I want to start talking about quantum mechanics now. We haven't talked about quantum mechanics.

We've talked about classical physics in a world where what we can call the state space, the space of states is discrete. We've talked about hopping around from one state to another, and how information is counted in bits, how with time you can jump from one state to another, and so forth.

That's all classical physics, or it's physics based on classical logic. Classical logic means that the space state is a Boolean space, so that everybody know what a Boolean space is.

Because if not, all it means is a bunch of points in some space, and that each point represents a state, and all of the logic is classical logic. Classical logic that Aristotle would have understood, that Newton would have understood, and so forth. Whereas quantum mechanics makes use of an entirely new kind of logic, which is in a very basic way different.

To illustrate it, I'd like to start with a basic idea, first of all, of a single quantum bit, a qubit. We talked about the classical bit, which is just the heads tails type distinction.

Here we can, I have my prop already. This end is different than this end. If I point it up, that's one state. If I point it down, that's another state. That's the two state system, and that's all there is to it.

Nothing in between, assume nothing in between. That's the basic classical bit, either up or down, with nothing in between. And of course, you can have many bits. Ooh, that's fun. You can have many bits, two bits, three bits, four bits, whatever.

And pretty much all of physics. Anything in physics can be represented, or at least approximated, by a system of classical bits. Anything in classical physics. Quantum physics, the concept of a bit is very, very different. The concept of a bit of information, and that's what we want to talk about today.

The basic, simplest example is the spin of an electron. Now, we don't have to know very much about what spin means. Does it have something to do with spinning or not? This is not very important.

What is important is that every electron has associated with it a vector. Now, we're going to get into big trouble, because there are two things that I'm going to wind up calling vectors. One of them is abstract vector spaces, which we're going to come to in a little while. And the other is vectors in ordinary space.

A vector, this I took with me just to illustrate the idea of a vector. You know what a vector is in ordinary space. It's an object with a length and a direction. And I put a nail in the end here to indicate the arrow so that you'll know which way it's pointing. It's not pointing at me, it's pointing up in the corner of the room there.

That's your naive elementary physics concept of a vector. A thing in ordinary space that has a direction. Of course, it also has components. For example, this vector here has a component in the horizontal direction. It has a component in the vertical direction. But all together, it's pointing in this other direction.

So, that's one concept of a vector. A pointer in space with a length and a direction. Now, there's another thing we're going to call vectors, which is a more abstract concept. And it is not necessarily, in general, will not be referring to things in space.

It's an abstract mathematical, I just wish there was a different word for it. We're going to be talking about vector spaces. I think sometimes when I teach quantum mechanics, I make up a new word for one or the other of them. Call it a schmector or something, I don't know. But it never really works.

So, we have to keep in mind that when I use the word vector, sometimes, maybe I should just make a new word. Some people say arrow for the old guy. Yeah, that's a good idea. Let's just call this an arrow. You know how long I will last before I start calling it a vector?

Not very long. Right. Now, the spin of an electron is an arrow. I was going to say vector. I can't help it. No, I cannot. I can't do it.

I'm going to call it a vector. I'm going to call it a vector. I can't help myself. Let's call it a spatial vector. How about a spatial vector? A spatial vector means one in ordinary space. So, if my fist is the electron over here, the electron has attached to it a mathematical vector. It's not a rod like this, but it's a sense of directionality. Really, what it is is a direction of twist.

But just think of it as every electron has attached to it a vector which can point, you might think, in any direction in space. In fact, in some sense, it can point in any direction in space. And, in fact, that vector can also be thought of as a magnet.

It can be thought of as a bar magnet with a north pole and a south pole. So, here's the north pole with a nail. The south pole is over here. Every electron is a little magnet. And the amount, the strength of the magnet, which is measured in some units, which I forget what they're called,

strength in magnet units, is some particular number. So, that means all electrons, they're space vectors that go with them. It's called the magnetic moment. The magnetic moments are all the same length. They may point in different directions, but they're all the same length.

And we can simply imagine that they are objects all of the same length. They're a common length, but different directions. Now, let's talk about ordinary classical magnets.

And I want to talk about the concept of preparing a state and detecting a state. We're still thinking very classically for the moment. Everything I explained here was completely with classical logic. So, here's our electron. And it has attached to it an arrow with a north pole and a south pole.

And let's say I want to prepare that electron in a configuration where its magnetic moment, its little vector, is pointing. This north there doesn't stand for its direction.

It stands for the north pole of the magnet. I want to have the north pole. I want the magnet to be pointed vertically upward. I want to prepare it that way. How do I do it? Okay, what you do is you simply create a large magnetic field. For example, you create a magnet.

This is not a very good pen. You create a magnet. You can do it as an electromagnet if you like. Turn on some current and make an electromagnet with a north pole over here, a south pole over here, and the electron is supposed to be in the magnetic field.

And what will happen to that electron? What will happen to that little bar magnet? Well, what actually will happen to the little bar magnet, if you know anything about how bar magnets move in magnetic fields, is it will precess. It will not just jump up into the right position, not unless there's some friction.

Instead, what it will do is it will precess like this around the magnetic field. Everything imagined is in a vacuum, so there's no friction or anything like that. The electron will precess around like this. That's what a little bar magnet would do. But sooner or later, it will get rid of that precessional energy.

Anybody know how it gets rid of the precessional energy? It radiates some radiation. It radiates a little bit of electromagnetic radiation, or a lot, depending on the circumstances. And the size of the magnetic field, and so forth. It will radiate away its energy

and come into the best energetic configuration, which is pointing straight up, where the north pole of the magnet is pointing toward the south pole, where the north pole of the electron is pointing toward the south pole of the magnet,

only which part I've drawn. So that's the preparation. And if we do it with a very, very big magnetic field, it will radiate that energy quickly, and very quickly come to equilibrium with the north pole of the electron pointing vertically upward. If you like, if we don't want to get,

I was going to use the word entangled, but embroiled with the question of where the electron is, just imagine in your head that I hold the electron absolutely fixed in position so that the only thing that can do anything interesting is the direction of the magnetic field. Okay, so after a small amount of time,

that electron will be pointing in the up direction. We could, if we liked, we could prepare the electron in a different configuration by rotating the magnetic poles to something like this, south-north, we could prepare the electron initially

so that it's pointing in that direction. So we start with a electron pointing in some direction. Now we want to measure, we want to do a measurement to find out

which direction, this is the second kind of experiment, it's the detection. First was the preparation, and then we do some things to the electron, whatever it is, and next we want to find out, we want to detect which direction the electron is pointing in. Is it pointing in that direction?

Is it pointing in some other direction? Mainly we could ask, what's its angle relative to the magnetic field? And to answer that question, we do exactly the same thing that we did to detect, to prepare the electron. We stick it into the large magnetic field and wait for it to radiate some radiation.

