The Science of Everything Podcast - Episode 83: Advanced Quantum Mechanics Part I

Episode Date: June 26, 2017

An examination of some more advanced concepts of quantum mechanics, focusing on describing Dirac's bra-ket formulation of quantum theory. I discuss the formulation of quantum mechanics in terms of vectors in Hilbert spaces, Hermitian operators as corresponding to observables, orthogonality of eigenstates, incompatible observables, and Schrodinger's equation. Recommeded pre-listening is Episode 14: Principles of Quantum Mechanics.

You're listening to The Science of Everything podcast episode 83. Advanced Quantum Mechanics. I'm your host, James Fodor. So in this episode, we're going to take a bit more of a deeper dive into quantum mechanics than we did in the previous episode, way back in episode 14, principles of quantum mechanics, which is the recommended pre-listing for this, obviously. So basically, what I'm going to do in this episode is I'm going to try to explain the concepts of quantum mechanics as they typically introduced at an advanced undergraduate or

graduate level depending on exactly what institution you're at. So beyond the typical sort of introductory level that usually is that people's first exposure to quantum mechanics. So that means I'm going to introduce the concepts of bras and kets on Hilbert space as the and the related algebraic formulation of quantum theory with the related postulates of observables as permission operators and incompatible observables and so forth. Also, I'm going to talk about some more advanced ideas relating to quantum mechanics, particularly NERTHIS theorem, particle statistics, so the difference between bosons and fermions, and I'll look a bit at perturbation theory, which is an approximation technique for calculating in quantum mechanics,

and I'll also talk a bit about the EPR paradox and Bell's inequality. So quite a bit to get through, possibly this may extend over two episodes, I guess we'll see how we go. And obviously, a lot of what I'm talking about in this episode is quite mathematical in nature, and I can't go through the mathematics in an audio podcast, obviously. But what I'm trying to do is explain the key conceptual basis behind the mathematics to either get you interested in further studying quantum mechanics or to help if you're already studying quantum mechanics with understanding

the sort of conceptual apparatus behind it and so forth. So this is not obviously meant to be a substitute to the sort of rigorous mathematical study. This is a, if you like, a taster of that and a guide to the concepts behind it, because the concepts do matter as well. If you try to do maths without understanding what you are doing, you're liable to end up getting nonsense. Okay, so let's start with working through the basic concepts, the formulation of quantum mechanics in terms of vectors on Hilbert spaces. So first we need to understand what we need to understand what. we're talking about in terms of a Hilbert space. Physicists, when they're talking about quantum mechanics, don't worry too much about the formal mathematical definition of a Hilbert space, and I'm not

going to either. Likewise for some of the other mathematical concepts that I talk about later throughout this episode. I'm not going to worry about the precise mathematical formulation, because that's not important for us. What I'm going to present is a way of understanding conceptually what the key ideas are. In terms of a Hilbert space, the key idea there is a Hilbert space is very much like a vector space. Now what's a vector space? Well a vector space is just the space where vectors live essentially. Now the easiest way to think about this I think is in terms of three dimensions. Of course they're not limited to three dimensions but we'll think about three dimensions to begin with. So a very simple example of vector space is

just the set of Euclidean vectors in three-dimensional space. So basically this means that you can imagine a vector, so a vector is just a mathematical object with a direction and a magnitude. So it points somewhere. And we're looking at three-dimensional vectors, so the vector points in some direction in three dimensions and it has some magnitude representing how far it's pointing, or the magnitude of the force, whatever it is, it doesn't matter. The vector space just represents essentially all of the different possible ways that the vector could point along with in some sense the possible magnitudes that it could have. So, I mean, you can visualize a three-dimensional Euclidean vector space essentially

is just like three-dimensional coordinate space. You can point in any direction and you can point as far as you like in that direction and that represents a vector in the vector space. The vector space is, in some sense, the set of all possible vectors you can have in the defined space. So obviously there's vectors that point in the X direction, in the Y direction, in the Z direction, or any combination of those that I like. A vector space is just a generalization of that Cartesian three-dimensional space

into arbitrarily many dimensions. Or fewer, you can have two-dimensional vector spaces, but you can also have infinitely many dimension vector spaces. Now, Hilbert space is again, we're not going to go into the exact mathematical definition. For our purposes, it's good enough to say that a Hilbert space is, basically it's a vector space, but it has a length defined, or in a product technically, but we'll call it a length scale defined. So that is, you can always define the length of a vector in a Hilbert space. So it's basically just a vector space with a bit of extra structure. But I'll probably just talk about

vector spaces because it's a little bit easier potentially than Hilbert spaces, which just sounds a bit more technical. Okay, but we need to bring this back a bit, because I I've been going off on to all these mathematical concepts of Hilbert spaces and vector spaces and whatnot. What does this have to do with quantum mechanics? Well, the best way that I can describe this is that if you recall either the previous episode in principles of quantum mechanics or just other education that you've had on the subject, one of the key ideas in quantum mechanics is that of superposition. That is, that a given physical system before it's subject to a measurement,

does not necessarily exist in a single well-defined state. It can exist in a superposition of many different states, each with its own probability of being measured. And this is such a key principle of quantum mechanics that it's necessary to incorporate this in a formalism in the technical apparatus that we construct to describe quantum systems. And essentially vector spaces are the way that we do that. So to understand this idea, let's consider a really simple example,

which is that of a spin-half system. Now, if you don't know what that means, it doesn't matter. The basic idea is just that it refers to the two different possible orientations of the spin of an electron or something like or a similar particle. But really, all that's important for us is that there are only two possible classical states that the system can be in, and we'll call them up and down. So classically, after it's been measured, we say the system's either in state spin up or it's in state spin down.

are the only two possibilities. It can't be anything else. It can't be in spin sideways or it's been mostly up or anything like that. It's either up or it's down. Now that's classically, however quantum mechanically, that is before we measure it, remember that any quantum mechanic system can exist in a superposition of the possible states. So in particular, before it's measured, the system will exist in some superposition of up and down. And we can represent this on a single axis. If you imagine drawing a line and if one extreme of the line, that represents spin down, the other extreme represents spin up, classically, you can only be at one of those two points, all spin up or all spin down.

