qm2.eng.srt

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Stanford University

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Well, let's begin, I'm gonna review very quickly, I hope quickly 

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We describe a simple system consists of one qubit

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I'm gonna change that terminology a little bit

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I'm gonna start call the qubit a spin

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Don't worry about what it has to do with spinning at the moment 

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Of course, more important is that it's a little pointer

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Now I use the term pointer, 

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I began to use the word pointer to refer the vectors in three-dimensional space

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while I was writing I realize that's awfully awkward unfamiliar terminology

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And I decided it to replace it with another term, the standard term three vectors

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Vectors in three-dimensional space 

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I may sometimes use the term pointer 

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but I think I more likely to speak of three-vectors

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By three-vectors I mean vectors in a real space

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The more general concept when I simply say vector

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more likely I'm speaking about the abstract notion of a vector space 

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and the elements of the vector space

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Just to warn you and keep the language clear

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Now we've talk about this qubit which has now become a spin

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it has some of the character of being a little three-vector in space

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There are some sense or other in which it point into that direction

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And that we can confirm by an experiment 

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As I said before, once we start thinking about quantum mechanics 

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We really do have to think about system and their connection with the apparatus

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Apparati? Is that the word call apparati I'm not sure? Apparatus apparatucee?

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Anyone know the plural of apparatus?

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Or just apparatus? Okay, just apparatus

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We do have to think about the connection with the apparatus

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That measure them

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And we don't get away with just ignoring it

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What's the difference? First of all I tell you 

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what basic difference in classical physics and 

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quantum mechanics and quantum physics

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In classical physics measurement can be arbitrarily gentle

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Arbitrarily gentle means you can find out 

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any thing you want about a system in principle

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by inter reacting with something that changes anything about it 

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by the arbitrarily small amount

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For example you want to locate a location of a particle in classical physics

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How do you do it? You shine some light on it

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You shine some light on it and you make an image

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Of course you have to use light which in small enough wavelength

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That you can resolve the object, this is true

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But there is no limit of how weak a light signal can be at a given wavelength

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You can make the light signal as weak as you like

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So it carries as little energy as you like, as little momentum as you like

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Let it interact with the thing you are interested in

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Form image, and after the image has been form, it can't be magnified

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Not magnified in the sense of making it bigger

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But what's the right word for amplify?

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It can't be amplified 

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So the experiment itself can be arbitrarily gentle

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And still find out anything you might want to know about the system

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The result of that is experiment can always be done

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which have not effect what ever on future experiment 

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On subsequence experiment for example right afterwards 

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That's classical physics

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Quantum physics is not like that, 

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quantum physics is such that experiments inevitably change systems, 

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any experiment to measure something will change something else

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It's almost as though the experiment that measure coin

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was such that every time I look at the coin

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the coin jump up in the air and randomly flip itself

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We know that doesn't happen,

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we can measure the state of the coin gently without changing it

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Quantum mechanics, not so clear, well it is clear

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It is clear that cannot be done

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Okay, let's but we are gonna call it a little bit 

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before we move on to more about vectors spaces and operators

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and mathematics of a quantum theory 

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I want to talk a little bit more about the logic of it

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And in what way logic in quantum mechanics is different 

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than logic in classical physics

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Let's go back to our spin and our apparatus

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The spin is a, we draw a little arrow 

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whether they are truly correspond to anything looks like a classical or not

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is irrelevant, we have to draw it on the blackboard and

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Anything I draw on the blackboard will be classical

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Just because ink and pens are classical or approximately classical

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The apparatus has a window that has a number that comes out of it

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Numbers appears in the window tell result of the experiment

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And it all start to happen to have a direction in it

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which tell you which side is up

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How to orient it, in fact I might have a whole set of three-axes

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to tell you how to orient it

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But that's not important just the up back is what's important 	  

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and when the apparatus is pointing up

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In fact no matter what the direction the apparatus is pointing in

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When interact with the spin, it produces a number in the window

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And that number in the window is either plus one or minus one

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No matter what the apparatus is pointing in 

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No matter what has happen to the spin before it()

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When the apparatus is pointing up we will say that what it has measured

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is the z component of the spin

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Let's pus some axes in here

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Z up, x to the right and y out of blackboard

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And when the apparatus is pointing up, 

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we will say that it produces an experimental number

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Which measures the z component of the spin

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It's not particularly interesting to think of it as having a directionality

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when we say it's the z component of the spin

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Imagining it that the object here has a sense of directionality to it

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How can we test it has a direction to it? A sense of directionality to it?

