talk about those in a moment. But the key point here, and this is what Nathis theorem says, is that each symmetry corresponds to the conservation of some property.

intuitively this makes sense because a symmetry is something that stays the same after you do something to the system like flip it left to right for example where and conservation over time means well it doesn't change over time if I look at it at one time and look at it again later that quantity will be the same so both symmetry and conservation relate to something staying the same so it's perhaps not completely surprising that there's a connection between the two of them but so far I've been pretty abstract especially talking about symmetries as doing something to something and it's staying the same. Well, what does that mean exactly? Let me give then some examples of where NIRTH's theorem applies and why it's interesting. By the way, another word that we use when we talk about NERTHIS theorem is invariance.

That means essentially the same after you do something to it. So it's directly related to the year of a symmetry. Symmetries occur when a system is invariant to some change that you make to it. Again, like flipping left to right. So let's look at then some examples of NERS's theorem to illustrate what the point of it is and why it's interesting. One example is that physical systems are invariant with respect to spatial translation. That is, the laws of physics don't change with location in space. I mean, they can change if you move around in space, of course, but that's always because of other factors.

Like if there's a strong gravitational field in one place than another or an electric field, obviously, then you'll experience different forces. But that's not what's being spoken about here. This is just purely in terms of spatial coordinates. If I add 50 to all my spatial coordinates, if I jump a light year in one direction, one direction in deep space where there's nothing around me before or after the jump, do the laws of physics change? Do I experience some new force or something? The answer is no. We don't expect that. We don't observe that. There's no reason to think that. So we say that physical systems are invariant with respect to spatial translation. Nothing changes if you just move everything a meter to the left or a light ear to the left. So this is a symmetry, right? We can shift everything

in one direction in any of the in any of the cartesian directions up down left right whatever we can shift our whole physical system arbitrarily far in in one direction in whichever direction we like and nothing should change so this is a symmetry this is an invariant property of the system notice theorem says that there should be a corresponding property that is conserved over time and you can actually it's a theorem right so northus theorem is a mathematical theorem you can actually derive what the conserved property should be, which is a really powerful result, right? Because in the cases that I'm going to list, we kind of knew the answer beforehand. We know that all of these properties that I'm about to discuss the conserved beforehand,

so Nuthysdium doesn't tell us anything that we didn't know. But if we start looking at new exotic cases where we don't necessarily understand the physics beforehand, especially when there's no classical analog, like the weak nuclear force, for example, or strongly nuclear force, there are no classical analogs to those. we might not know what all the symmetries are beforehand, and so NERTHOS theorem provides a very powerful tool for figuring out what the symmetries and what the invariant properties are.

So, in the case of invariance to spatial translation, it turns out that NERTH's theorem gives you that the linear momentum will be conserved, that is, linear momentum of the system won't change over time. That's pretty cool. That tells us that there's a connection between spatial translation and linear momentum. Another example is rotation. Space, as far as you know, is isotropic in the laws of physics. That is, it doesn't matter if I'm facing one direction or facing the opposite direction. The direction that I'm facing, this is not my position anymore. This is the direction that I'm facing. My rotation in space should not affect any of the laws of physics or any of the things that I measure or observe. So this is what we say, invariance with respect to rotation. The fact of invariance with respect to rotation, that's another symmetry.

I rotate any physical system by any angle that I want, it shouldn't change anything. So there should be a corresponding conserved quantity, and it turns out that in this case that conserved quantity is angular momentum, which again we knew is conserved, angular momentum is conserved. This tells us that there's a connection between angular momentum and rotation. That's pretty cool. The other two main ones that I'll talk about here, that is symmetry or invariance connected with a conserved quantity,

is invariance with translation in time, That is, it doesn't matter whether a physical process happens today or 100 years ago. You can translate it in time, you can move it about in time, and it should act in the same way. You should measure the same things, as far as we know again. This leads to conservation of energy. Now that's pretty cool. Turns out that conservation of energy is not just a brute fact of the universe. It actually follows from invariance to time translation.