But quantum mechanically, the system can exist at any place between those two extremes along the line. So 30% up and 70% down, or 50-50, or 90% one and 10% the other, whatever. Any linear combination like that is a possible superposition. So in order to describe the possible range of states that the system can be in, we use the vector space. For each possible additional state that the system could be in classically, so that is after a measurement, the number of possible states that the system could be determined to be in, we need to add another dimension. So in the case of a spin-half system, there are only two possibilities up and down.

So one dimension is enough to characterize all the possible combinations that the system could be in. But if there were three possible combinations, if there were three possible outcomes, you would need two dimensions to fully describe all of the possible superposition states that the system could be in. And for many variables in quantum mechanics,

there are actually an infinite number of possible outcomes that could be measured, or that classically that the system could be in. So, for example, position or momentum to things we often want to measure of a system, there's an infinite number of possible values. essentially you can have any position in the universe theoretically or any momentum from zero to infinity again theoretically So we need to have an infinite dimensional space that represents the

infinite possible combinations that that you could have of your superposition that is we have to be able to say well there's some probability of our particle having being in position one some probability of it being in position two some probability of it being in position three and some probability of it being in position three and so so on and so on for actually an infinite number of different positions. Of course I've described the positions as discrete there, in reality they're going to be continuous, but that's just to illustrate the point. The point is a vector space is a way of representing all of the different combinations, weighted combinations of classical states that the system can be in. The reason we need vector spaces in quantum mechanics is precisely because of this superposition property,

that the system is not just in one specific sense. specific state but is in fact in general in a superposition of all of the different possible states that it could be in of course with with varying probabilities of each of the particular states but in order to represent that we can represent it as a vector in a vector space the and the direction that the vector is pointing in this abstract vector space essentially represents the relative probability of the system being measured in those different states so then the basic idea

of this formalism of quantum mechanics is to represent any quantum system. So the quantum system could be an electron orbiting an atom, or it could be a bunch of atoms in a crystal lattice somewhere, or it could be really anything you could imagine. You can describe anything using this quantum mechanical process. The issue is it just becomes very complicated for large systems. But we often talk about electrons around atoms. That's a convenient way of thinking about it. But it doesn't have to be that. It can really be anything that's analyzable quantum mechanically. That's what we mean by a quantum state. A quantum state is represented as a vector in Hilbert space

and these vectors are given a special name just so we can have something specific to call them and they're called kets. The usual notation for a kett is a vertical line and then some sort of symbol, it could be a letter or a number or a Greek letter or whatever to just name the ket, and then an angled bracket that points to the right.

Now if you sort of have seen that before, you'll know what I'm talking about. If not, just look it up. You'll see how a KET is represented. A Ket, as I said before, simply represents a vector in a Hilbert space. It is a representation of the quantum state of a given system. Is a vector in an abstract space, the orientation of which represents how all of the different possible final experimental outcomes, or measurement outcomes of the system, how they are superposed together in the quantum system before it's measured. Of course, it's possible that the KET could just be in a pure state. That is, if we look back in our spin-half example, the Ket could just be in the spin-up state. So in that case, it would be a very

simple Ket, essentially. It's just, there's no superposition. It's just in one final state. But in general, a quantum system is going to be in a superposition of many states. So the Ket is going to point in a direction that's not orthogonal to any of the axes that correspond to a measurable outcome. So in other words, the vector's not going to point in direction 1, 2, or 3, it's going to point somewhere in between them, which represents a superposition of the states 1, 2, and 3. So again, that's the whole point of this vector representation. It's just a way of encoding the fact that quantum systems exist as superpositions of possible outcome states, that is states that you can actually observably measure after you conduct a measurement.

So a ket represents that in the formal mathematical sense. Now each state ket also has a corresponding bra, BRA, BRA, which represents the same underlying physical state. And the usual notation, the way a bra is represented is by an angled bracket pointing to the left, and then some symbol or number or something, and then a vertical line. So it's sort of like a mirror image of a kett. So a bra has a corresponding kett and a kett and a kett corresponding bra. They're like mirror images of each other. That's not the formal mathematical definition obviously but that'll do for us. Okay so so far we've got this Hilbert space and a given Hilbert space is defined by the quantum system that you're concerned with. If it's a spin half system there's only going to be one dimension

there. However and the dimensionality depends on the dimensionality the number of possible outcomes that you could measure. So in the case of measuring the momentum of the system, for example, effectively there's an infinite number of momentum that you could potentially measure, so your Hilbert space there is going to be infinitely dimensional, and each dimension is going to correspond to a potential value, a potential measured value of the momentum of whatever it is you're considering, whatever particle or what you're considering. So that's what defines that the Hilbert space and the dimensionality of the Hilbert space is just the system you're considering. But so we've got this Hilbert space and a particular

KET in that space represents a particular quantum system that is a superposition of possible states, measurable states in that system. So what do we do with that KET? At the moment, it's just an abstract description. How does that actually help us to understand what's going on and to perform calculations? Well, the next important thing to understand, and the next sort of postulate of this formalization of quantum mechanics, is that permission operators on this Hilbert space correspond to measurable variables of a system. Now an operator is basically like a function. It operates on something.