Now, if this, if all the physics were classical, and this were really a perfectly classical set up, a classical compass here instead of a quantum mechanical electron,

then the, again, the electron would eventually point its way itself up, but one measure of the angle would be how much radiation it emitted. For example, if it was almost perfectly upward to begin with, if it were almost perfectly upward,

then it only has a little bit of extra energy, perfectly upward has the minimum energy, when the north pole is pointing toward the south pole. That's the minimum energy of that little pointer here. Rotate it away a little bit, you give it a little bit of magnetic energy, and of course,

after you've put it into the, after you've let it come to rest again, it does so by giving off that little bit of energy. Alright? If you were to go to a more extreme situation, where you pointed the electron in some, horizontally, then by the time it got vertical,

it would emit even more energy. Finally, if you pointed it straight down, it would emit the maximum amount of energy in the form of radiation, and so you could measure at least the angle relative to the magnetic field by measuring

how much radiation comes out. And the answer would be a nice continuous function of the angle of the electron, so that when the, so when the electron was pointing up, no energy comes out. This is angle,

horizontally, energy emitted, if the angle is zero, that's when it's pointing straight up, no energy, if the angle is downward, maximum energy, and then as you turn it over all the ways, back to zero energy, and so forth. So, you would measure then, the same way

that you prepare, you measure the spin, the spin direction of the electron, by letting it emit a photon. By letting it emit some radiation, excuse me, and counting up the amount of the radiation. Now, I've told you completely the wrong story for a real electron. This is not the way it works.

This is not the way it works for a genuine electron. Something else happens, and I'll tell you now what it is. And it's very weird. It really does illustrate the weirdness of the quantum world. First of all, no matter which

way you create the electron in here, I won't even try to give a classical picture. If I try to give a classical picture, I will prejudice the story. You do something to the electron, and you put it over here. Whatever you do to the electron, one thing you could do to the electron is

initially prepare it in some funny angle, but whatever you do to the electron, you put it over here, you turn on the magnetic field, and one of two things happens. Only two things can happen. One is it emits no photon,

no electromagnetic radiation. The other is that it emits one quantum, one photon, of electromagnetic radiation of a very particular frequency. It's the particular frequency that corresponds to the energy of jumping

from down to up. Whatever that energy is, you can emit a photon of just exactly that energy and no other energy. So there seem to be, it's almost as if the electron had only two possible configurations.

You jumble it up, you don't know what you've done to it, you stick it in here, and it seems there's only two things that can happen. One is that it behaves as if it were pointing up, in which case it gives off no photon. The other is that it behaves as if it were pointing downward

and gives off the photon. And those are the only two things that happen, that can happen. But, depending on, now that seems a little weird. Why? Because supposing you prepared the electron by putting it into a magnetic field

that would have the effect of freezing it into a direction, let's say at 45 degrees, or even better, at 90 degrees relative, in other words horizontal. Then we would think to ourselves, well that electron is now pointing in some other direction.

Now when I stick it into the vertical field you would think that the response would be something different than if it were just plain up or down. But no, the response is always just one of two responses. Either it does give off a photon, or it doesn't, and always of the same energy.

So it seems that in some sense there are only two states for the electron. Either it's pointing up or it's pointing down. But wait a minute. What about if you put the electron into a magnetic field pointing in yet a different direction? Then you're going to find

that there are only two states for the electron either pointing this way or pointing that way. Something's very confusing. How many states does the electron actually have? Is it only up or down? Or can you make it point in some other direction and if you do make it point in some other direction, how come when you stick it into the

up-down detector here the detector that detects the direction in the up-down direction that you only get two possibilities? How come there isn't a continuum of possibilities in between? That's the puzzle, or that's one of the puzzles of quantum mechanics that it behaves as if there are only two states

up or down when you measure it but you can nevertheless prepare that electron pointing in any direction by using a magnetic field in the appropriate direction. Now, the one thing though that is true is if you started with the electron

pointing upward let's suppose we created the electron pointing upward by putting it in this very strong magnetic field shut off the magnetic field for a while and then turn it back on. The answer is we will get no photon, the electron is pointing up.

Supposing we first create the electron in some other direction pointing along the 45 degree axis here and then we do the vertical experiment. We turn on the vertical magnetic field we will either discover that it's up or down

but with a probability distribution. It will not be definitely up or definitely down. We can do this experiment repeatedly over and over and over many many times identically. Sometimes we will find a photon emitted sometimes we won't.

The more the electron was initially pointing vertically upward the less the probability is that we will emit that single photon. The maximum probability for emitting that photon will be if we were originally pointing down. Somewhere in between the probability will

be half for a photon to be emitted. In fact that's exactly the case if the original electron were prepared horizontally by a horizontal set up and then put into the vertical field there will be a probability of a half that it emits the photon and a half that

it doesn't emit the photon. So there's something extraordinary going on here. In some sense it seems there are only two configurations. Every time you measure that electron you will either find it's up or it's down. Nothing else. Nothing in between. But there are situations where it

will be probabilistically if you do the same thing many many times you'll find there's a probability distribution and it's that probability distribution which has some memory of which way you originally created the photon. So a quantum bit is a confusing thing. It looks like it has only two

states up or down but then you can make it point in some other direction so that it looks like there's other states possible again when you measure it you find only two possibilities. It's a confusing entity, a quantum bit.

Any questions so far? Yeah. Say it again. Not the magnet that's attached to an electron is it? Right. Oh yeah the magnet that creates the field is what?

Yeah. If one would try to do this to test and create the field to interact it with small objects like other electrons that's when we're going to start to talk about entanglement between electrons. But let's do it one at a time

and then we'll come to the two electron system. The two electron system is the one that's really interesting because that's where entanglement starts to happen. So let's not try that one yet. Little steps for little feet. Yeah, question. To get this probability distribution do you have to do the

whole experiment over and over again, prepare it and then measure? If you want to get it experimentally. Or can you prepare it once and then... Well, there's no way that with one experiment that you can accumulate a probability distribution. Prepare it once prepare it this way measure, turn the field off

measure again turn the field off, measure again or do you have to prepare it each time in between? No, you have to prepare it again each time in between. Right. Yes, yes, yes you can prepare a lot of electrons at once but I think the question goes something like this. Supposing you prepared it this way and then measured it.

And then measured the same one again and then measured the same one. No. Once you discovered this was up the first time it will be up every time afterwards and so what you will discover in the second experiment the third experiment, the fourth experiment no photon. So once the experiment

has fixed it to be up, that's it. It has no memory anymore of this. The other hand has... Preparing means waiting to see what happens and if it emits you assume Right. So you're just doing that again

That's what I'm calling it. Yeah, it's the usual way. I'm talking about an electron not an atom. Well, I'm talking about an electron perhaps which is in some potential which nails its position

down so that we don't have to worry about it moving. We're just concentrating now purely on the spin of the electron so imagine driving a nail through the heart of the electron and nailing it to the blackboard. It still has the ability for its spin to move. Is the magnetic field

or one electron constant? Is the electron field The magnetic field is constant Is it constant in time? Oh, okay. So here's what we're imagining. Imagining I want to do an experiment so I take an electron which is there originally I don't know how I got it there

anyway I like without the magnetic field being there. Now I want to measure that electron so I very quickly turn on a very large magnetic field How do you do that? If it's an electromagnet, it's very easy you flip a switch and all of a sudden there's a large

electric field, sorry, magnetic field pointing upward and if that electric if that magnetic field is large enough it will very quickly cause the electron to radiate. Okay, so I would say no, we want to be thinking about turning on the field to do the measurement or to do the preparation

Yeah? Is it possible to prepare facing south and then put it into the magnetic field system and then the outcome is facing north That's right If you start turning over the whole device you start with

south and the bottom where south belongs and north and the top where north belongs then the electron will point down eventually now you remove the magnetic field you remove the magnetic field or suddenly reverse just turn off the magnet

and reverse the polarity you can do that by reversing the current in the electromagnet so that suddenly the south pole is on top and the north pole is on the bottom the electron was down in a very short amount of time, it'll flip up and give off a photon. Was that the question?