So if the laws of physics essentially are constant in time, we would expect to see conservation. of energy. So that's a fascinating result. And the final example that I'll give here is conservation of electric charge follows from invariance with respect to change in the phase factor of the complex field of a charged particle. And that's a little bit more of a mouthful,

but all quantum systems have a phase associated with them that generally we don't observe because what we observe are the probabilities, and phase is essentially a complex component. So you need to square the wave function to get the probability, essentially, and therefore the complex

components always cancel out so you don't observe those phase factors. They're the bits that you don't see essentially, the imaginary parts that you don't see. But they can make a difference to certain outcomes and in this case the fact that a system is invariant with respect to changing that phase factor because you don't observe the phase factor leads to conservation of electric charge which is also pretty cool. So again that's not just a brute fact of the universe it actually follows from something even more fundamental which is this invariance with respect to changing the phase factor.

So that's the reason I wanted to highlight NERTHIS theorem is because it's a very powerful result, which in fact is not directly, explicitly, just a quantum phenomenon, it applies more broadly, but it's very useful in understanding why some of these conservation laws apply. It's because of underlying symmetries. Okay, so that's NERTHIS theorem. Now I want to move on to talk about particle statistics. This is a much more explicitly quantum mechanical phenomenon here, but it's very important to understand.

In classical mechanics, all particles are, in principle at least, distinguishable. This means that you can track an individual particle. In principle, you can distinguish particles either by labeling them in some way, so identifying a property of them that is distinct, and then just measuring that property and saying, okay, well, that's this one. Or you can just follow the trajectory through space. So I could find an electron and just watch it as it moves about its trajectory and goes wherever it goes, and then see, at the end of the trajectory, okay, well, it's...

I've kept track of it throughout its trajectory, so it's got to be the same electron. Classically, you can always do that. You can always follow the trajectory of a bus or of a billiard ball or of a person or a planet or whatever. So you can always, in principle at least, follow where it's been and identify that it's the same one or if it's a different one than you were looking at earlier. In quantum mechanics, however, we cannot do that. Essentially this derives from the Heisenberg Uncertainty Principle, right, which relates to the incompatible eigenstates of position and momentum that I talked about in the previous episode.

But it boils down to the fact that the classical trajectory of a quantum system is not well defined. You can't simultaneously define the position and the momentum of any quantum system. And so its trajectory, which essentially is just a specification of the momentum and the position at every given moment in time, is not defined. So you can't just follow the trajectory of a quantum system over time. and thus there's no way of determining whether the electron that you're looking at today is the same exact electron as the one you looked at yesterday. The only other way, if you can't follow the trajectory, would be to identify some property that is different about the electron. But of course all electrons have identical intrinsic properties, by definition really.

Charge, mass, spin, etc. They're all the same for all electrons. So you can't distinguish it on that basis either. So all fundamental particles, and in fact many non-fundamental particles in quantum systems as well, like Adam, and molecules and so on, are indistinguishable from each other. It's not just that we can't tell whether one is the same as the other, it's that, as far as we know, according to the laws of physics, there is no fact of the matter about whether they're the same or different. It just makes no sense to ask that question.

They're just two electrons, and there's nothing to say about whether the one I'm looking at today is the same electron as I looked at yesterday under this microscope, or whether it's a different one. All I can say is that it is an electron and it is now in such and such a state. So this is what we mean when we talk about indistinguishable particles. And fundamentally this is different from the classical case where particles are always in principle distinguishable from each other. Because if all else fails, you can just keep track of it where it is at all times. You can't do that quantum mechanically because not because our measurements are limited, although they are,

but it's because the position and momentum are not simultaneously defined. and so there is no defined trajectory of the particle over time. And so there's nothing to keep track of, and so no fact of the matter about whether it's the same particle yesterday or today. So, quantum particles are indistinguishable. Now, this has very important effects when it comes to theory and also calculating empirical results. If you don't factor in this consideration

that quantum particles are indistinguishable, you'll get the wrong results, and there are some famous examples from when this has happened. Now the reason it matters is because we're often interested in keeping track of configurations of the system. This relates to entropy, for example, so entropy is determined by the number of microsets, the number of underlying configurations of the system, and counting how many of those there are that correspond to a specific macroscopic energy state and so on. However, in the case of a macroscopic system where all of the underlying particles are distinguishable,