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Well, we can prepare the spin 

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in the state of what we think as pointing up

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by simply letting it interact with the apparatus

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And when the apparatus appears with a plus one sign in it

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We say, aha, the spin is up

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And if we do it again with the same apparatus pointing at the same direction

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with the same spin, we will get up again

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And up again and up again, in that sense quantum mechanics 

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are out(allows)for reproducible the experiments 

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If you measure the same thing several times in a row

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with the same apparatus

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You will get the same answer over and over

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So things are not that weird 

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That's also important because we have to have some

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concept of reproducibility of experiment to result

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in order to make any sense out of things

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so this is a good thing 

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When you measure something and you measure it again again and again

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you get the same answer, yeah

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>> so how is that (inaudible)

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You are gonna come, we are gonna come to it

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Well I didn't say that experiments are so necessarily so ungentle

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Then when you measure anything you affect everything

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The answer is you affect something 

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In this case when you measure something you don't affect that same thing 

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if you measure the same thing again

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You will get the same answer 

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Okay now you could say that now I have detected the spin is up

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Let me turn the up of the apparatus completely over

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And point down

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If this were a thermometer measuring the temperature

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I don't think we will expect 

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it to detect minus temperature if we turn it over right?

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No with any kind of decent thermometer

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So therefore to say the temperature does not have a sense of orientation

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Certainly not at incident

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On the other hand if there is a true spin

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And we turn the apparatus over 

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If initially the apparatus told us that the spin was plus one 

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after we have turn it over and do the experiment again

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It will register minus one

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So it seem to indicate that there is a sense of directionality

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And of course, 

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the same thing is true no matter what direction we hold the apparatus in

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If we hold the apparatus in the horizontal direction

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And come along and detect a spin we know nothing about		

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We have not yet made a measurement on it

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Maybe somebody else has made a measurement on it

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Maybe some agent available to us who some secret person

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has done something to that spin

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But we don't know, okay we don't know about it

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We come in with our detector and we make a measurement

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What will we measure again we will measure plus one or minus one

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And if we are beginning to think of this thing 

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as having a sense of direction to it

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We might say that we are measuring now is this x component of the spin

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Whereas previously we were measuring the z component of the spin

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Let's give them names

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Sigma z which is the component of the spin along the vertical axis

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At the moment yet I'm not asking you to think this notion of component 

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is the same as the notion of component in the classical physics

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Just the name for the moment 

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And sigma z when measure is plus one or minus one

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The same happens when we turn the apparatus on it's side

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And we measure the same procedure 

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we measure something we can now call sigma x

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And sigma x is always plus or minus one

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And the same thing if we turn the apparatus around 

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So that it's facing front or in fact 

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no matter what direction we orient the apparatus

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It always give us plus or minus one

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This will be a little bit strange more than a little bit strange 

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If thie object really were a classical vector 

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Perhaps a vector of unit length

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If it were a vector of unit length

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And this detector is really measuring the component of spin along it's own axis

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There is no way that it will just give plus one or minus one

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A component of the vector 

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even if the vector is required to have exactly unit length

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Here's a vector with exactly unit length a point,

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a three-vector exactly unit length

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If we turn it and do other things with it

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So the x component and z component can be any number 

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between minus one and one

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So this is a funny vector if it's a vector at all

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But it still have a sense of orientation and 

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the result of the experiment do seem to change when you rotate the detectors

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In particular if you begin by orienting the spin along the z axis

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in the up direction, let's call this up

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That means sigma z is equal to plus one

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And then rotate the apparatus	 

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Detect it again, you won't necessarily get plus one

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In fact you may get plus one, or you may get minus one with equal probability

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You can do the experiment over and over and over again

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Half the time you will get sigma x equals to plus one 

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half the time you will get sigma x equals to minus one

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Even though you know sigma z was exactly equal to plus one

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In which case, so, ah sounds funny?

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But what is true is the average value of sigma x will be one

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Sorry would be zero, 

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if the initial starting point of spin that was up along the z axis

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Same for the y axis

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If the initial value of the spin was prepared to be up along the z axis

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And you rotate the apparatus to the right orientation

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Again you will have equal probability

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So the average value of sigma y would be zero

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You can even go beyond that

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You can say let's start with the apparatus not vertical

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But let's start the apparatus along the x direction 

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Use it to prepare the spin 

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so that it's known to be pointing horizontal along the x direction 

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and sigma x is equal to plus one and do the y z experiment

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Always we will get random but equal probability for up and down

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Or plus or minus one, you go a little bit further

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Of course I'm quoting 

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what will be the result of the experiments on spins

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I haven't done such an experiment I don't think I could 

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But I have no doubt if I have a student to do it,

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this is what will happen

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>> suppose you start out we have not prepare the spin at all

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Doesn't matter, okay go ahead

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>> so it could be up or down or x or y or z

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>> we don't know what it is, right?

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>> We set it up so that the apparatus on the side along the x axis

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okay

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>>and then we do measures

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Yeah, we first do one measure before we do anything else

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Yeah, let me come back for a second, let me come back for a second

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You are right, we won't do a lot of experiments 

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and a lot of measurement in quantum mechanics

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There are two distinct way to do quantum mechanics experiments

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So it really doesn't matter which we do

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We can imagine having a large number of identical spin 

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prepared from the beginning

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however they are prepared 

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And we can do experiments with a large number of them

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Or we can do a repeated experiment on the same spin

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There are logic stand to doesn't matter which we are talking about

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I will tell you when it matters

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But at the moment all we are talking about is one spin

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You said, start the apparatus horizontal

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stick the spin to it and detect it, what you get?