In this case, it operates on a ket or a bra. So an operator, basically, it's just like a function that takes a ket or a bra and then outputs something. And we'll talk about what the something is in a moment. So it's an abstraction that allows, to perform some sort of calculation. And when we say it's a omission operator,

I'm not going to get into the details of what a emission operator means exactly. I emphasize that it needs to be a emission because that's a very important mathematical requirement of the theory. But if that's confusing to you, then just forget about it. Basically, an operator on the

Hilbert space corresponds to a measurable variable of the system. So there is, another way of saying that is that for each measurable variable of the system, for each different thing you can measure, there is a different operator that we can define. The operator is an abstract mathematical device obviously, but it corresponds in one-to-one way with some physically measurable observable. So for example, for a quantum system you can measure the

position of a particle, you can measure its momentum, you can measure its linear momentum, that is, you can measure its angular momentum and you can usually measure its spin as well if it has a spin. So there's a four different physical observables. Each of those has its own corresponding operator. So there's a position operator, there's a linear momentum operator, there's an angular momentum operator, and there's an intrinsic spin operator. And for some systems there are other operators as well. Another important operator is what's called the Hamiltonian, which is the energy operator, gives you the total energy of the system. The key

point here is simply operator directly corresponds to physical observable. Now, the exact form that the operator takes, that is, like, how do you actually calculate the thing? Well, that depends on the representation you use for the system in question, and we'll talk a bit more about that. But at the moment we're just talking about abstractly. Abstractly, there's some mathematical object that does what we want it to in the sense of, we give it a ket, and it outputs the measured variable corresponding to that ket. Now, you might be thinking, but hang on a moment. A ket, as I just said, is in general a superposition of a set of a set.

different possible measurable outcomes. So it's a superposition of up and down. So how can I have an operator that gives me, say, the spin of an arbitrary Ket? If the arbitrary Ket doesn't have a particular spin, it's like 70% up and 30% down. Well the answer is that's exactly what the emission operator does. It will tell you what percentage the Ket is or the bra in each of the possible states of the system, measurable states of the system. So if it's in a pure state, so if it's in a pure state, So if your KET is in the up state, for example, that's in 100% up, that means every time you measure it, you'll find that it's in the up state,

then the permission operator will just take the KET and then basically it will output up. It'll tell you, yes, it's up, 100%. It's always in the up state. On the other hand, if your Ket is in a superposition of 50% up and 50% down, that means that half of the time when you measure it, you'll get up as the result, and half the time when you measure it, you'll get down as the result. then your permission operator will tell you that when you put the KET through the operator, essentially it will tell you that it's 50% up and 50% down.

So the operator will output the particular superposition of final measurable outcomes corresponding to whatever superposition the Ket is in. So really what operators are doing is just they tell us a particular thing about the system. Often there's more than one thing we might want to know. For example, we might want to know position and momentum and energy. So it's three different things. One operator will tell us one thing. then we might want to know. So there's a momentum operator. It tells us the momentum. It doesn't tell us the energy, for example.

That's a different operator. So that's the key thing at the moment. Okay, so so far, we've got kets and bras. Each kett has its own corresponding bra. So generally, I'll just talk about kets from now on, but just know that each has its mirror image. We've got kets living in the Hilbert space. We've got operators, which will tell us different properties of the Ket, say it's measurable properties of the KET, so it's momentum or its angular momentum, etc. Now I need to introduce something that's called an eigenstate of an operator, or an eigenket of an operator. Now this relates directly to what I was saying before about the fact that an arbitrary KET is a superposition of different measurable states.

And by measurable state, what I mean there is a state that you could possibly get as the outcome of a measurement. So, consider our spin-half system. If I measure the spin of this system, there are two possible outcomes. It could be up or down. I can't measure it as being 50% up and 50% down.

That's not a possible measurement outcome. However, you can have a ket that is in the state of 50% up and 50% down. It's just that as soon as you measure it, it collapses into one of those two states. We talked about that in the previous episode, the idea of the collapsing of the wave function. Our operator needs to be able to deliver that 50% outcome

if the operator is to be able to tell us the correct answer as to what state the ket is currently in. Not the state it is in after collapse, obviously we don't know that, but the state that it's currently in. So when the permission operator does that, there's a specific set of possible states which are called eigenstates or eigenkets,

and these are special in that they correspond to a possible measurable outcome. So let's go to our spin-half system again, our favorite example. And let's consider the intrinsic spin operator. So again, this operator just takes the ket and tells us what the spin is of that ket. Suppose I have a ket corresponding to a spin-up. So this is my pure 100% spin-up ket. Now this is a possible measurable outcome.