in that case, if I interchanged the position of two arbitrary particles, let's say people in a room, let's consider people in a room, clearly those are macroscopic and distinguishable. If I interchange the position of two people in a room, I've changed the underlying configuration of the entire system. It's a different state now. Now, instead of people, let's think of electrons, if I just swap the position of two electrons in whatever system that they're in, whatever chamber that they're located in, that does not change the configuration of the electrons. the system because the two electrons are identical. In fact it doesn't even really mean anything to swap two electrons because that sort of implies that they have their own

distinct identity which they don't. But you could imagine if we were to label them in the initial state and then swap the labels between two of them. That's sort of not literally swapping them around but in imagination land swapping them around. That does not produce a new configuration of the system because our labels are purely arbitrary and it doesn't reflect any actual physics of the matter. So the actual number of states, the number of states, the number of of distinct states that the system can occupy is different if, or changes, with indistinguishable compared to distinguishable particles. And so that's why it's so important to know what you're

dealing with. But it turns out that quantum mechanically there are two different types of indistinguishable particles. These two different types aren't relevant to distinguishable particles because you can always tell the difference between one particle and another. But for indistinguishable particles, you get two distinct flavors, if you like, or varieties of behavior. These are called bosons and fermions. An example of a fermion is an electron, an example of a boson is a photon, but that is not so important for our purposes. What we want to understand here is what the difference between the fermions and the bosons are.

And the key point that I've emphasized so far is that it all comes down to the fact that they're indistinguishable and we can't in principle identify what one photon is or what one electron is as distinct from a different one. There's no fact of the matter about whether it's the same as the one I looked at yesterday. Okay, now to see how this works, let's now consider a situation where I have two particles in two different possible energy states. So this is a two state system with two possible particles. If those two particles are distinguishable, let's call them one and two, particle one, particle two. Now, if they're distinguishable, then there are four different states that the system could be in.

I could have particle 1 in state A and particle 2 in state A, they're both in state A, or I could have them both in state B, or I could have particle 1 in state A and particle 2 in state B, or the other way around, particle 1 could be in state B and particle 2 could be in state A. Each of those, unless I have information to the contrary, is going to have the same probability, let's say 0.25% probability, so they add up to 1 as they should. That's the distinguishable case. That's the classical case. That makes perfect sense, right? You know, there are four possible outcomes, both of the same A, both of the same B, or one in A and one in B, or one in B, or one in A. So flipped around. That's the distinguishable case.

Now let's think about what would happen if my particles become identical, that is indistinguishable. What happens? Well, I can still talk about my case where they're both in state A and both in state B, because then I can distinguish not one particle from another, but I can distinguish whether the particle is in state A and state B. There's a difference there about distinguishing what state the particle is in and distinguishing which particle is which. So I'm okay with my both in A and both in B states. So those carry over from the distinguishable case, those are unchanged. However, recall the other two distinguishable cases,

where I had particle 1 in state A and particle 2 in state B, and then the flipped version, particle 2 in state A and 1 in state B. In order for those to be different states of the overall system, I have to be able to tell which one is particle one and which one is particle two. If I can't do that, then it becomes meaningless to make the distinction. And in fact, in the indistinguishable case, I can't make the distinction, and I have to describe both of these as the same state. So the way I do that is I talk about a superposition.

If you recall in the previous episode, I talked a lot about superpositions of states. This is a similar sort of idea here. In the indistinguishable case, then, I have to say, instead of four possible states of the system that I had in the distinguishable case, there are only three. There's when they're both in state A, both in state B, or when I have one particle in state A and the other in state B. But I can't split that up and say whether particle 1 is in state A and 2 is in B or vice versa. I can't divide that up anymore. All I can say is one is in one of the states and the other is in the other state.