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What you get will either be plus one or minus one

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You can then take the spin out of there

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Give it to your friend who does magic with the spin

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you don't know what it is, you don't know what the hell he does with it

251
00:15:57,251 --> 00:16:00,829
He flips it around juggles it, puts it in his pocket, takes it out

252
00:16:00,868 --> 00:16:03,261
gives it back to you and tells you to do it again

253
00:16:03,876 --> 00:16:05,923
Again, you will get plus one or minus one 

254
00:16:05,979 --> 00:16:08,620
And maybe that he did nothing, 

255
00:16:09,180 --> 00:16:12,230
in which the second time you get the same answer as the first time

256
00:16:13,531 --> 00:16:17,359
Or maybe he turn it over which you will get the opposite answer

257
00:16:17,399 --> 00:16:19,526
But whatever you get, there will be plus one or minus one

258
00:16:21,822 --> 00:16:25,405
then if you do a repeated experiment 

259
00:16:25,469 --> 00:16:28,572
knowing that nothing else was done to the spin in between

260
00:16:28,615 --> 00:16:31,391
You will get the same answer at the same experiment 

261
00:16:31,471 --> 00:16:33,828
Okay, that's

262
00:16:36,411 --> 00:16:38,662
Okay, let's be a little more general, 

263
00:16:38,737 --> 00:16:45,118
let's suppose we orient the axes of the director in any direction

264
00:16:45,158 --> 00:16:47,744
So let's make a notation for the direction

265
00:16:47,860 --> 00:16:49,981
A unit three vector

266
00:16:50,045 --> 00:16:51,655
A unit three vector

267
00:16:53,844 --> 00:16:55,814
Unit vector in three dimensional space 

268
00:16:55,909 --> 00:17:00,523
are always indicated by putting a little hat up of the symbol

269
00:17:00,589 --> 00:17:05,158
So this is a unit vector along a particular axis

270
00:17:05,198 --> 00:17:09,534
We start with the detector along with the axis m

271
00:17:09,591 --> 00:17:14,072
not horizontal not vertical not in or out

272
00:17:14,684 --> 00:17:25,553
But just the cardicorter() in some complicated way

273
00:17:25,593 --> 00:17:31,308
That's m, that's m

274
00:17:35,518 --> 00:17:38,895
We start with the spin, let's start with the spin

275
00:17:39,513 --> 00:17:45,399
Previously having been oriented or prepared along a different axis

276
00:17:46,136 --> 00:17:49,097
The n axis, how do we do that?

277
00:17:49,177 --> 00:17:52,545
Well, I should have gone, I should have gone a little slower

278
00:17:52,585 --> 00:17:56,687
We start the detector along the n axis

279
00:17:56,714 --> 00:18:01,128
Do an experiment and discover that along the n axis

280
00:18:01,208 --> 00:18:04,119
the spin comes up plus one 

281
00:18:04,174 --> 00:18:09,302
Then we rotate the apparatus into the m direction

282
00:18:10,382 --> 00:18:12,945
And we do another experiment on the same spin

283
00:18:14,123 --> 00:18:18,582
It will either come up plus one or minus one along the m axis

284
00:18:18,623 --> 00:18:20,132
Never in between, 

285
00:18:20,798 --> 00:18:29,142
but the average value of the spin along the m axis here will not be one

286
00:18:29,207 --> 00:18:30,881
It will not be minus one

287
00:18:31,007 --> 00:18:35,990
What will it be? It will be the cosine angle between m and n

288
00:18:36,047 --> 00:18:41,535
Another words, the component of m along n

289
00:18:42,651 --> 00:18:44,203
How do we write that, let's write that 

290
00:18:44,683 --> 00:18:46,323
Let's write a formula for that 

291
00:18:47,044 --> 00:18:50,212
The symbol of averages in quantum mechanics 

292
00:18:50,269 --> 00:18:53,286
are parallel angular brackets like this

293
00:18:53,388 --> 00:18:56,805
This one is called the bra, this one is called the ket

294
00:18:56,877 --> 00:18:58,019
And the whole things make a bracket 

295
00:18:59,596 --> 00:19:01,343
which is British for bracket 

296
00:19:08,182 --> 00:19:11,788
Okay the average value of sigma m

297
00:19:12,933 --> 00:19:18,614
Knowing that it's originally oriented along the n direction

298
00:19:21,069 --> 00:19:23,248
And measure along the n direction

299
00:19:24,446 --> 00:19:27,038
There is a cosine of angle between m and n

300
00:19:27,574 --> 00:19:30,401
Which can also be written as n dot m

301
00:19:33,253 --> 00:19:35,094
The ordinary dot products of the vectors

302
00:19:35,174 --> 00:19:39,653
Or just the component of n along m

303
00:19:40,470 --> 00:19:42,650
The spin is oriented along n direction 

304
00:19:42,689 --> 00:19:47,842
you measure along the m direction and the answer is plus or minus one

305
00:19:48,212 --> 00:19:53,833
But on the average it's what a classical physicist might have expected 

306
00:19:54,228 --> 00:19:56,831
if you measure the component of the spin 

307
00:19:56,865 --> 00:19:59,710
which is oriented in one direction along some other direction