The probability of that is one The probability of that is one, yeah The probability that it will give off yes, in that circumstance the probability that it will give off a photon is one If on the other hand this weren't perfectly aligned and you were at some cattywampus angle

then the probability would be less than one and it's those probabilities which constitute the experimental yeah So it must put energy into the electron in order for it to keep emitting in all these experiments, right? So somehow it must get charged after No, it doesn't get electrically charged it has electric charge

turning on turning on the the electric current going through these magnets you'll have to do a little more you'll have to put a little more energy into the magnet because the electron is there in other words, originally

the energy comes from the energy that you used to turn on the magnetic field One more In the case that the electron actually emits a photon are we supposed to assume that that's instantaneous, or what happens in that transition? Well, the point, yeah the language is probably wrong

Here's what I can tell you if you make the magnetic field sufficiently large the electron will very quickly emit a photon Now, the correct answer in general is that the photon will be emitted at a more or less random time

in the same way that a radioactive atom will decay it decays with a half-life it decays with a half-life so if you put the electron into the magnetic field in the wrong direction it will, with a certain half-life decay decay means give off a photon

and write itself in the other direction but if you make the magnetic field large enough that half-life will be very short and so it will very suddenly flip

One of the things we've learned in quantum mechanics is to ask questions only if we know how to make the measurement If you can tell me in detail how to measure when the electron emits its photon then we can try to answer it

I would say that's one of these questions that you're not supposed to answer There's a thing called the energy-time uncertainty principle and if you know the energy of the electron you're uncertain about the time

but let's come to uncertainty principles The bigger you make the magnetic field the more energy you put in the more quickly it will Absolutely More quickly in the sense of a smaller half-life

Right Okay, so first of all we're dealing with a theory that has a probabilistic description and we will see ultimately that there is really there's probably no way around

the fact that the description has to be probabilistic at least at some level The mathematics of the state space of the concept of a state the state is different in quantum mechanics, it's clearly different if it's

clearly different It's a system with only two states you think but then again it seems to be able to be oriented in any direction The mathematics of quantum mechanics and the states and in particular the states of a quantum bit

are not the mathematics of a set of two points They're not the Boolean mathematics of a set of two points The thing which describes the mathematics describes a quantum state is a vector space Now I use vector in a different sense It is no longer an arrow in ordinary

space, it is a mathematical abstract vector space whose character you'll come to understand if you persist and persist at it a little bit you will begin to understand what this vector space is and how it characterizes

the possible configurations or the possible states of an electron and how it can be consistent with this fact here Okay I'm going to spend a little bit of time now just doing some abstract mathematics, very abstract mathematics and then we'll seek to interpret it and to interpret

the state of the electron I'm going to tell you what a linear vector space is, what a vector space is A vector space is a collection of objects and here are the rules for an abstract vector space As I said, do not confuse this with

the pointer that is a vector in ordinary space. This is just an abstract completely abstract concept First of all, a vector is an object and we'll label it like that It's an object with a set of rules, it's a collection

of objects, it's a collection of objects with a rule, first of all a rule that any vector can be multiplied by a constant to get a new vector This gives a new vector, we could call it

let's call it a prime So there's a concept of multiplication by a constant Now first of all, let me tell you that the constants we're going to be talking about are complex numbers not real numbers Anybody here not familiar with the concept of a complex number? If so, raise your hand and be humiliated

Okay Alright now If you really don't know what a complex it may be that you know it by another word Now if you really don't know about complex numbers, look them up

because you're going to need them Alright, a complex, remember remember this number I I is the square root of minus one Now minus one has no number that's there's no number ordinary number whose square

whose square is minus one So there is no square root of minus one Any number times itself is always positive, right? Three times three is nine Minus three times minus three is also nine, alright? But mathematicians many many many years ago invented the abstract idea of an imaginary number They called it imaginary

and said let it be that I squared is equal to minus one invent a new number Okay, we know how to use those numbers I'm not going to I'm not going to spend any time

at it Whoever raised their hand Are they familiar with I equals the square root of minus one? Yes, alright Now, you can have complex numbers. Complex numbers are what happens if you take two two real numbers, ordinary numbers

Let's call it A and B Different A plus IB In other words, if you add a real number to an imaginary number, you get a complex number. That's the whole definition of a complex number You can multiply complex numbers

and all you have to remember, you use the ordinary rules of arithmetic, except you have to remember that I squared is minus one So for example, if you're going to Well, as an exercise you can square this. What's the square of A plus IB? I won't tell you You can work it out Alright, so that's a complex number Often written as

Z. Sometimes Z You can use any letter that you like But it's a complex number with both a real part and an imaginary part Another concept is the complex conjugate. This is very important You'll need it And it's very simple

It's A minus IB is the complex conjugate of Z So if you take a real If you take a complex number and you change the sign of the imaginary part, that's called the complex conjugate of the number The complex conjugate of the complex conjugate gives you back the original number

These are important mathematical concepts They're very elementary Let me give you an example Supposing you multiply Z star by Z What is that? That's uh Or Z times Z star It doesn't matter either way

It's A plus IB times A minus IB Let's see what we get Just ordinary multiplication We open up the brackets here We get A times A, which is A squared We get A times IB with a minus sign minus IAB

plus IAB from this one times this one And then, what about IB times IB What does IB times IB give? Minus B squared Minus B squared, but there's a minus sign here already So it's minus I times I times B

And so that gives plus B squared Altogether, we get A squared plus B squared These pieces cancel So that's an example of using, I mean, that's an example We're not using it for anything It's just an example of a definition and then using the definition of complex arithmetic

to calculate what Z star Z is Notice that Z star Z is always real, and it's always positive It's always real and positive A squared, A and B are real numbers A squared and B squared A squared are both positive So Z star Z is a real number and it's positive

It's thought of more or less as the magnitude of the complex number Alright, so first of all, for the kind of vector spaces we're going to be thinking about They're called complex vector spaces And one of the rules of one of the operations that exist in the complex vector space is multiplication

by a complex number Now, a complex number includes real numbers incidentally, so you can C could be 2, or it could be I, or it could be 2 plus I, or anything like that There's a notion, if there's a vector you may multiply it by a constant And you get a new vector

That's the first operation And there's only one other operation that's of interest to us It's adding vectors There's a rule for adding vectors Or, assume there's a rule for adding vectors So if I have two vectors A and B I can construct a vector called