So I represent that by writing the two possibilities and adding them together and saying, that the um and essentially dividing by two basically so that the probability is normalized but that's not important basically I just add them together I say well I've got particle one in state A and particle two in state B plus particle one in state B and two in state A and that whole thing is a possible state of system because I can't tell the difference between those two even in principle so they're actually the same state now when I perform an experiment again absent any other factors

each of these three possible states, now again we're looking at the indistinguishable case, each of these three possible states, both in A, both in B, or the mixed state, has equal probability. So now the probability of obtaining any of those states is 1 in 3 instead of 1 in 4, as it was previously. So the probabilities have actually changed

because of the distinguishability case. So it's very important to know and factor in this indistinguishability. distinguishability. Now, remember I was talking about fermions and bosons, how do they fit into this picture? Well, it turns out that what I just said with those three different distinct states, both in A, both in B, or one in A and one in B, but can't tell which is which, that three-state case was actually only applies to bosons. So particles, or it really goes the other way around, particles that obey that behavior are called bosons.

But what's the alternative you might ask? Well, you remember I said that, When we had the one of each case, one in state A and one in state B, I just sort of took each of the possibilities and added them together. I put a plus sign between them. Why a plus sign? Where did that come from exactly? Well, for reasons that I won't get into here, one of them that relates to the probability conservation, it's got to be, it can't be three because I can't change the total probability. So because of theoretical considerations like that, the number that I have to put in there has to be either plus one or minus one. there are only two options. The case that I talked about initially was the boson case where I choose the plus sign. In that case I get these three possibilities. Let's suppose I chose the minus sign.

That is, let's suppose my mixed state is particle one in state A, particle two in state B, minus particle one in state B, particle two in state A. So instead of having a plus sign between them, there's a minus sign between them. What difference does that make you might ask? Well, let's now think about what would happen if I consider, essentially I take this, this is called the anti-symmetric case because there's a minus-sign instead of a plus-sign, plus-sign symmetric, minus-sign, and asymmetric. Let's consider what happens if I take this minus-sign case

but then just put both of the particles in the same state. Let's say I put them both in state A. In this case what happens, essentially I get the expression particle 1 in state A, particle 2 in state A, minus particle 2 in state A, particle 1 in state A. Now because those are indistinguishable, I can't tell the distance with them, so they're the same. I've got a minus sign here. I get zero. They subtract off, they cancel each other out. The same happens if I put them both in state B. It doesn't matter which, whether they're in state A or in state B. The point is, I can't have them both in the same state, or to put it slightly differently. if my particles behave in the antisymmetric way with that minus sign there,

and I try to put them both in the same state, it turns out that that's the same as them not existing at all. So that's just the way of saying they can't both be in the same state. Particles that obey this, it's called statistics, this form of the statistics, but the antisymmetric, the minus sign in between them, are called fermions. And because of this property, two fermions cannot simultaneously exist, in the same quantum state. This is the Pauley Exclusion principle that I talked about

in the original episode on quantum mechanics. This is the reason for the Pallel Exclusion Principle. It's not just an arbitrary fact of nature. It actually is, the reason for it, derives from the different statistics that bosons and fermions operate under. Bosons are not constrained by the Palli Exclusion principle. You can put as many bosons in the same state as you like. That's because they have this plus sign in between the two indistinguishable states, such that if you put both of the particles in the same state, they just add together, and you can stack as many on as you like. But in the anti-symmetric case, there's a minus sign in between those two states that I can't tell apart, so that if I try to put both particles

in the same state, say both in state A, then they'll subtract off, and I'll get zero, which is a way of saying you can't do that, that it's impossible, and hence the Pally Exclusion Principle. So the Palli Exclusion Principle is really just an application of this more underlying fact of the different statistics that identical particles can operate under, depending on whether they follow the anti-symmetric case, in which case they're called fermions, or whether they follow the symmetric behavior, in which case they're called bosons. Now, there's something else related to this called the spin and statistics theorem,

which says that the intrinsic spin of a particle is directly related to the statistics that it obeys. I won't explain why that's true here. it relates, you need to consider relativity in order to show that this is true, but the upshot of the theory is essentially that fermions can be defined in two different ways. One way of defining a fermion is that it has