308
00:20:00,537 --> 00:20:03,448
This is exactly the component of m along n

309
00:20:05,887 --> 00:20:09,925
If you do the experiment many many times and collect averages

310
00:20:10,809 --> 00:20:16,134
It seems that average quantities do behave like vectors

311
00:20:16,161 --> 00:20:17,913
Or like three-vectors

312
00:20:17,953 --> 00:20:21,241
But the individual experiments are very resistant 

313
00:20:21,281 --> 00:20:23,931
to giving any answers other than plus or minus one

314
00:20:25,714 --> 00:20:27,017
Okay, so this now

315
00:20:27,122 --> 00:20:30,658
>> The apparatus doesn't know how to give the answer (inaudible)

316
00:20:33,421 --> 00:20:35,163
Doesn't know how to get any

317
00:20:35,949 --> 00:20:38,973
You might try to build another apparatus does better

318
00:20:39,049 --> 00:20:39,989
But it won't be better

319
00:20:41,068 --> 00:20:47,765
>> I'm not sure I understand the set up is actually prepare the 

320
00:20:48,651 --> 00:20:50,360
the spin along the n axis

321
00:20:51,910 --> 00:20:53,959
You measure it along the m axis

322
00:20:54,831 --> 00:20:58,746
Then you get a random answer but random-biased 

323
00:20:58,847 --> 00:21:03,573
Biased in such a way that the average is in that

324
00:21:03,653 --> 00:21:07,031
For example what it means when m and n are the same direction?

325
00:21:07,086 --> 00:21:09,069
That you didn't rotate it at all

326
00:21:09,124 --> 00:21:11,131
What is n dot m indicates?

327
00:21:13,433 --> 00:21:17,213
One, right, and that would just be the indication 

328
00:21:17,310 --> 00:21:20,851
that you got the same answer if n was up, 

329
00:21:20,895 --> 00:21:23,506
then you did the experiments to get it up and up and up

330
00:21:23,563 --> 00:21:24,674
Over and over and over again

331
00:21:24,732 --> 00:21:25,892
Okay,

332
00:21:27,980 --> 00:21:32,613
>> What is ... here? Would it be more accurate 

333
00:21:32,653 --> 00:21:37,813
>> say the expected value of the n times m goes to average?

334
00:21:37,847 --> 00:21:43,332
No, no no, my friend I love to say expected value 

335
00:21:43,371 --> 00:21:45,630
It is not the expected value 

336
00:21:48,723 --> 00:21:52,044
If I gave you a probability distribution which look like this

337
00:21:53,459 --> 00:21:58,556
What would be the expected value of it? How?

338
00:21:59,708 --> 00:22:03,246
At least the expected value would be the one right in the middle

339
00:22:03,284 --> 00:22:06,028
But the average value would be the one right in the middle

340
00:22:08,518 --> 00:22:15,835
Expected values and average values are very different in general

341
00:22:16,348 --> 00:22:18,435
And this is the average value, 

342
00:22:18,869 --> 00:22:21,338
the average if you average the quantity 

343
00:22:21,378 --> 00:22:26,818
then you get the standard sense of averaging, standard average

344
00:22:29,068 --> 00:22:34,019
It's also called the expectation value but again that's a misnomer

345
00:22:34,074 --> 00:22:37,438
That's not the value you expect necessarily 

346
00:22:38,596 --> 00:22:40,970
>> So when they try, if I think about this right

347
00:22:41,033 --> 00:22:44,389
>> the spin, the only we got the spin is 

348
00:22:44,445 --> 00:22:47,417
>> when we measure you get plus one or minus one right?

349
00:22:48,603 --> 00:22:50,927
Right, it's all we know until we do a lot of experiments

350
00:22:50,981 --> 00:22:53,704
>> the z component that has to do with 	

351
00:22:54,439 --> 00:22:58,814
>> there is nothing to do with the figure

352
00:23:01,350 --> 00:23:05,171
Well, there are different experiments that you can do on a spin

353
00:23:06,443 --> 00:23:10,804
The result of these experiments can be use

354
00:23:10,868 --> 00:23:13,483
Or the particular experiment you do

355
00:23:14,173 --> 00:23:17,236
Can be used to label what it is you measure

356
00:23:19,430 --> 00:23:23,252
>> guess what, for what point are you doing with the spin

357
00:23:23,756 --> 00:23:26,106
>> antagonly, it's up and down

358
00:23:30,448 --> 00:23:32,928
>> plus one or minus one whatever that means we don't know

359
00:23:32,960 --> 00:23:35,034
>> there is some geometric meanings

360
00:23:35,034 --> 00:23:38,858
Yeah, there is no geometric meanings to begin with

361
00:23:39,634 --> 00:23:42,761
But we did discover that by turning this thing over 

362
00:23:42,801 --> 00:23:44,413
we always get the opposite side

363
00:23:44,488 --> 00:23:47,433
Well, let's not even tell us that there is some difference 

364
00:23:47,459 --> 00:23:49,617
Between the apparatus up and apparatus down

365
00:23:49,657 --> 00:23:52,067
It's not like a thermometer measuring temperature

366
00:23:52,093 --> 00:23:57,447
Some notion there that is different to the up and down

367
00:23:58,693 --> 00:24:02,296
And then if we turn on the side we find out in some sense 

368
00:24:02,313 --> 00:24:05,274
it does behave on an average like a vector

369
00:24:06,064 --> 00:24:07,135
On the average

370
00:24:10,115 --> 00:24:11,826
But it is what it is

371
00:24:11,855 --> 00:24:16,118
>> that vector changes in prepared?