I can measure the outcome of the system as being spin-up. So that means that if you think formally about what's happening, I can take the operator, apply it to my ket, which is in the state up, 100% up, and what will I get back? What will the operator tell me about the spin of that ket? Well, it should tell me that it spin up, because I just told you that it is. And in fact, that's what it does. So the idea there is that operator, when it acts upon special states, so-called eigenstates of the system, it just delivers that same state back

multiplied by a particular number, which is called the eigenvalue. And the eigenvalue for every eigenket will just be the value of the observable that corresponds to the particular eigenket in question. So, for example, if the eigenket is the spin-up eigenket, then the eigenvalue will just be half because that's the value of the observable

that corresponds to the particular. eigenket. So eigenstates are special because when an operator hits them, when an operator acts upon them, you just get that state back. Now, note, this is not the case for just some arbitrary kit. Let's consider our ket that's 50% up, 50% down, so it's in a superposition of, an equal superposition of those two states. So I can write that ket, and I apply my spin operator to it, and what I'll get back is essentially some representation that tells me 50% probability up and 50% probability down. Now that's not what I started with. What I started with was just one ket. What I got after applying the operator was essentially two kets added together, each multiplied by a 50% probability. So that's a 50%

chance of the up, getting the up-ket and a 50% chance of getting it in a down state. So that tells us that this 50% superposition of up and down is not an eigenstate of this spin operator. Or in other words, that it's not a possible measurable outcome, which we knew. We already knew that. We know that it's not possible to measure a spin-half particle and get the outcome 50% up, 50% down. That's not possible. It's either up or it's down once you measure it. It's only when it's in the quantum superposition state that it can be prior to measurement, that it can be in that 50% state. So these eigenstates are really special because each eigenstate corresponds to a possible measurable outcome. And what we say, is that an arbitrary KET, or an arbitrary quantum state,

can always be written as the weighted sum of all of the possible eigenstates of the system. And the coefficients of that weighted sum correspond to the probability that the state will be measured in that eigenstate. So in the 50-50 case that I've been talking about, I can represent that Ket just by writing my spin-up Ket and then putting essentially a coefficient in front of it and then adding a spin-down kett and then putting another coefficient in front of that spin-down-ket. And those two coefficients are basically that correspond to the probabilities of the system being in each of those states.

Now, of course, those probabilities have to be equal, so those two coefficients must be equal. Turns out that the square of that coefficient is actually the probability that the state will be measured in that eigenstate. So those two coefficients should be the square root of 100. half, obviously, because the probabilities have to add up to one. So when I square them, I get half, each, and then I add them, and I get one. So we've just found a representation of the ket that corresponds to 50% up and 50% down. And the representation of that ket is just 1 over root 2 times the up ket, plus 1 over root 2 times the down ket. And what we've done

is we've expressed this ket as a weighted sum of eigenstates. Now this is a particularly simple example because the weight is just the same for both kets, but it doesn't have to be. It could be, one could have a 90% weight, say the spin-up could have a 90% weight, while the spin-down could have only a 10% weight, and then my coefficients would be different. And of course, this is just our simple spin-half example. If there were more than two possible states,

then I could have a third eigen-ket that I add in here, a third eigenstate that I'm adding in, or four, or infinitely many, in fact, if I'm talking about something where there are infinitely many possible eigenstates, like, for example, position. In that case, you don't add over them, you integrate over them. If that's confusing, don't worry about that, but integration is basically just like the continuous version of addition. So, in the case of position and momentum, you integrate over all of the possible eigenstates. But the principle is the same. Any arbitrary ket can be expressed as the weighted sum, or integral, if it's a continuous, of the possible eigenstates

of that particular observable, whatever it is, be it spin, position, angular momentum, whatever. Now, you might say, well, so what, why is that important? Well, it's actually very important because this is an extremely useful result. Suppose I have an arbitrary quantum system, you know, an electron, I don't know its energy, for example, or, you know, whatever, it doesn't matter. An arbitrary quantum system. I might not know anything much about it except, I'll need to know what type of particle that is, say, what thing I'm trying to measure about it. But, you know, I'm trying to measure its momentum. I know it's an electron, but apart from that, I don't know anything about it. Nevertheless, I can write down an expression for its unknown momentum or unknown spin or whatever

in terms of a sum or integral over all of the possible eigenstates of that system. Generally, I know what those are. If I know what the system is, I know what the possible momentum value, momentum value are, for instance, or I know what the possible spin values are for an electron. I don't know what spin or momentum this particular one has, but I know what all the possible ones are. And then I just have to find these coefficients. I just have to find how much spin up and how much spin down it is, or how much of the momentum is in the high values and how much in the low values, or whatever it is. And that allows me to be much more specific about what I'm trying to do. It provides a structure for solving problems. Instead of just saying, well, I have no idea,

what this quantum system is, it now becomes, well, I need to find these unknown coefficients about what the relative weights of the different possible eigenstates are. So one of the really useful things about this formalism is that we can use it to calculate the expected value of some measured observable. So in this case, energy is a better example than spin. So say we have an electron, but we don't know exactly what its energy is in some given system. There's some probability of it being at different energy levels, let's say. And we want to calculate the expected value of its energy, that is

the average energy value that we get if we conduct many measurements of the same system. This is going to be very useful because that's something we can measure, right? We can compare our theory to the measured results. Well, how do we do that? How do we compute this using the theoretical apparatus that we've been discussing? Well the basic idea is that you take your measurement operator, remember, Remember, every observable, there's a emission operator that corresponds to that observable. In this case, energy, so it's the Hamiltonian, but the name doesn't really matter. It's just it's the operator.