A plus B And that's a new vector Which I guess we can call C But C is a different C than appears here This C stood for constant This C just stands for a new vector which is A plus B So every pair of vectors you can add them And I'm going to show you

some less abstract examples But, first of all the simplest example of a vector space like this is just the complex numbers themselves The ordinary complex numbers allow you to multiply

any complex number by another complex number A particular A constant particular complex number And it allows you to add complex numbers So just complex numbers by themselves are a vector space A complex vector space That's the simplest example But, let me show you

another example And the other example is something we talked about last time which we called column vectors Just represent an abstract quantity by a set of components

All the components are themselves complex numbers So, for every vector really what that vector is standing for I write equal signs, I probably shouldn't write equals Mathematicians would have a stroke if I wrote equals there

Um Is represented by a column vector The column vector has entries and it's just a table of numbers but it's a table of complex numbers A table of complex numbers Supposing you want to multiply that column vector here by a constant

The rule is very simple Multiply every entry by that constant So, C times A is represented by CA1 CA2 CA3 We just multiply all the components

by the same number That's multiplication by a constant How do we add two vectors together? Let's suppose we have two vectors, A and B A is represented by the column whose entries are A1, A2, A3 B is represented by B1, B2, B3

Where these are, A's and B's are just numbers And when we add them We just get a third column vector whose entries are A1 plus B1 A2 plus B2 A3 plus B3 So column vectors are an example of

a vector space And they are the basic example that we will use over and over again As I said Mathematicians would have a stroke if I called the columns vectors They would say the columns represent vectors And you would call these the components

or a set of components of a vector Okay That's the idea of an abstract vector space The very strange thing is that the states of a quantum system, in particular a quantum bit, oh, incidentally you're not restricted to three. This is a particular

case. This is a three dimensional vector space We could have a two dimensional vector space. That's actually going to be more interesting to us for the moment A two dimensional vector space just has two entries. That's the next simplest thing if you like to the numbers The next simplest vector space is a two

dimensional vector space. Then there's a three dimensional vector space and a four dimensional vector space and so forth And they all make sense Okay That's You'll get used to this. Sooner or later you'll start thinking in terms of abstract vector spaces if you pursue this subject

There's no way to learn quantum mechanics honestly and correctly without going through this mathematics So it's absolutely essential Now If we were talking about the simplest vector space just complex numbers then every complex number

has its complex conjugate That means that there is a second vector space which is the space of the complex conjugate numbers Alright? That the concept of complex conjugation is an important concept and it also exists

for other vector spaces All you have to do basically is write the complex conjugate of the entries But a rule When you're writing the complex conjugate of a vector I want to write the complex conjugate of the vector A You draw it the other way

This way That stands for the complex conjugate of the vector A and furthermore you represent it by a row vector The components of the row vector are the complex conjugates

A1 star A2 star So whenever you see a row vector, always think of it as the complex conjugate of the corresponding of the corresponding column vector If I tell you there's a

row vector A which is in correspondence with a column vector A always remember that there's a complex conjugation operation Okay? This is the basic idea of a complex vector space written out in components Everybody happy with that?

Anybody? There's no asterisk at the A No Usually, that's right That's right The notation does not put a star over here But you remember that the star

is implicit in turning the symbol backward I'll remind you, this symbol is called a ket No, this symbol is called a bra This symbol is called a ket Do I have it right? I don't One of them is a bra, one of them is a ket, but it doesn't matter

It's completely symmetric between the two Remember that the complex conjugate of a complex conjugate is the original thing So that's the notion of a complex vector space And finally not finally but I'll put the pieces together You can multiply

two vectors The product is called We've discussed this before, I realize I'm not getting senile, I know I've talked about this before but we really need to do it right now There's the concept of the inner product of two vectors The inner product of two vectors the easiest way for me to describe it is in terms of components

I don't want to spend a lot of time giving you the most abstract definition Are there many mathematicians in the audience? Plug your ears, just plug them up and Right

Actually mathematicians also use components but I think Anyway The next concept is the concept of the product of two vectors but it's always the inner product of the two vectors and the concept is

really the product of a vector with the complex conjugate of another vector It's very much like multiplying a complex number by the complex by the complex conjugate of another number, could be the same number, could be another number, so I'm going to give you the rule

The rule I'm going to tell you in terms of components Here's the rule If I want the inner product between the vector B and the vector A and it's written like that It's got a bra ket, it's a bracket OK, that's where the term bra ket came from, half a bra

half a bracket is a bra, the other half of the other vector is a ket and again I forget which one is which This is the ket, right? Yeah, that's the ket, this one's the bra, OK, good Good The rule is You take the two vectors, the B vector is

a row vector and you write B1 star B2 star That's the representation of the B vector and the representation of the A vector is A1 A2 You do not star it And the inner product between them, this product, is gotten by multiplying the first component

of B with the first component of A B1 star A1 plus B2 star A2 First row with first, sorry, first entry here with first entry here plus second entry here, second entry here

Supposing OK, everybody That's the basic concept of the inner product It's just like multiplying two numbers together except we get a term for each, for each entry, one entry and two entry For example, if we take the inner product of a vector with

itself, let's take A with A The inner product of a vector with itself has the form A1 star A1 plus A2 star A2 Notice one thing about this A number times its complex conjugate is

first of all positive and second of all real Well, it couldn't hardly be positive if it weren't real It's a positive number, so the inner product of a complex vector with itself is always a real positive number It can be thought of as a kind of size of a vector It's actually the square of the size of the vector

If you like You can think of it as the square of the size of the vector The length of the vector It measures the magnitude of the vector So, this operation of inner product is absolutely essential, as we'll see, to the interpretation

And it's complex vectors like this which represent the states of a quantum bit Let me just digress, let me stop doing mathematics for a minute and talk about the quantum bit again

The quantum bit could be pointing straight up We could invent a symbol for the state for the configuration of the electron pointing up Let's call it, we could either, we could call it

up or we could call it little vector up or we could just call it plus I think I'll just call it plus to indicate that it's pointing upward Or we can have an electron which is pointing down Now remember, when we go to

measure the electron, we either find that it's up or down and nothing in between and that we can identify with the vector minus Just an abstract notation This stands for the state of an electron pointing up, this stands for the state of an electron pointing down Now in classical physics

we would never under any circumstances think of adding these two vectors or multiplying them by numbers We would just say, this means electron up this means electron down In quantum physics the general state of an electron

or an electron spin is a vector in a vector space which we could write A plus times plus plus A minus times minus In other words, it's a two dimensional vector space, we could also represent it

by A plus A minus And here's the rule So there are some coefficients that we can add them together with And here's the meaning of those

coefficients, or at least a partial meaning of those coefficients A plus star A plus is the probability that we find the electron up. Remember it's positive and it's real