half integer spin, so spin half, spin three over two, etc. Another way of defining a fermion is to say two fermions are anti-symmetric when you interchange two identical fermions, that is you add in this minus sign. So minus sign, half integer spin, turns out it's the same thing,

fermions either way. Bosons have integer spin and they obey symmetric. They are symmetric when you interchange two identical particles. That is, you have the plus sign between those two states that you can't tell apart. And that's really quite a remarkable fact. There's no, in principle, that is, before we sort of knew about the spin statistics theorem, there's no reason why there should necessarily be any connection between spin, which relates to intrinsic angular momentum, and the statistics that the particles obey. But it turns that they are actually directly related to each other, such that

that there's two distinct ways of defining bosons and fermions, either by their statistics or by their spin, and they're equivalent. So that's a pretty neat piece of physics, I think, there. And the Pallelixclosition principle is the fundamental reason why matter takes up space and has volume, because electrons can't all pack into the lowest energy space. They repel each other essentially because of what's called an exchange interaction. They're pushed apart by the principle that they can't all occupy the same lowest space, the same space at the same time. so they can't all go down to the lowest energy space.

They have to separate out from each other, and that's why atoms take up space, and that's why electrons don't all fall down into one lowest energy state, and essentially everything becomes a black hole. So power of the exclusion principle, very important, and it comes down to the different statistics that indistinguishable particles can undergo. So interestingly, if particles were distinguishable,

it's not clear that you would have the power of exclusion principle. I mean, it wouldn't be relevant to distinguishable particles. And so the universe would be a very different place if electrons were distinguishable from one another in principle. Okay, so that concludes the little section on particle statistics. I now want to talk very briefly about perturbation theory. I don't have too much to say on this. I just wanted to sort of include it in coverage of advanced concepts in quantum mechanics.

But perturbation theory is a set of approximation methods that is used in quantum mechanics to calculate things that are otherwise too hard to calculate. Most problems in quantum mechanics are too hard to calculate exactly. The maths is too difficult, essentially. Only a few toy problems can actually be calculated exactly. But in order to get around this, we use one tool that we can use is called perturbation theory. The essential idea of perturbation theory is that you can generally find the unknown Hamiltonian,

so the energy function, for example. Often we're interested in finding out energy levels of some system. So you can often find the energy levels of a system by representing the Hamiltonian of that system, the energy function, as the sum of two different components. One component is essentially the known, unperturbed Hamiltonian. That is, it's the Hamiltonian of a system that we know really well. Often that's something like a hydrogen atom or a harmonic oscillator, or something simple that we can calculate exactly. So that's the known unperturbed bit. The other bit to the Hamiltonian that we add on, that the plus something else,

the perturbation. That's the tricky bit if you like. So essentially perturbation theory is just about separating the Hamiltonian, the energy function, into the easy bit and the tricky bit. As long as the tricky bit, the perturbation is small relative to the easy bit or the unperturbed part, then what we can do is expand out the energy levels or even actually the wave functions as well, but we'll just focus on energy levels. So we can expand out the solutions to the true energy level as a series in increasing powers of the coupling constant, essentially, which we often call lambda, although what we call it is arbitrary.

But lambda is essentially a proportionality constant. It tells us how important the perturbation is. The bigger lambda is, the stronger the perturbation is. If the perturbation is too strong, then this method breaks down, and we can't use perturbation theory. But as long as the perturbation is relatively small, and therefore if lambda is relatively small, we can write out energy levels as a power series in lambda,

and essentially just neglect most of the terms in the series. So a power series is an infinite series, so there's an infinite many terms in the series. But if lambda is small, particularly less than one, if you square something that's less than one, it gets even smaller, and if you cube it, it gets even smaller than that. So the point is higher powers aren't going to add very much, because they're going to be tiny.