So we grab that operator, and we sandwich it in between the bra and the corresponding kett of whatever our system is. So for our electron system, we put it in between those two. And then we calculate what that is, that that sandwiching represents a number. Remember a kett and a bra, neither of those is a number, that's actually a vector in an abstract space. But when we essentially when we sandwich them either side of an operator like that, the result is actually a number. And the number corresponds to the expected value of whatever the operator is, in this case energy, because we said that it's the Hamiltonian operator. So if you have a system, so our electrons say that's jumping around energy levels,

and we take its kett and corresponding bra, and then we grab it, It's the appropriate operator for this system, and we sandwich the bar on the ket other side of the operator, and then we calculate that, we compute what the result is. It will always be a number, and the number will be, correspond to, the expected value of the energy of that system. Hey, that's great. Then we can compare it to the measured value and see how we go. Now, there's one thing you might have been wondering, which is that I've been talking about operators and bras and kets as abstract mathematical objects, which is what they are. They're abstract. objects and you can just write them down without necessarily having a mathematical form for them. I can write down my kit and write down my operator. But in order to actually compute anything to get that number is your answer. Obviously, I need a specific mathematical form for what I'm calculating. Otherwise, there's nothing I can do.

An important thing to understand here is that we can represent the abstract physical features of a system without necessarily having a specific mathematical formulation for what that looks like. So, to put it another way, there are many different ways of representing or writing down a mathematical function for a particular KET or for a particular operator, say for Hamiltonia, for example. And that's going to depend essentially on the coordinate system that you choose to use or the expansion in eigenstates that you use. So that might be a bit confusing, but think about it this way. If I have an electron, I know that I can write. its ket as the sum or superposition over all of the possible positions that that electron can be in. So that's one way that I can represent the Ket. A different way that I can represent the Ket as is as the sum or integral over the superposition

of all of the different momentum states that that electron can be in. So these are two different representations either in what we say is position space, coordinate space, or in momentum space, they're both different ways of representing the same underlying physical system. And so the exact nature of the calculation that you do depends on how you choose to represent the system, whether you're working in momentum space, coordinate space, you can also, there are other even more abstract ways of representing the systems as well, which I won't confuse them matter by getting into. But the point is there are different representations you can use depending on what your problem is

and what you're interested in solving. and the form that your operators take will depend on the representation that you've chosen. Obviously, the energy operator will look different if it's expressed in terms of momentum than if it's expressed in terms of position, or if it's expressed in terms of something else. So the specific form that your operators take depends on the coordinate system that you use as well. But when we're making these calculations, there's a few special properties of quantum systems that help us,

that essentially make these calculations possible. but also they contain important ideas about the system, concepts about quantum systems. So one idea is what's called the completeness property. That is, and the easiest way to understand this is just think about the question. What is the probability that an electron is measured to be at some position in the universe? Now the answer is one, right? The probability that an electron, a given electron, is somewhere in the universe is one. If it exists, it's got to be somewhere.

We might not know exactly where it is, but it has to be somewhere. So if we define the possible spaces literally anywhere in the universe, the probability is one. So another way of putting that is that the sum of the probabilities of the electron being in any specific location is one, if you add up over all of the locations. Likewise for momentum, we might not know what the momentum of the electron is, but the sum of the probabilities over all of the possible momentum is, of course, one, because it has to have some momentum. Likewise for spin, likewise for angular momentum, likewise for anything that we could measure about the electron or whatever the particle is.

If we sum over all of the possible outcomes that we could measure for that variable in question, the answer is always one because we have to get something. It has to have some spin or angular mimism or whatever it is. So that might sound trivial but it's actually very useful because it means when we're doing these calculations we can do what's called insert a complete set of states. That is, we can just essentially shove into the calculation. I mean literally, you can just write it there, the sum over all of the possible position states or spin states or momentum states or whatever it is. And the reason you can do that is because the sum over all possible states is always equal to one. And of course, you can always just add in one, well, not

adding, you can multiply anything by one and without changing the answer. And that's essentially what we do when we're inserting this complete set of states. We just say, well, multiply it by one, but let's make one be something useful. In this case, we'll call it a complete set of momentum states, or a complete set of position states, or whatever it is. And the purpose of that essentially is to change our calculation. Instead of then representing this abstract ket, we shove in a complete set of, say, position states, and then we can rewrite our arbitrary ket in position space. and now, and in that particular case, if we had, say, an electron that was represented by an arbitrary ket, and then we inserted a complete set of position states, we could then rewrite our abstract ket of an electron as the good old wave function

that you would remember, hopefully, from the first episode of principles of quantum mechanics, or likewise other study of quantum mechanics it might have done. A wave function is actually just one specific way of representing the system. In fact, you can have wave functions in different spaces. So you can have a position wave function where the variable in question is the location of the electron in space, but you could also have a momentum wave function where the variable that you have to put into the function is the momentum of the electron. And so you can talk about the probability of the electron being located somewhere in momentum space instead of in position space. You can actually do this for pretty much anything, again, any observable spin, angular momentum, energy, whatever. So the point to take away there is, and again it's a bit hard to talk about this without being able to write down some examples of these equations.

The basic point, though, is that this ability to insert a complete set of states is extremely useful because it allows us to represent the same system in many different ways, in terms of momentum, in terms of spin, in terms of position, in terms of energy, in terms of whatever is useful for our purposes. And then we can write specific functional forms, you know, things that have signs and cosy, and exponentials and pluses and multiplications in them that we can actually solve. So if you ever see some of this stuff, it's often represented very abstractly. You know, I've got a ket here in an operator. How do I actually compute anything that I can compare to measurement? Well, the answer is you first have to choose a particular representation

in momentum space or position space or spin space or whatever. Once you do that, generally everything falls out. Then you can write particular functional forms, and then you can also write your operator in terms of particular functional forms. and then you can actually compute things and work at your expected values, for example, that we talked about before, and compare to experiments. So that all comes from being able to write the abstract ket in terms of whatever particular representation we choose to use. One other related property to that, to completeness, so again, completeness is the idea that the electron has to be somewhere,

that the sum over all possible eigenstates, that the sum of the probabilities of all possible eigenstates is always one. the flip side to that is something called orthogonality. Now, that's a, again, has technical mathematical meanings that I'm not going to get into. But the basic idea here is that two eigenstates are always orthogonal to each other. What does that mean? It means that if the system is in one eigenstate, the probability of measuring it in a different eigenstate is always zero. That might seem trivial. Why would that be interesting?