A star times A is positive and it's real and it's the magnitude or square of the magnitude of the coefficient of the up configuration And then there's the down configuration The probability of finding the electron down is A minus times A star

minus That's the probability to find it down That's the interpretation or that's part of the interpretation just to give you some orientation where we're going The quantum state of that electron which when we measure it we find with a probabilistic distribution

is either up or down and nothing in between that's represented by saying the electron when it's pointing in some funny angle is given by a complex vector, complex vector space which can either be represented abstractly like this it can be represented concretely as a column vector

and the square or magnitude of the square of the entries here are simply the probabilities to find it up or down Now presumably probabilities add to one So, one of the rules about quantum mechanics

is that the vectors that represent, we have a vector space of some kind, an abstract vector space but the vectors which actually represent the physical states of a system are the word is normalized

Normalized means that the sum of the probabilities is one So a plus star a plus, plus a minus star a minus should be set equal to one This is like considering only vectors of unit length

That doesn't mean that in the vector space itself there aren't other vectors of other magnitude there are, but the physical states of a quantum system have to be normalized, which is the same thing as setting the sums of the probabilities to one So normalized vectors in a vector space

represent the states of a quantum system Now that's a very abstract concept Why do we get driven to such very abstract concepts? Why can't we visualize this the same way we visualize classical physics? Well, it's because we don't have the wiring and we have to rewire ourselves and we have to rewire ourselves to learn

to think about the quantum states of a system being an abstract vector space Once you get this idea of an abstract vector space and how it fits together with the states of a system, then you're flying and you can understand all of quantum mechanics So any questions about this?

The entries, the coefficients or equivalently the components of the vector when you square them, or better yet when you multiply them by their own complex conjugate, give you the probabilities for the two possibilities

That represents a quantum state if A star A In fact, we can say it this way The inner product of A with itself That's given by A plus star A plus plus A minus star A minus That should be one

So, legitimate quantum states should have inner product with themselves which is equal to one And that's just a statement That's just a statement that the sums of the probabilities of all the probabilities should add up to one Oh, plus and minus are now just

to indicate that one of them corresponds to spin up and the other corresponds to spin down And we could have given them different names I could have called them A up and A down Or I could have called them A1 and A2

They do have They are the coefficients of the ket plus and the ket minus So let's go back a step What about the ket plus? How should we represent that? In some basis, let's At the moment

I don't want to discuss the ambiguity in basis vectors We'll come to that But for the moment we have a basis Don't worry about it, you don't know what I'm talking about This vector will be represented by one zero It'll be represented by

a one in the upper place and a zero in the lower place What about the minus ket? The minus ket is represented by a zero and a one This is just some arbitrary correspondence Now

then if I multiply this by A plus the numerical constant A plus then we have to multiply this by A plus We don't have to multiply zero by A plus because zero times anything is zero If we multiply this by A minus then this becomes A minus here

and if we add them together A plus times the plus ket plus A minus times the minus ket we just add these together and we get adding them gives us A plus A minus

Does that answer the question that was asked? Yes, certainly the plus and minus notation instead of one and two, here I used one and two They're just labels They're just labels. One and two is just a label

to label the two entries I could have used up and down I could have used Joe and Pete I could have used anything I want to label them Here I'm using plus and minus The two possibilities for the orientation of the electron of the quantum bit I'm labeling them by plus and minus

Good. Okay, so now we have the basic concept of a vector space the inner product

Does minus cat plus make sense? The minus cat. Minus one times plus cat. No, the minus cat is not minus one times the plus cat. No, no, no, no, no, no. No. I know that. No.

There's minus. Yes. Yeah. You're asking whether the, whether this object makes sense. You're allowed to take any vector and multiply it by any complex number. In particular, a complex number is minus one.

In fact, but this, but this is not, it is not the same as minus. Right. This has a, okay, let's make that clear. The minus here does not mean that it's minus the plus

vector. It's just a label. No more than eight, than the, than the, eight two means twice A1.

Yeah. Yes. That's correct. That's correct. Wait, wait.

This cat here has probability one in the plus state. Which one? Which one there? That's right. That's right. Both of these. That's right. Absolutely.

Absolutely. We're going to come to that. Very good. Very good. Very good. Very good. Well, you're, you're pointing out something that I was about to come to in a moment. I'll tell it to you right now. Well, okay. If I multiply it by any complex number of magnitude one, it, it's just, that's right. That's correct.

That's correct. So, physically we would not distinguish this state from, from this state. I won't write equals because in the mathematical vector space they're not equal, but they represent the same physics. All right. However, however, when I add them, A plus and minus A

minus are not the same. That is a different state. All right. That is a different state with a minus sign here. But we'll come to that. We'll come to that. It was something I was going to talk about, but not yet. I want you to digest just the idea that the components of

these abstract vectors, when squared, correspond to probabilities. Does it make physical sense to change the minus ket to a bra and multiply those two together? Yes. And bra is minus, is minus k plus.

Let's, let's talk, I think you were asking whether it makes sense to take the ket vector plus and consider what I'll call its conjugate, which is the bra, the bra vector plus. In this case, the, the, there's no change because the

Well, except that you would write it as a row. What about the inner product? The components don't change. Hmm? The components don't change. That's right. In this case, in this case. Right. The course one is a real number. We don't have the complex conjugated.

Okay. So here's a, here's a, a little exercise. What's the inner product of the plus vector with itself? One. Because it's just multiplying one zero times one zero. One times one is one.

So this is one. What about minus minus? Also equal to one. What about the inner product of plus with minus?

Let's just check that. For those who don't see it so quickly, let's just check it. The plus bra is one zero. The minus ket is zero one.

All right. So when we multiply them together, we get one times zero plus zero times one, which is altogether zero. All right. What about minus plus? Also zero. What's the word for two vectors whose inner product is zero?

Orthogonal. Orthogonal. They're called orthogonal vectors. So the two vectors representing the two distinct states of the electron, one up and one down, are two orthogonal vectors in a linear vector space of states.

A linear vector space of states. OK. That's the basic set up. And the coefficients that appear there, their squared magnitudes are the probabilities for the two configurations. The probability for up and for down.

Now, as you can see, there are more possible physical states than just up and down. There are all the linear combinations. It's the linear combinations of them that correspond to the electron point

having been prepared in different directions. So in other words, an arbitrary combination like this with an A plus and an A minus there, when this is not one and zero or zero and one, correspond when you consider all the possible complex numbers to the different directions that you might have prepared the electron.

But still, all there is is up and down and the probabilities for up and down. So a state, for example, let me give you an example. A vector, let's write one in both slots.

That's not allowed. Why isn't that allowed? Probability doesn't add to one. So the probability would add to two here. One squared plus one squared is two.

We'd have to divide it by square root of two. Now each of the probabilities is a half and they add up to one. Parentheses where? You're right. I meant to write the column.

That's a state with equal probability, namely probability a half to find the electron up and now this one has probability a half to find the electron down.

There are more states than just up and down. And in fact, this one corresponds to what you would create if you created the electron pointing in the horizontal direction. If with a magnetic field in the horizontal direction, you froze the electron. Let's take a break.