So you just throw away most of those higher powers and only consider the first order or second order approximations. So this is a very powerful tool in calculating otherwise, impossible to calculate energy levels or other things in quantum mechanics. Basically you just take an easy problem and then find a, hopefully you can write your Hamiltonian as the perturbation to that easy problem, write it as a series, throw away most of the higher order terms that don't add much to it, and then just calculate the first or second order terms

that serve as corrections to the simple easy to solve problem. So that's a really useful concept to have. Pertubation theory is used particularly in quantum-futable. field theory, which I'm hoping to do an episode on soon. So it's important to have some idea about what it is and how it works. Okay, we're almost at the end here. There's just one last topic that I want to cover,

which in some sense is the most advanced topic. And this is the EPR paradox and Bell's inequalities. The EPR paradox stands for the Einstein, Bedolsky and Rosen paradox. These are the three people who came up with it. And yes, it's that Einstein. Einstein did not like quantum mechanics. He said that God does not play dice with the universe and had various other philosophical and theoretical issues with it.

One of the ways that he attempted to argue his case was by developing the EPR paradox essentially, which was designed to show that quantum mechanics cannot be a complete description of nature, that there must be something else beyond mechanics in particular, one idea about what this other thing might be, is hidden variables, things that we can't measure, but that determine the outcome of quantum experiments. In particular, what Einstein really hated was a one. One of the things you really hated was this idea that if a quantum system starts out in a superposition of different states, say superposition of up and down, then you conduct a measurement, then there's some sort of, well, almost miraculous process in the sense that we don't have any explanation of how it occurs, a process of collapse whereby the system sort of just instantaneously jumps from being in the superposition of states to one single eigenstate that we then measure. It's not really miraculous.

We don't understand how that happens. There's no real theory for it. and so therefore. But Heidzine really hated that, particularly the probabilistic nature of that. So he wanted to say that it can't just be that, there has to be something more going on, even if quantum theory doesn't, can't describe what that is yet. And he developed the EPR paradox, he and his colleagues developed the EPR paradox as a way of sort of showing that or attempting to show that. Now, to understand the EPR paradox, you have to understand the principle of quantum entanglement, which is the idea that two different particles, or groups of particles,

we'll just call them particles, can be connected to each other in ways such that the state of one cannot be described independently of the state of the other. And when I say connected, I don't necessarily mean like physically connected, although they could be, but more generally it just means that there is, well, literally you can't describe one without describing the other, you can't describe them independently of each other. If we jump back to our previous episode where we talked about KETs in Hilbert space,

two different particles are entangled if you have to describe, them, if you have to describe the whole system of those two particles as a single ket. If you can't split out each particle and describe it using its own ket, then those two particles are entangled with each other. You have to have a Ket for the whole thing. If you can describe each particle with its own Ket, and then just describe the whole system by multiplying the kets together, then those particles are independent of each other. You can measure stuff about one, and it won't affect the other they're independent of each other. Quantum entanglement is when that is not the case,

when you have to have a Ket for the whole thing, when the whole thing, when the whole thing is a single quantum system and therefore everything within that quantum system affects everything else in that quantum system, at least potentially. So you can have two different particles which are entangled, meaning that let's have a concrete example here. So suppose this was not by the way the original formulation of the EPR paradox but it doesn't matter. It's equivalent to it and is easier to understand. Suppose we have some process that generates electrons but particularly it generates pairs of electrons such that one is all spin up and the other is always spin down. So there's a correlation between the two.

They're entangled. We can't describe a single one of these electrons with a ket where essentially there's a 50% chance of finding an electron that goes to the left as up and the electron that goes to the right as down and a 50% chance is finding it the other way around. But notice in another case we always have the one up and the other down. It's just we don't know which is which. So this is an entangled system. the properties of one electron are directly linked to the properties of the other and you can't separate them. At least not until you make a measurement, but at that point you break the entanglement. So before then they are entangled, they're connected to each other in a sense.