Of course, if it's in one eigenstate, it can't be in another one. But note that this is not true for arbitrary superpositions. Now, again, let's jump back to our spin-half case of 50% probability of up, measuring up and 50% probability of measuring down. We said before that a ket in that state is not in an eigenstate. The two eigenstates of this particular system are spin up and spin down. Those are the only two eigenstates. 50% up and 50% down, or 50% up plus 50% down,

if you want to think of it that way, because it is a linear combination of the two, that is not an eigenstate. So, if I ask, given a ket in this 50-50 state, what is the probability of it being found in the up state? The answer is 50%. Because essentially the projection of the ket that I start with, the 50-50 ket onto the up-ket is 50%, essentially if you think about it in probability terms. 50% of that initial kett, the initial superposition ket, is in the direction of the up-ket.

Of course, the other 50% is in the down-ket. So this initial 50-50 kett is not orthogonal to the up-ket. It's not orthogonal to the down-ket either, because it has components that are in both the up and the down directions, essentially. However, if I started with an eigen-ket, an eigen-state, that was all in the up-direction, and then ask how much of this is in the down direction, or how much of this is, what is the overlap between this up and the down? The answer is zero, because if it's up, it's not in the down bit.

Hopefully that makes sense. If I'm in a superposition between multiple eigenstates or multiple eigenkets, I can project my superposition state onto any of those of the corresponding eigenstate, sort of pure measured states that you could be in, and there'll be some positive probability of being measured in that particular state. So if we consider another example where there's four different possible energy levels that my particle could be in, and imagine that my particle was in a superposition of three of those states. So it's in a superposition of one, two, and three, but not four.

So I would say that this superposition state is orthogonal to the energy state four, because if I ask, what's the probability of measuring this superposition state, of getting a measurement of it being in state four, the answer is zero, because there's no component of this initial superposition state that is in the direction of this fourth energy state. So it's impossible to ever get that as a measurement outcome of this state. I can, however, get one, two, and three as possible measured outcomes, which tells me that this initial superposition state of the three energy levels is not orthogonal to one, two, or three, because it's made up of one, two, and three, it's made

up of a combination of those three. And this is where, where, where the that vector space formalism comes in really handy because we can see directly how useful it is. If we talk about orthogonality and we say a given state is orthogonal to another state, it means literally in that sort of geometric sense it's pointing in a perpendicular direction, such that if I ask what proportion of my one vector goes in the direction of my other vector, the answer is zero if the two vectors are perpendicular to each other. It's like asking, if you want me to walk five metres forwards, how many metres sideways do I need to go? Well, the answer is zero, because those directions are perpendicular to each other.

However, if I need to walk five directions sort of diagonally from my current position, then I'm going to need to walk some number of meters to the left and some number of meters forward, because I can project out that diagonal direction partly onto, in a sideways direction, and partly into a forwards direction, and then walk a little bit in both in the sideways and in the forwards and I'll eventually get to the diagonal position that I'm supposed to be at. So it's exactly the same for super positions of states. Now, jumping back to what I originally started this little section with, I said that eigenstates are orthogonal to each other.

And again, that just means that if you're in one eigenstate, there's no projection of that state onto a different eigenstate. They're separate, different, distinguishable, measurable outcomes. The only case where you do have that projection, where there is, where there isn't orthogonality, in other words, is when you're in some superposition of multiple states. And that's a way of saying, well, you could end up in one state,

but you could also end up in the other state. If you're already in an eigenstate, then there's no could or about it. You'll always be measured in that eigenstate. So if I start off with my electron 100% in the spin-up state, I'll always and every time measure it in the spin-up state. And so there's no proportion of the time when it's measured in the spin-down state.

The only time when I get proportions in the up and the down-state is if I start off in a superposition state where there's some proportion of it's in the up state and some in the down state. So because eigenstates are always orthogonal to each other, that often greatly simplifies the calculations because whenever we find, whenever essentially we're multiplying the bra of one eigenstate

with the Ket of a different eigenstate, the answer is always zero, because they're orthogonal to each other. There's one more important concept that I need to discuss with respect specifically to bras and kets, and observables, which is the idea of compatible and incompatible observables. Compatible observables are those that I can measure both at the same time to arbitrary precision. And formerly the way we say this is that they possess simultaneous eigenstates.

The two observables have simultaneous eigenstates. So that means the system can be in a well-defined state of both observables at the same time. Now you might wonder, well, what's the alternative, right? I mean, if the two things are different observables, then why can't I measure them both at the same time? That seems a bit odd, right? And that's because classically, all observables are compatible. Classically, you can measure the momentum and the position of an object to arbitrary precision whenever you like. So all classical observables are compatible in that way.

That's not the case in quantum mechanics. If you recall from the previous episode of quantum mechanics, I would have talked about the uncertainty principle, Heisenberg uncertainty principle, that is you can't determine the position and the momentum of, say, an electron or any quantum system, to arbitrary precision. If I know the one more precisely, I have to know the other less precisely. And in fact, it's not an issue of knowing, it's an actually, it's an issue of the more precisely defined. One is the less precisely defined the other is.