If you froze the electron into place with a magnetic field in the horizontal direction, in the horizontal plane, one of the possibilities would be this state here, which would have half a probability of being up and half a probability of being down. Half the probability of up and half the probability of being down

actually corresponds to some configuration where the electron is lying somewhere in the horizontal plane. So we have plenty of vectors around with all kinds of complex numbers out of which we can build, and we'll see that we can do this, we can build the space of electrons which in some sense are pointing in any direction,

but whenever we measure them, all we get is up or down with probabilities that are governed by these coefficients. Let's take a break for 7 minutes, 7 minutes and 30 seconds. Now more than one person is probably a little bit confused about the two notions of vectors.

One is having to do with directionality in ordinary space, and one having to do with this abstract concept and this notion of components of vectors labeling upstate and downstate.

The question is what's the connection between the notion of directionality in space and these vectors here. Now what I want to say first of all is they are not simply related. They are related, but not simply related.

For example, how many components does a vector have, an ordinary vector? Three. And they're real numbers, right? X, Y and Z. They're all real numbers. These vectors have two components and they're complex numbers. That means, how many real numbers does it take to describe one of them?

Four. So they're not the same thing, but there must be some connection. We're not going to do that connection just yet, but we will come to it.

The next concept, which we already went into a little last time, is the concept of a matrix. How do you know what vector space describes a particular quantum hypothesis?

It's an experimental question in a sense. From the fact that there are only two possible answers to the way that when you do this experiment on the electron, you either get that it's up or down and nothing in between, that tells you that there are two possibilities and it tells you that you should be dealing with a two-dimensional vector space.

All two-dimensional complex vector spaces are mathematically the same. So they're all the same. A two-dimensional complex vector space is a two-dimensional complex vector space. There's not more than one of them mathematically.

So you count up the number of possibilities. Somebody asked me before if I had a spin-one particle instead of a spin-a-half particle, which is what an electron is, how many components would there be? Then there would be three.

But it's an experimental question. How many possibilities are there? How many states are there? How many distinct possibilities are there? In the case of the electron spin, in the case of the spin of the electron, there are two possibilities. And so it's a two-dimensional vector space. And they're all the same. There's no distinction between different two-dimensional vector spaces.

Other than the fact that we could consider real or complex vector spaces. Okay. Let's move on to the concept, to the abstract concept of a linear operator or the concrete concept of a matrix.

Let me tell you where we're going. We've talked about states, but we haven't talked about the things that you actually measure. The things that you measure or that you can measure are called the observables.

The observables of a system are the things that you can measure and get answers for. Position of an electron is an observable, but we're not doing anything as complicated as that. If we were to label the up and the down state as plus and minus, we could invent an observable which would have two possible values.

It could either be plus, plus one or minus one, and it would be an observable that we could measure. If the electron's up, we will assign it the number plus one. If it's down, we'll assign it the number minus one, also called an observable.

Anything that we can measure that has a numerical value, a numerical value means a real number. Anything that we can measure that when the measurement is recorded, it gives rise to a number, a numerical number, is called an observable. So as I said, in the case of the electron spin, if we measure the, whether it's up or down,

we could assign up the number plus one and down the number minus one. And in that way, have an observable whose numerical value when we measure it is either plus one or minus one.

Let me talk about, before talking about observables in quantum mechanics, let's talk about observables for a minute in classical mechanics, in classical theory. In classical theory, the states of a system are just a set of states which we represented by points.

How many do I have? One, two, three, four, five, six. These could be the six possibilities for throwing a dice, a die, a single die.

The one, two, three, four, five, six. And we could assign some numbers to each one of these points. We could label this one one, two, three, four, five, six. We throw the die and we look at it.

That's the measurement. We throw the die and we look at it and we get an answer. The answer is either one, two, three, four, five, or six. That measurement is the measurement of an observable and it's an observable which has six possible answers. There are other observables that we could concoct.

For example, we could concoct an observable which is zero everywheres except at one value. One, zero, zero, zero, zero. Now, this means that if we flip the die and we get a one, we assign the number one to the observable.

If we get any other number, we assign the value zero to the observable. So there are many observables that you can make and basically they correspond to any functions of these points. Any functions of these points, assign any function to the points.

Function points mean assign any numbers you like to these points, real numbers. And then when you flip the die and you see what comes up, you say the value of the observable is whatever the numerical value of that observable was when you flip the die.

There are many, many observables that you could think about in this simple system. But they're all rather trivial. I mean, they just correspond to the basic question of whether the die is one, two, three, four, five, or six.

It's interesting just not because... Yes, it is interesting to define the observable as a function of which state we're talking about.

F sub N. N here represents which of these states we're talking about. And F is a function which could be any number for each of these states. That's called an observable. When you flip the die, you look at what you get and you assign the number F to the particular state.

F sub one, F sub two, through F sub six. That's called an observable. And it's a sort of overkill concept in this classical situation here. But let's continue. Now, let us suppose that for one reason or another, in classical physics,

we haven't been very careful in doing our classical physics measurements and so forth. And we don't know exactly what state the system is in. All we know is some probabilities. For example, we may have a loaded die. A loaded die, which is an unfair die, like the kind dirty gamblers use.

And you flip it and it lands on the table. There may be a probability distribution for different values of the one, two, three, four, five, or six. Let's label that P sub N also. That's the probability, the a priori probability, let's say, that when you flip the die, you get the Nth state.

Then what is the average if you flip the die many, many times, and each time you measure F, you flip it many, many times, you measure F, what is the average value of F after many, many, many identical experiments?

The answer for the average value of F, or we're assuming, of course, that the sum of the P sub N's is one.

In other words, the sum of the probabilities of one. The average value of F is the sum of all the possibilities of the probability

for the Nth configuration times the value that the function has in the Nth configuration. So what you do is you add up all the configurations weighted according to their probability. Weight them according to their probability. For example, if all the probabilities are equal, then you just add up the F sub N's.

If they're all equal, incidentally, they're not one, they're one over N, because they're one over the number, one over six in this case. But this is the average over many, many, let's take an example, let's do a coin, a simple coin, up or down, heads or tails.

Let's say there's the probability for heads, and we're going to call heads plus. There's the probability for heads, which is P plus, there's the probability for tails, which is P minus.

If we have heads, we assign the number plus one. If we have tails, we assign the number minus one. What is the probability for that observable, which is either plus one or minus one? It's P plus, the probability for plus, minus the probability for minus.

Because every time you get a minus, you give it a value minus one. In other words, every time you get a tail, you give it a minus one. Every time you get a head, you give it a plus one. The average is P plus minus P minus.

The difference between the probability of head and tail is the average of the headsness or the tailsness. So this basic formula here, which is just the most elementary formula of probability of the notion of an average, that you average over all the possibilities, weighing it with the value of the quantity, the observable, whose average you're taking,

and you weigh it according to the probability of that particular configuration. And that's the average value of an experiment which consists of many, many repeated experiments.

What's that? Expected value, sometimes called the expected value, sometimes called the expectation value, sometimes called the average value. And that's what we're going to be interested in. We're going to be interested in the average values of things that we can measure, the average values of observables.

Now, what I haven't told you is how you represent observables in quantum mechanics. And the representation of observables in quantum mechanics is more intricate and complicated than just saying we assign to each state a numerical value that we call the observable value in that state.