Not physically, but it seems that the way we have to describe this as them mechanically there's a connection. Now, suppose that we measure one of the particles, that we measure the spin, the intrinsic angular momentum of one of the particles along the z-axis. We can measure whatever, along what we measure, along whatever axis we like. The Z-axis is sort of traditionally the one we measure spin along, but you know that's just convention. So suppose we measure one of the particles along the Z axis and find that it's spin up. Now instantaneously we know that the other one must be spin down because they're entangled, we know that

there's always this relationship between them. Now this holds even though the other particle could be many light years distant. This does not mean that we can allow information to travel faster than light because there's no way to sort of determine beforehand where where the the first one we measure is spin up or spin down. So there's no violation of causality here, there's no violation of special relativity. But it certainly is odd, this idea that previously we didn't know

what the spin of this other particle was, even though it was light years away, potentially, if we'd let it travel for a long time after they were admitted. But now, instantaneously, we can say what it is because we measured this one particle that's light years away. That's counter-rituitive, and Einstein thought that that was an unacceptable result that showed that quantum theory had to be under-determining what was happening.

There must be some underlying process that's determining the spin of both of my particles. It's just that we don't know what it is. Now, in 1964, physicist John Bell proved was that certain local hidden variables, certain theories of local hidden variables. So these hidden variables are the things that Einstein thought was under the scenes determining the outcome of these experiments. We couldn't see them, but he thought deterministically they were, determining what was happening. And local means that essentially you avoid this weird

action at a distance thing where my measurement now affects something that happens instantaneously light years away, which is non-local. So these hidden variables are supposed to be local, so there's none of this weird action at a distance stuff. Because I don't like either of those. He didn't like the probabilistic side, he didn't like the action at a distance side of it. So local hidden variables solved both of these problems. You say there's some purely local phenomenon that's determining responsibly responsible for these measurements, it's just that we don't know what it is, quantum theory doesn't describe it. So what Bell showed, again, this is after Einstein died, so this is sometime later, what Bell showed is that

for local hidden variables, certain experiments could be performed, and the outcome of those experiments would satisfy what are called Bell's certain Bell inequalities. Now, I don't even really understand these inequalities, they're confusing, basically. This is really advanced stuff. We're not going to worry about what these inequalities are there, particularly mathematical inequalities, that the point is that certain experiments, if you perform them, would have to satisfy these inequalities if these local hidden variables existed. So he was able to show this. That's what the real insight of his theorem was. He didn't directly actually show that local hidden variables existed. He showed that if they existed, then certain experimental results would have to be a certain way. They would have to

satisfy certain bell inequalities. Now, over, so that was like 50 years ago now. Over the succeeding decades, many people have conducted experiments to test these Bell inequalities. And the evidence has progressively coming, you know, the first experiments weren't that good, and subsequent ones have been better and the more rigorous. And basically, there have been very strong, very clear evidence of violations of these inequalities, up to many, many standard deviations. That is very high certainty that Bell's inequalities can be shown to be violated in certain experimental setups,

which means that there cannot be local hidden variables. So that's why it said that Bell showed that local hidden variables didn't exist. He didn't show that exactly. He showed that if they existed, Bell's inequality would have to hold, and subsequent experiments showed that Bell's inequalities don't hold in certain experimental setups. Now I say certain experimental setups, because there are still philosophers and physicists

who aren't too happy with these results who like local hidden variables, and so they look at what are called loop-poles. in these experiments because there are still certain ways you can get around the results of the experiments if you look at these technical loopholes and argue well what if such and such based on this loophole. Most physicists don't think that these loopholes are too convincing. We won't talk about what they are here. Most of them are very technical. But the basic idea is there's still wiggle room around if you really want these local hidden variables to hold. You know, no experiment could be 100% definitive essentially. But most

physicists find these experiments pretty compelling and don't think local hidden variables are possible based on these results. But notice that I only said local hidden variables. Non-local hidden variables are still possible. Bell's theorem doesn't apply to those. It's only local ones. So it's still possible that hidden variables are a viable interpretation of quantum theory. It's just that they must be non-local. So you can't avoid this action at a distance thing that Einstein didn't like. So really what the results of this is saying is that Einstein, wouldn't be happy. Maybe he'd be one of the ones looking for these loopholes in the experimental results. But he didn't like

this action at a distance thing where you can tell what's happening light years away by measuring one thing, by measuring one of these particles, and then it's entangled particle, you can infer what state it must be in. He didn't like that. But he also didn't like the non-deterministic nature of quantum mechanics, whereby that you don't know what outcome you're going to get exactly until you make