So it's actually an issue of whether one is defined, not whether you can measure it. but of course if it's not defined you can't measure it. So essentially what we're talking about here is the Heisenberg uncertainty principle but in broader terms because it doesn't just apply to position and momentum it applies to other pairs of of variables as well. One example is the Y and the Z components of angular momentum. I can measure the the Z component of angular momentum as precise as I like but I can't simultaneously measure the Z and the Y components of angular momentum precisely as I like. I can measure one or the other, or I can sort of have a fuzzy measure of both,

but I can't measure them both to arbitrary position. So it's not just position and momentum that have this relationship. It's any two variables that do not have simultaneous eigenstates. So if they do have simultaneous agonistates, then go ahead. You can measure whatever you like to arbitrary position. If they don't, then you cannot. And we say that the two observables are incompatible. Now, that doesn't mean that you can't measure them at all. It means you can't measure them both to arbitrary precision in the same quantum system. And the more formal way of putting this is that it means that if you have two incompatible observables, then if one observable is in an eigenstate, that means it's defined precisely,

you know exactly what it's going to be, then the other must be in a superposition. So let's take momentum and position as sort of the canonical ones, linear momentum and position. Suppose that I know my position exactly. That means the position of this particular system, is in an eigenstate of the position operator. I can tell you exactly what the position is, and it's always going to be that position. That's what an eigenstate is, right? It corresponds to exactly one possible measurable outcome, and that's always the measurable outcome you get for that system, obviously until the system is changed, but then it's a different system. So we're talking about the one system without it being interfered with by anything. You always get that one outcome. That's what an eigenstate is.

So suppose the system is in one single position eigenstate. Now, question is, what will the momentum representation of that system be? Remember I said that you can represent the same KET in terms of many different variables, or different coordinates, different representations, depending on what you're interested in. So let's say I initially started off with my Ket being represented in position space, and I found that it was in an eigenstate, so exactly one position that I know where that position is. Now let's suppose that instead I want to transfer and look at it in momentum space. The question is, what will the KET look like, what will my system look like in momentum space?

In particular, can it be in a momentum eigenstate as well? It's in a position, I guess state. Could it also be in a momentum eigenstate? Could I know exactly what the momentum is, and always get that same measured momentum every time I measure the system? The answer is no, I can't. And that's because position and linear momentum are incompatible observables. They do not have simultaneous eigenstates, which means that whenever I'm in, an eigenstate of one, in this case position, I must be in a superposition of the other. So, if I'm in a position, I can say, I must be in a superposition over multiple different

possible mementa, or vice versa. If I was in a momentum, I guess state, I must be in a superposition of different position eigenstates. And this is why you can't arbitrarily measure to any precision the position and momentum, an object, or any quantum system, say an electron, at the same time. because both cannot simultaneously be in a single eigenstate. If one is in an eigenstate, that is you can measure that one, say, position precisely, then you can't measure the other one because it's in a superposition. So sometimes when you measure it, you'll get one answer, and another time, other times when you measure it, you'll get a different answer,

and therefore you'll get variability in your measurements, and you won't be able to say precisely what the momentum is. So being in a superposition of states corresponds directly to having a spread in your measurements, if you repeat the measurement over and over again with the same system prepared in the same way, you'll get different answers. You'll get a probability distribution, so it's not like you'll get just whatever, that there'll be some answers that you tend to cluster tightly around, but you won't always get exactly the same answer

because the system's in a superposition. It doesn't have a single well-defined value. So this is incompatible observables. When you can't simultaneously measure to arbitrary precision the value of two observables, that's because the operators that correspond to those observables, do not possess simultaneous eigenstates. Okay, the final thing I want to talk about in this initial overview of the basic concepts

is to see where Schroenger's equation fits into this scheme, because Schroenger's equation is central to the study of quantum mechanics, and certainly you'll have seen it if you've ever studied quantum mechanics before, and I would have talked about it in my previous episode on the principles of quantum mechanics.

Schrodinger's equation just gives the mathematical description of how a quantum system evolves over time. There are many different forms of Schroenger's equation depending on the details of the system. The most general form can be expressed in the language of bras and kets that we're talking about in this episode here. Essentially what it says is, so Shrownger's equation in this form says that the rate at which a quantum state changes so that it's particularly this is the partial derivative of a kett with respect to time so but that's the rate at which the

quantum state is changing, is proportional to the expected energy of that state. Now, that doesn't tell you precisely the form of the equation, which is, again, not really possible to do over an audio podcast, but I'm giving you the sort of gist of what the equation, what Schrodinger's equation means in this context. It tells you, Schroedger's equation, tells you the rate at which the quantum system, the KET, changes over time. And it says specifically that the rate at which changes over time is related to, or is directly determined by, the energy of that state. So states with higher energy change more rapidly, essentially, very crudely put. It also is a differential equation, that is, the rate at which a ket

changes depends on that particular ket. So you'll get exponential solutions. So Schroeneges equation is alive and well in this formalism. It may look different if you look it up, and the reason is because you can express Schroenger's equation in much more abstract forms than the typical algebraic form where you have particular functions of x and your position coordinates and so on. That's actually just Schrodinger's equation for in the position representation and often for a particular potential, like for example an electron about an atom. But the most general form of Schroedges equation doesn't specify what the energy function has to look like, and it also doesn't specify what coordinate system you're using and what representation you're using. It really just says that the rate at which a quantum system changes over time depends on the energy of that state.