372
00:24:17,191 --> 00:24:20,598
>> it's that when you prepared, you may have 

373
00:24:23,946 --> 00:24:28,506
>> if you got average it one way, and then you prepare the average another way

374
00:24:28,562 --> 00:24:32,122
>> it may have a z component, is that correct?

375
00:24:35,051 --> 00:24:38,073
Yeah, so it is what it is, so I've told you a fact

376
00:24:38,827 --> 00:24:43,946
Facts are if you first prepare by putting your apparatus along n direction

377
00:24:43,986 --> 00:24:47,413
And then measure with the apparatus along the other direction

378
00:24:47,493 --> 00:24:51,413
The average value of the results would be n dot m

379
00:24:51,469 --> 00:24:55,651
But ever single event would be plus one or minus one

380
00:24:55,690 --> 00:24:58,082
Nothing left to say, say it any way

381
00:24:59,601 --> 00:25:04,461
>> if n is in the same direction or opposite direction with m

382
00:25:04,486 --> 00:25:06,757
>> Then it's not just an average, it repeats itself

383
00:25:06,784 --> 00:25:08,749
>> what if you takes the other

384
00:25:15,621 --> 00:25:19,915
But if a number or result can only be plus one or minus one

385
00:25:20,867 --> 00:25:24,621
And the average is plus one, it mean every event is plus one

386
00:25:29,385 --> 00:25:35,263
Okay, let's talk about logic a little bit

387
00:25:38,391 --> 00:25:41,878
A problem of quantum mechanics is that logic 

388
00:25:41,919 --> 00:25:48,536
the logic of a quantum system is different than the logic of a classical system

389
00:25:49,508 --> 00:25:55,257
but our brains are classical systems incapable of thinking in quantum logic

390
00:25:55,298 --> 00:25:59,090
Nevertheless we can get a glimmering of what quantum logic is like

391
00:25:59,116 --> 00:26:01,143
By thinking about this experiments

392
00:26:08,545 --> 00:26:11,896
We can either talk about the calculus of these propositions

393
00:26:11,960 --> 00:26:15,729
Or the nature of the states of a system

394
00:26:15,795 --> 00:26:17,156
Space of states of a system

395
00:26:17,866 --> 00:26:20,033
Let's start with the space of states of a system

396
00:26:20,132 --> 00:26:22,738
Space of states of a system being classically for a moment

397
00:26:23,547 --> 00:26:27,051
They are set, they are set of possible states

398
00:26:27,090 --> 00:26:34,354
Here they are, heads and tails, six five four three two one for the die

399
00:26:36,041 --> 00:26:40,057
Particle on a line would have infinite number of states

400
00:26:40,098 --> 00:26:42,429
But the state form is set as some type

401
00:26:43,441 --> 00:26:49,986
The proposition about the state of a system is basically a subset

402
00:26:50,012 --> 00:26:53,475
It's a subset of states for which the proposition is true

403
00:26:55,059 --> 00:27:01,858
So for the die, my proposition that the die shows a even face

404
00:27:02,986 --> 00:27:07,691
That's a proposition and it's true for half this states of die

405
00:27:07,716 --> 00:27:12,002
And so it correspond to the subset of two four six

406
00:27:14,562 --> 00:27:17,323
The contradictory statement, the NOT statement 

407
00:27:18,445 --> 00:27:21,634
NOT the die shows an even face

408
00:27:21,672 --> 00:27:24,407
That's obviously the equivalent to the statement 

409
00:27:24,432 --> 00:27:25,847
that the die shows an odd face

410
00:27:27,369 --> 00:27:35,168
And so in classical logic, the NOT of the statement 

411
00:27:35,197 --> 00:27:37,625
The NOT means the contradictory of a statement 

412
00:27:37,664 --> 00:27:40,710
If this is the statement mean the same is true 

413
00:27:40,751 --> 00:27:42,548
for the subset of states in here

414
00:27:42,587 --> 00:27:44,964
But the NOT statements is the statement 

415
00:27:45,060 --> 00:27:51,788
is the component of collection of things which are not in there

416
00:27:53,400 --> 00:27:56,135
Okay, let's go to our spin

417
00:27:57,776 --> 00:28:01,033
Proposition, the spin is up along the axis

418
00:28:02,144 --> 00:28:03,576
What's the not of that?