the measurement. So he wanted hidden variables and he wanted them to be local to avoid both of these problems. But the violations of Bell's inequality seem to show that that's not possible. And so you've got to give up at least one of these things. You've got to give up either hidden variables or locality. Or you've got to try and look for loopholes in these experiments,

which is what, as I said, some people are doing. But generally the view is that quantum mechanics is correct, that Bell's inequalities are violated, and that therefore there are no, or there cannot be local hidden variables, and that, in fact, the general view is that there isn't really a paradox here. All that there is, in the EPR paradox, all that there is really is a limitation. or a clash with our classical intuitions.

In particular, and I think this is a real crux of it, and a helpful way of thinking about this is that we tend to think of two particles if they're located light-ears apart and they're separate particles. Classically, we think of those as like billiard balls, light-ears apart. We think of those as distinct objects, which shouldn't have any effect on each other if they're so far apart. Not instantaneously, obviously, if they're that far apart. But that's just the wrong way of thinking about it.

Quantum mechanically, particles are not like billiard balls, which are separate distinct objects. We know quantum mechanically that everything that is a particle is a wave and everything is a wave is a particle, loosely speaking. So to begin with, particles are smeared out. They're not localized in the way that billiard balls are. If you measure the precise position of a particle, which you can do, then its position is localized, but then its momentum is more smeared out. So there's always this sort of smearing. You can't just define, the position and momentum of particles, quantum particles, like you can, a billiard ball, which is a classical object. So thinking about particles in that way is mistaken. But the other side of it is that if I have two, just because I have two particles doesn't mean that I have two different things, two different distinct objects or physical systems. Classically, this would generally be the case, especially if I'm putting, if I have two particles and I put them, you know, light years apart from each other, they'd essentially be distinct independent objects.

but quantum mechanically it's not the case. If they're still described in terms of the superposition of eigenstates in the same Hilbert space, then they're the same quantum system. They are one quantum system. Just because one bit of the quantum system is light-eis away from the other does not undermine that fact. They are still connected quantum mechanically. So I think the problem with the EPR paradox is in thinking about the two particles as distinct entities. Really they're not.

They are spatially separated parts of a single quantum system. system and they remain such as long as that entanglement is in place. Once you measure the system or interacted in some way that breaks the entanglement, then they're no longer part of the same quantum system. But until that point, they're interacting and part of a single entangled system, and you can't separate them or meaningfully even talk about them as distinct entities. So there's no real spooky action at a distance here, and certainly no conveying information faster than light. It's counterintuitive, that's to be sure, but fundamentally it comes down to this fact that quantum mechanics is counterintuitive, the idea of superpositions of

states and collapsing of wave functions, wave particle duality and tunneling and all of this stuff, it's all counterintuitive. We don't see these behaviors in the classical world, and so the manifestations of these behaviors in the quantum world can often seem bizarre to us. And the EPI paradox gives a nice example of when that's the case. But it's not generally thought to be a genuine paradox by most of the physics community. today. Okay, that concludes this episode, part two of advanced quantum mechanics. Hopefully you found this interesting. Again, heavy material, so don't be too disheartened if you found this two episodes particularly hardgoing. This doesn't represent a shift in the podcast. I have a few more advanced

physics topics that I'd like to do. As I said, I'm planning on doing one on quantum field theory. That will be a bit more heavy going as well, possibly two episodes. We might do one in the future on particle physics, but these are just for listeners who are interested in that sort of thing and are, you know, ready to take on the material. If not, don't worry, there will be plenty more episodes coming that are a bit more accessible on sort of more typical subjects that aren't quite a sort of math-y in the conceptual side of things. Anyway, so if you enjoy the show, please leave a review on your favorite podcast aggregator. iTunes is still popular, but there are many others as well. You can also email me if you have any feedback or

suggestions. My address is Fods12 at gmail.com. That's F-O-D-S-1-2 at gmail.com. Always like to hear from my listeners. You can also look up the podcast on Facebook and like the Facebook page. Thank you for listening, and I'll talk to you next time.