And that's a very powerful result that we can then use to compute a lot of things in quantum mechanics. So that's all I have time for in this episode. I only got through some of the basic concepts of advanced quantum mechanics. In the next episode, we'll look at some of the more advanced ideas that I mentioned at the start of this episode. Perhaps just before I finish up, I'll briefly review the key concepts that we've been through. Because this has been a pretty heavy episode, and hopefully I've been moderately, clear about explaining the key points. The key points have been, first of all, that quantum states are represented as vectors in Hilbert space, and these vectors are called kets. Purpose of representing

quantum states this way is to represent the fact that an arbitrary quantum state is a superposition of different measurable states. Second point is that each of these possible measurable outcome states is called an eigenstate. And if the system is in an eigenstate, then you'll always get that particular measured corresponding measurable outcome, whether it's position or spin or whatever, you'll always measure that outcome. Conversely, if it's not in an eigenstate, then the system's in some superposition of eigenstates, then when you measure it on different occasions, you'll get different outcomes with probabilities depending on essentially how much of the superposition is in that eigenstate. Third point, all of the possible measurements that we can make on a quantum system

have a corresponding emission operator on the Hilbert space. So it's a function that operates on the kets in that Hilbert space. The particular form of that function depends on the representation I choose, whether I choose position or momentum or whatever else. But the operator is just an abstract algebraic object that operates on the ket and outputs the corresponding variable that corresponds to the operator. So it could be energy, it could be angular momentum,

it could be linear momentum, position, or whatever else. The next point is that different eigenstates are always orthogonal to each other. That is, if the system is in one eigenstate, then there's never a proportional chance of being measured in a different eigenstate. The eigenstates represent the possible measurable outcomes of the system. So if the system is simply in an eigenstate, then it's always measured to be in that eigenstate. Only if it's in a superposition over the eigenstates, over the possible measurable outcomes, then do I measure a range of possible values. Next point is that multiple different observables are said to be compatible

if they possess simultaneous eigenstates, which means that we can simultaneously measure both of those variables to arbitrary precision. Position and momentum are not compatible. That is, they are incompatible observables, which means that if the system is in the eigenstate of one of those, say position, then it cannot also be in the eigenstate of another one. And therefore we will always have a superposition of one or the other of these two incompatible observables. And therefore when we measure those, we will always get a spread in at least one of those possible observables,

which explains why we can't measure them both to arbitrary position, because we can't get them both in a simultaneous eigenstate. Essentially because there is no defined simultaneous eigenstate. Final point is that Schroeneg's equation describes how a quantum system, or Ket, changes over time and in particular the rate at which it changes with respect to time is directly related to the energy of the system and the particular form that that equation takes again depends on the representation that I choose for the system the complete set of states that I insert in

so just to finalize in some sense the whole point of this formalism that we've talked about which is called Dirac Brachet notation by the way Darak came up with it but involving bras and kets and abstract Hilbert space and so on In some sense the whole point of it is to keep track of these eigenstates, because these eigenstates are what we measure. The system can only ever be measured to be in an eigenstate. And what we say is that when we conduct some sort of physical measurement, the system that's initially, the quantum system that's initially in some superposition of eigenstates, collapses into a single eigenstate with some probability depending on essentially the projection of the initial superposition onto that specific eigenstate.

how much was pointed in that particular direction in the abstract Hilbert space. That collapse only occurs upon measurement, so we can only ever measure a system in an eigenstate, but when the system is not being measured, or prior to it being measured, quantum systems exist in a superposition of possible eigenstates. Now, sometimes that superposition can just consist of one eigenstate. That's, you know, case I gave before,

in my spin-half's case when you can just have a ket that's in the spin-up direction, then it's already in an eigenit state, and you'll always measure it to be in the spin-up state. But in general, quantum systems are not like that. They don't generally hang around in eigenstates. In general, they exist as superpositions over different possible eigenstates with varying probabilities of being in each eigenstate, depending upon essentially the projection of the vector that represents the system in the direction that corresponds to the eigenstate. Each direction or dimension of the abstract, HILBET space, vector space, that the KET lives in, exists in, defines a different possible eigenstate or a different possible observable, observable value or measurable outcome of that system.

What we really care about are these observables, but the whole Hilbert space and vector space and orthogonality and the superpositions and also the operators acting upon that, all of that is to keep track of the superpositions and the possible combinations of these eigenstates that are possible because of the quantum mechanical weirdness of the system. That's essentially just a fundamental postulate that quantum systems exist in this superposition. At the end of the day,

we're interested in the probabilities of going from a given superposition to a particular measurable outcome, and that's essentially what all this formalism is about, keeping track of those probabilities of superpositions going to particular measurable outcomes for a particular observable. Hopefully that's been moderately clear. you enjoyed listening to this episode, please consider visiting the podcast Facebook page, which you can

find by just going to Facebook and typing in the Science of Everything podcast, where you'll get updates about the show, links to new episodes, and also I sometimes post a visual material to accompany the show, to accompany the episodes. So if you could like that, that's a bunch of appreciated. It's a way of exposing the show to a wider audience. You can also jump onto iTunes or a podcast aggregator of your choice and leave a favorable review of the show, which I greatly appreciate. If you'd like to contact me with questions or suggestions or just other feedback, my email address is FODS12 at gmail.com. That's F-O-D-S-1-2 at gmail.com. Thank you for listening and I'll talk to you next time.