419
00:28:03,616 --> 00:28:07,789
The not of that is if we measure the spin along the z axis

420
00:28:07,816 --> 00:28:09,991
We get something else other than one

421
00:28:10,845 --> 00:28:12,663
The only other possibility is minus one

422
00:28:12,718 --> 00:28:16,238
So the spin equal minus one along the z axis

423
00:28:16,301 --> 00:28:21,861
Is the NOT statement of the spin is up along the axis

424
00:28:21,901 --> 00:28:25,391
So logic make some sense

425
00:28:26,509 --> 00:28:29,209
But we begin to find very peculiar difference 

426
00:28:29,302 --> 00:28:34,776
when we start talking about the operations of combining proposition

427
00:28:35,426 --> 00:28:37,951
to the use of AND and OR

428
00:28:38,072 --> 00:28:42,434
AND and OR are different in quantum mechanics 

429
00:28:42,474 --> 00:28:44,066
than they are in classical mechanics

430
00:28:44,855 --> 00:28:46,143
I wanna lay that out

431
00:28:46,816 --> 00:28:58,024
It's so interesting and so rarely explained that's sort of like to explain it

432
00:29:02,377 --> 00:29:06,360
>> in the classical example like the die

433
00:29:06,417 --> 00:29:11,535
>> the universal set is finite in terms of the elements in it

434
00:29:11,563 --> 00:29:12,604
number of states

435
00:29:12,683 --> 00:29:16,043
>> then you jump to quantum mechanics 

436
00:29:16,069 --> 00:29:20,357
>> and things like that is called plausible time

437
00:29:20,382 --> 00:29:21,844
>> Is that make a difference? 

438
00:29:21,901 --> 00:29:23,031
All the possible of what?

439
00:29:23,107 --> 00:29:25,987
>> All the possible experiments that you could do, not limited

440
00:29:26,044 --> 00:29:29,927
Ya, you are getting at something important which we are going to get to

441
00:29:31,665 --> 00:29:36,445
It's clear that in some sense the number of states of this spin is finite

442
00:29:36,485 --> 00:29:39,898
It's just up or down, if we measure it along the axis

443
00:29:40,717 --> 00:29:45,986
On the other hand it also looks like we can tilt the apparatus in any direction

444
00:29:46,049 --> 00:29:51,222
And prepare the state in any orientation along any direction

445
00:29:51,262 --> 00:29:55,438
So it looks like there is infinite ambiguity in the states of a system

446
00:29:55,496 --> 00:30:00,409
This is what, this is the whole point of trying to understand the state of system

447
00:30:00,464 --> 00:30:02,693
Understand the connection between 

448
00:30:02,770 --> 00:30:06,651
the continuously different directions of electron could point in

449
00:30:06,679 --> 00:30:10,708
and in fact any time we measure it along we are getting axes only up or down

450
00:30:10,787 --> 00:30:11,826
But we will come to it

451
00:30:11,890 --> 00:30:14,314
It's a good question, but we will come to it

452
00:30:15,314 --> 00:30:21,140
Okay, let's talk about the calculus of propositions

453
00:30:21,202 --> 00:30:23,436
I gave you two propositions A and B

454
00:30:24,226 --> 00:30:29,674
I can form two combination of them A OR B and A AND B

455
00:30:29,715 --> 00:30:33,409
Let's just see what does that mean in terms of sets

456
00:30:35,099 --> 00:30:38,099
If we have two proposition this one is A this one is B

457
00:30:39,859 --> 00:30:49,148
Then A AND B means the set of states which belong both to A and to B

458
00:30:50,610 --> 00:30:54,022
So it's the intersection of the set A and the set B

459
00:30:54,050 --> 00:30:58,237
called logically the intersection or set theoretically the intersection

460
00:30:58,275 --> 00:31:01,804
It's basically the AND operation

461
00:31:02,546 --> 00:31:06,421
This point is in A and it's in B

462
00:31:06,484 --> 00:31:08,018
Okay, that's straightforward

463
00:31:09,224 --> 00:31:15,316
And given any two propositions, you can always form

464
00:31:17,660 --> 00:31:23,533
And the, the right way to say it, is this is the point of set here

465
00:31:23,615 --> 00:31:28,374
from which the proposition A and B is true

466
00:31:28,413 --> 00:31:32,829
A AND B is true here and it's true no where else 

467
00:31:34,998 --> 00:31:43,058
What about these two? These two subsets A and B

468
00:31:44,740 --> 00:31:51,450
A AND B is still a proposition, but it's always false 

469
00:31:52,276 --> 00:31:57,649
There is no state for which A AND B is true

470
00:31:58,475 --> 00:32:02,937
Alright, so there are As and Bs that you can put together

471
00:32:03,001 --> 00:32:09,416
I mean, my friend san jade() over here is weight 600 pounds

472
00:32:09,457 --> 00:32:12,740
And my friend uh, let's get another one

473
00:32:15,500 --> 00:32:19,898
There are combinations which A AND B is always false, obviously 

474
00:32:20,379 --> 00:32:25,050
Alright, in particular if you take A and NOT A, it's always false

475
00:32:25,844 --> 00:32:30,829
But still, the definition of A and B is the intersection 

476
00:32:30,910 --> 00:32:35,090
And it's true for all states in the intersection