It's a more complicated concept and a more tricky concept, the notion of an observable or measurable, measurable quantity. And it's related, not related, but is, well, related, related to the concept of a linear operator or a matrix.

For our purposes, the same thing. Again, the mathematicians will object to me saying a matrix is a linear operator. I don't know. Anybody object to my saying a matrix is a linear operator?

There is a question over there. Can I just, in this example, f of plus is one and f of minus is minus one. The expectation value, though, is zero.

If the probabilities are equal. I said the probabilities are equal. My point is that the expectation value ends up being a number that the underlying system is never in that state. Oh, that's true. Yeah, yeah, right. That's why people don't like calling it the expected value because it is a value which is very definitely not expected.

Everybody used to call it the expectation value. Now, that's a little weaker than saying the expected value. Then my friend Murray Gelman started calling it the expected value.

Well, it ain't, as you point out, the expected value. It's worse than that because you have to divide by, you really have to divide by n. No, the sums of the p's add up to one. Assuming the sums of the p's add up to one. Okay, but when you think of it as dividing by the overall sample size, in this case it's one.

Yeah, that means that p is a half and a half. That means that you end up with the typical person having two and a half kids or whatever. Yeah, right. So it's not really the expected, you know, you have to guess how many kids you're going to have.

You probably wouldn't guess two and a half. It's a payoff. In the gambling you can actually wind up with zero. So it's a value you get, it just doesn't correspond to the point. But it is the average value. It's by definition the average value.

So that's right. I'm glad you pointed that out. That if p plus and p minus were both a half, then the average value would be zero, but zero is not a possible answer. So the average value is not the expected value. There is no expected value in this case.

But it tells you something. You get something like zero in your average value. It tells you that there's a lot of divergence in the sample space. Say it again. It tells you that there's a lot of divergence. Divergence? There's a lot of, the variance is too wide apart.

Well if I think a coin and I flip it, a fair coin and I flip it, I think it will get half times, heads at half times tails. You said something about, did you say something about temperature? Variance. Variance. I mean in the case of coin pulsing of course it tells you that it's a 50-50 percent chance.

Yeah, it tells you you're basically completely ignorant. If you're completely ignorant there's a 50-50 chance and if you do the experiment over and over with a fair coin and you average the result, the average had better be zero because it can't be positive and it can't be negative

because of equal balance. On the other hand, if you have an unfair coin, let's say three quarters of the time it comes up heads and one quarter of the times it comes up tails, then the average is what? It's three quarters minus one quarter which is a half.

All right, so if you have this unfair coin which three quarters of the time comes up heads, one quarter comes up tails, then the average is a half. Again, it doesn't mean that you can measure a half, it just means that that's the average value. Average values are important.

Let me give you one observable, one kind of observable which is special. Supposing I take this special class of observables which is one at one state, F is one for one state over here and zero for all the others.

What's the average value of it? Just the probability, right, just the probability. So if I had an observable which I invented which was one at one place and zero every place else, then there would be only one term in the sum because F is zero for all the others

and for the one case where it's not zero, we just get P. So if you knew the rules for calculating averages, you would also know the rules for calculating, for all possible observables, you would also immediately know the rules for calculating probabilities.

Another way to say it is if you know how to calculate the average value of any observable, you can reconstruct from it the probability distribution, the probabilities for that observable, the probabilities for the different possibilities.

So that raises the question now, what is the mathematical representation of observables? And that is matrices, matrices or linear operators. So let's talk again a little bit about matrices and, where am I, I don't know where I am.

Let's talk about observables or let's talk about matrices first.

A matrix is a thing that you can multiply a vector with or act on a vector with. It does something to a vector. It's an operation on a vector. But it's not just any old operation on a vector. It's a linear operation.

I'm not going to explain that because we won't need to explain it. But whatever it is, call it M, M for matrix. And in the abstract notation, it's a thing, it's an object

which you multiply a vector by and you get a new vector. For example, one simple example is just multiplying by a complex number. Multiplying by a complex number is an operator. It's a very simple operator. You just multiply the vector and you get the number 2.

It just takes every vector and doubles its size. But the general operators that we're going to consider are ones which are represented by matrices. So if we have our vector A, which we represent by a column vector,

and I'm only going to write two dimensional column vectors, but you'll immediately deduce what to do if you have a few more entries here. We can multiply it by a matrix, and a matrix is a square array of numbers M11, M12, M21, M22.

And in general, these numbers are all complex numbers. In general, they're all complex numbers. The As and the Ms in a complex vector space can all be complex numbers. And the rule for matrix multiplication, which we discussed last time,

for multiplying a matrix by a vector, is just, if you want the top entry, which you can think of as the top row, you take the top row and you multiply it by the vector. You take the inner product of the top row with the vector,

which will be M11A1 plus M12A2, and then I'm down here, M21A1 plus M22A2. So it's another vector. I had to draw it pretty wide

because I was adding some numbers here, but it's just a vector. It's a column vector with one column, and it's constructed by taking this row times the column, and this row times the column, and those are the two entries.

That's it. That's all the matrices are. Let me give you a couple of examples, a couple of simple examples, from a very ordinary vector space. A very ordinary vector space is just the pointers or the arrows that you can draw on the blackboard. You can multiply a vector by, in this case, a real number.

You can double it, you can triple it, you can multiply it by minus one, and you can add two vectors. So vectors that you draw on the blackboard are a vector space. Let me give you, and they have, what are their components? Their components are the X component, here's X, here's Y,

and the vector has an X component and a Y component. So we represent this vector by its X component and its Y component, and we could write it in the form X component of vector, Y component of the vector. Of course, now I'm only doing two-dimensional space.

These would be real numbers on the blackboard. They wouldn't be complex numbers, and this would be a two real-dimensional vector space. Now, let's think about some operations. The first interesting operation, which is the most easy, is to stretch the vector space. For example, multiply every vector by two.

Multiply every vector by two will sort of magnify the whole space by a factor of two. Any vector will get stretched out to twice its original length. That's all that happens. Here's the matrix that represents that operation.

It's just the diagonal matrix 2, 2. Diagonal means that its entries are along the diagonal. There are two diagonals in the matrix. This is called the principal diagonal, I think, if I remember. The other one is called the unprincipled diagonal. I don't know what it's called.

The good diagonal and the bad diagonal. This one is the bad diagonal. Let's just calculate what we get. In the upper entry, we get 2 times VX plus 0. The upper entry just becomes 2VX.

The lower entry, we get 2 times VY. As advertised, this is the operator which simply doubles the length of every vector. Very easy. Here's an almost similar thing, but supposing I just wanted to double the Y component.

What would that do to the vector space? It would take every vector and stretch out its Y component by a factor of two without stretching out its X component. So it would be a stretch in the Y direction by a factor of two, but no stretch in the X direction.

How would we represent that? Again, a diagonal with a 1 here in the X place, in the XX, and the Y Y place, the factor 2. Let's check it. 1 times VX gives us VX. 2 times VY gives us twice VY. So that's a stretching of the vector space in the Y direction.