477
00:32:35,130 --> 00:32:37,516
OR is a little bit more complicated, 

478
00:32:37,548 --> 00:32:39,281
at least there are two versions of OR

479
00:32:39,337 --> 00:32:42,385
There is the exclusive OR or inclusive OR

480
00:32:42,425 --> 00:32:46,268
In the English language when we usually use OR

481
00:32:46,324 --> 00:32:48,803
we usually means the exclusive OR 

482
00:32:48,869 --> 00:32:58,481
When we say that, I'm Jewish or I'm Catholic 

483
00:32:58,538 --> 00:33:03,500
I ordinary mean that I can't be both, alright that I can't be both

484
00:33:03,538 --> 00:33:08,666
So the usual OR is the exclusive OR, the one you normally think about

485
00:33:08,695 --> 00:33:12,324
But in logic the term OR is the inclusive OR 

486
00:33:13,035 --> 00:33:19,472
It means A, it's true if A is true, it's true if B is true, 

487
00:33:19,540 --> 00:33:21,667
it's true when both of them are true

488
00:33:21,761 --> 00:33:28,666
It's the inclusive OR , so I can correctly say

489
00:33:29,442 --> 00:33:35,400
I am a physicist and I am a father, that's true 

490
00:33:35,463 --> 00:33:38,722
But I can also say I am a physicist OR I am a father 

491
00:33:38,780 --> 00:33:42,211
And it's also true, why? Because both of them are true

492
00:33:42,291 --> 00:33:44,276
So that's the inclusive OR

493
00:33:45,056 --> 00:33:49,376
Inclusive OR correspond to the union of two sets

494
00:33:49,416 --> 00:33:51,867
If I have A and I have B, 

495
00:33:52,656 --> 00:33:56,354
then the inclusive OR is everything in both sets

496
00:33:57,668 --> 00:34:01,572
Or either sets, anything - not in both sets

497
00:34:01,598 --> 00:34:04,171
Anything in either sets or in both sets

498
00:34:04,850 --> 00:34:09,896
The exclusive OR with correspond to things in one set

499
00:34:10,336 --> 00:34:12,256
but not the other set

500
00:34:13,062 --> 00:34:18,711
The exclusive OR include, we are not always 

501
00:34:18,775 --> 00:34:23,657
When I was speak of the OR, I always mean the inclusive OR

502
00:34:24,375 --> 00:34:29,542
So that's AND and OR, what else can we say about it

503
00:34:29,572 --> 00:34:30,464
They are not very much

504
00:34:41,864 --> 00:34:47,660
We are now going to use that concept to try to formulate a set of AND and OR

505
00:34:47,688 --> 00:34:49,260
and statements about a spin

506
00:34:49,316 --> 00:34:53,569
And then ask how would we verify it, how would we verify them

507
00:34:53,633 --> 00:34:58,256
How would we verify them, what does it mean when we verify them

508
00:34:59,276 --> 00:35:02,988
Okay so let's take two proposition A and B 

509
00:35:03,044 --> 00:35:15,329
to be sigma z is equal to plus one, and sigma x

510
00:35:15,369 --> 00:35:21,300
This is A and this is B, sigma x is equal to plus one

511
00:35:22,156 --> 00:35:26,139
They are both propositions, we will know how check either one of them

512
00:35:26,180 --> 00:35:32,026
If we want to check sigma A, we will turn our apparatus vertically upward

513
00:35:32,082 --> 00:35:34,757
And we measure the spin, we will check it true or false 

514
00:35:35,601 --> 00:35:38,178
If we wanted to find out if B is true of false

515
00:35:38,249 --> 00:35:40,290
We will turn the apparatus on it's side 

516
00:35:41,514 --> 00:35:45,617
In classical logic, if there are two propositions

517
00:35:45,684 --> 00:35:49,306
you can form a combination A OR B

518
00:35:49,386 --> 00:35:52,690
Let's start with A OR B, so we will take A OR B

519
00:35:57,859 --> 00:36:00,209
And let's think about checking it's true value 

520
00:36:00,254 --> 00:36:01,708
Is it true or is it not?

521
00:36:01,748 --> 00:36:07,137
Alright who design an experiment and try oh,let's circle back the steps

522
00:36:07,177 --> 00:36:13,844
Let's imagine a hidden ghost in the world has previously unknown to us

523
00:36:13,884 --> 00:36:20,214
Unknown to us, has originally oriented the spin upward along the z axis

524
00:36:23,202 --> 00:36:24,389
First we know, we know nothing, 

525
00:36:24,447 --> 00:36:29,750
but the ghost in the world has oriented the spin vertically upward

526
00:36:29,816 --> 00:36:33,089
Just, it doesn't matter, but we could start that way

527
00:36:33,130 --> 00:36:37,169
So the spin is upward, and let's check that if A OR B is true

528
00:36:37,209 --> 00:36:42,513
How do we begin? Let's begin by measuring the spin along the z axis

529
00:36:42,578 --> 00:36:47,283
We put our detector or our apparatus in vertical direction

530
00:36:48,174 --> 00:36:49,557
And we measure it, what do we get?