Likewise, we could put the 2 here and the 1 here. That would be a stretching of the vector space which stretches it out in this direction. One or two more. These are very easy examples. So far, they've only involved diagonal matrices.

Let me give you another one. The matrix which corresponds to rotating every vector by 90 degrees. Take any vector and rotate it by 90 degrees. In other words, this is the operation which rotates the plane by 90 degrees.

I'll tell you what it is and then we'll check and see if we can see why. It's off-diagonal, or 1, 1. 1 minus 1, excuse me.

Oh, let's do this one first. Let's do what this is. Before I do the rotation, let's do this. Let's see what this does. This takes 0 times VX, 1 times VY. It interchanges VY and VX. It interchanges VY and VX.

Anybody see geometrically what that corresponds to? A reflection. It's a reflection of the vector space about this diagonal. It takes every vector and reflects it about that diagonal. Just flips it about that diagonal.

That's this matrix. Here's another matrix where you put a minus 1 down here. What does this give? This gives VY minus VX. VY minus VX. It interchanges the X and Y, but then throws in a minus sign for one of them.

That is an operation. I'm going to leave it to you to prove. We don't have time, but I'm going to tell you what this does. It rotates the vector space by 90 degrees. Does it go the other way?

It's an easy way to see all this. It's going to rotate the other direction. It rotates every vector by 90 degrees.

What's that? The Y component. The X component is going into minus. The vector pointing in the X direction is going into minus the vector pointing in the Y direction. It's rotating by 90 degrees. Oh, clockwise.

Rotating clockwise. Yeah, that's what I drew here. Yeah, yeah, yeah. It rotates the vector by 90 degrees. So you see what matrices correspond to is they correspond to transformations of the vector space. Stretchings, rotations, more complicated kinds of things.

Another example would be a shear. This is interesting to try to work out the matrix that corresponds to a shear motion. Let me tell you what a shear motion is. What a shear motion is. A shear motion takes every vector.

Well, it slides stuff this way. So it slides you to the right by an amount proportional to how high you are. It takes every point and shifts it to the right by an amount proportional to how high you are.

There's a matrix that describes that motion that forms the plane, tilts this axis over is what it does. I'm not going to write that one out. I'll see if you can find it the next time I'll tell you what the answer is for a shear motion.

But in general, matrices correspond to transformations, special transformations. Not all transformations are represented as matrices. These are the linear transformations to be specific. But these are the ones we're going to be interested in. And we're going to be interested in a special class of them.

Now this is very abstract and for the moment you will not see why. This is just definition but I've got to give you some mathematics and then intersperse it with some physics. I'm going to give you the mathematics today and next time I'm going to show you what the point is. There's a notion of a Hermitian matrix.

The notion of a Hermitian matrix corresponds to the notion of a real number. It's a kind of concept of reality versus imaginary but it doesn't mean that the entries are all real.

What it means is that if you take an element of the matrix, Mij, that's some element of the matrix. It could be some big matrix. I don't know how big it is. It's somewhere over here.

And then there's Mji. Mji is the reflected matrix element. If this is M12, it'd be over here and then M21 would be over here. M35 might be over here. M53 would be over here.

So interchanging rows and columns is a kind of reflection of the matrix about the diagonal here. A Hermitian matrix is one that if you reflect it, complex conjugates. In other words, Mij is equal to Mji*.

If this matrix was just a number, let's just take a case of numbers. And I tell you I have a number which is equal to its own complex conjugate. What does that tell you about the number? That it's real. That it has no imaginary part.

Now, for a matrix, that's not what it says. It doesn't say that the entries are real, but it's a kind of reality property. Let me give you some examples of matrices. One which is Hermitian. Everybody know how to spell Hermitian?

H-E-R-M-A-S-H-U-N. Hermitian. Hermitian. H-E-R-M-I-T-I-A-N. Hermitian.

Some people write it with an E. Would you like to write anything else for us? Yeah. No, I wouldn't. When you say some people write it with an E, it's got an E.

E-A-N. E-A-N? In England, that's not what it says. Hermine ends with an E. Hermine ends with an E. Hermine is a kind of little bug that eats her. It eats her.

It eats polynomials. Yeah. That's right. It eats polynomials. Okay. In other words, when you flip it, you get the complex conjugate. Here's some examples. Oh, that says one thing. It says the diagonal elements are real.

Because M11, what does it say? It says M11 is M star 11. If you flip the rows and columns on the diagonal, you get the same answer. For example, I equals 1, J equals 1. M11 equals M11 star. So, first of all, it says the diagonal elements are real.

Real numbers here. 7 and 3. But then it says that the off-diagonal elements are complex conjugates of each other. So, here's a matrix, for example, that is not Hermitian. 4 and 2.

That's not Hermitian because 2 is not the complex conjugate of 4. This is Hermitian. 4 is the complex conjugate of 4. So, if a matrix is real and it's Hermitian, it's also symmetric. But here's another one which is Hermitian.

4 plus i and 4 minus i. 4 plus i is the complex conjugate of 4 minus i. So, this is the notion of a Hermitian matrix. And now I'm going to tell you that we're going to find next time, or we're going to postulate next time,

that Hermitian matrices are the quantum version of observables. The classical version of observables, functions of the state points. The quantum version of an observable, a Hermitian operator,

which means a Hermitian matrix for the moment. That's where we're going, yeah. What you'd like to do is you'd like to kick this vector with that matrix, and it's a state of the electron system.

And, well, the sum of the probabilities is 1. Is this enough so that when you kick it or... No, when you kick an electron you want to kick it with a unitary operator, not with a Hermitian operator. Which step? We'll come to it. I'll tell you this. When a unitary operator is one which doesn't change the length of any vector,

which is I think what you were asking about. Right, right. A Hermitian operator does not have that property in general. So a unitary matrix is one which doesn't change the length, and it's the result of an operation that you actually do to the electron. You kick it, you hit it, you do something to it.

I'll tell you very quickly now. Well, no, I won't bother telling you, because we'll get into it next time. Matrices, or more generally linear operators, are the basic quantum concept of the observable, the thing that you can measure. And we're going to then discuss what kind of matrices correspond to the Z component,

the up-down component of the spin. What kind correspond to the X component of the spin? What kind correspond to the Y component? What kind of observable, what kind of matrix, corresponds to the component of the spin along some arbitrary direction?

And so we're going to see then that there's a connection between these vectors, matrices, and so forth, and real directions of space. But that's going to take some time. It'll take a little more pieces. Without all of this, there's no way that you can understand honestly what entanglement is, what Bell's inequalities mean,

what the basic setup of quantum mechanics is. So if I had my way, I would love to be able to teach this without all this mathematics, but I can't. There's no way. You want it honestly, you want it correctly, you want the basic ingredients that go into quantum mechanics,

we have to go through this bit of mathematics. Once we're through it, permission operators, vectors, and so forth, then the rest is kind of smooth sailing. It's relatively easy stuff. But so far, we've had to spend more time on mathematics than I might have liked.

Good. Okay. The preceding program is copyrighted by Stanford University. Please visit us at stanford.edu.