531
00:36:51,285 --> 00:36:53,342
We get plus one, we got plus one 

532
00:36:53,398 --> 00:36:56,621
because the ghost in the world has made them to begin with

533
00:36:56,651 --> 00:37:00,174
We don't have to go any further, A or B is true

534
00:37:00,878 --> 00:37:04,820
A or B include the possibility that both A AND B is true

535
00:37:04,861 --> 00:37:08,392
But all requires to be true is that one of them is true

536
00:37:09,041 --> 00:37:11,416
And we've just discovered that A is true

537
00:37:12,110 --> 00:37:18,880
So A or B is definitely true, so let's do the experiments in another way

538
00:37:18,909 --> 00:37:24,241
in order which would not matter if we are doing classical physics

539
00:37:24,303 --> 00:37:27,638
And the reason it doesn't matter when we do classical physics 

540
00:37:27,678 --> 00:37:31,304
It's when we do an experiment that has no necessary affect 

541
00:37:31,321 --> 00:37:32,973
on the outcome of latter experiments

542
00:37:33,008 --> 00:37:34,539
The experiment is gentle enough

543
00:37:34,619 --> 00:37:38,523
So let's do the experiment on the opposite order

544
00:37:39,225 --> 00:37:43,964
Instead of first finding out wether A is true

545
00:37:44,044 --> 00:37:46,336
Let's first ask if B is true

546
00:37:46,872 --> 00:37:49,251
And then we will come back and ask if A is true

547
00:37:49,658 --> 00:37:53,281
Alright, secretly the spin is pointing up at the z direction

548
00:37:53,314 --> 00:37:58,341
So the ghost knows that, but we don't

549
00:37:59,050 --> 00:38:01,693
So we come along, knowing nothing about the spin

550
00:38:01,723 --> 00:38:05,178
And why should I measure it? Why should I measure A first?

551
00:38:05,207 --> 00:38:06,468
Let's first measure B 

552
00:38:07,428 --> 00:38:10,113
Let's first measure the x component of the spin

553
00:38:10,154 --> 00:38:11,067
What do we get?

554
00:38:12,507 --> 00:38:14,659
We get either plus one or minus one

555
00:38:15,755 --> 00:38:19,706
We get plus one or minus one with a fifty percent probability

556
00:38:20,466 --> 00:38:25,330
So there is a fifty percent probability that B is plus one

557
00:38:26,099 --> 00:38:31,187
But if B is not plus one, it doesn't mean A or B is wrong

558
00:38:32,148 --> 00:38:34,522
We still have a chance that when we measure A

559
00:38:34,811 --> 00:38:35,886
It will be plus one 

560
00:38:37,131 --> 00:38:41,775
Okay, the problem is in measuring B first time around

561
00:38:44,166 --> 00:38:48,182
We orient, we prepare B along the x axis	 

562
00:38:48,198 --> 00:38:52,192
Whatever we can, whether it is this way or this way

563
00:38:54,245 --> 00:38:58,124
We have prepared that spin along the x axis 

564
00:38:58,933 --> 00:39:02,629
What would that experiment give when we measure along z axis?

565
00:39:04,566 --> 00:39:09,149
Fifty percent probability of being up and fifty percent probability of down

566
00:39:09,852 --> 00:39:17,048
That means we have a 25% probability of finding both 

567
00:39:17,563 --> 00:39:24,185
that sigma x was equal to minus one and sigma z is equal to minus one

568
00:39:24,904 --> 00:39:28,554
We have a 50% probability in the first experiment 

569
00:39:29,365 --> 00:39:31,859
that sigma x is not equal to one

570
00:39:32,529 --> 00:39:33,816
And then in the next experiment 

571
00:39:33,859 --> 00:39:37,395
we have another 50% probability on top of that

572
00:39:37,436 --> 00:39:40,235
That sigma z was not equal to one

573
00:39:40,275 --> 00:39:44,220
We have a net probability of one quarter 

574
00:39:44,260 --> 00:39:47,178
that both A and B are false

575
00:39:47,890 --> 00:39:50,953
When we did the experiment in one order

576
00:39:51,016 --> 00:39:53,483
When we do the experiment in the other order

577
00:39:53,523 --> 00:39:56,212
It is certain that A or B is right, right?

578
00:39:56,268 --> 00:39:57,804
Because A is right

579
00:39:59,545 --> 00:40:03,585
So whether it's true in classical physics

580
00:40:03,641 --> 00:40:07,825
That the idea or A and B is symmetrical between A and B

581
00:40:08,577 --> 00:40:13,937
The idea or the OR statement is A or B

582
00:40:13,977 --> 00:40:19,796
If A or B is true, then B or A is true

583
00:40:19,824 --> 00:40:22,300
And we find that in quantum mechanics 

584
00:40:23,305 --> 00:40:28,746
The concept of A or B at best can only be defined 

585
00:40:28,786 --> 00:40:33,010
in a way that the order should determine A or B

586
00:40:35,120 --> 00:40:38,548
>> the propositions are incompletely specified

587
00:40:38,599 --> 00:40:40,017
>> in the quantum mechanical case, 

588
00:40:40,098 --> 00:40:41,714
>> because you have to give order right?

589
00:40:41,754 --> 00:40:45,700
>> It doesn't like the particle has certain spin

590
00:40:45,740 --> 00:40:49,433
>> which said that's right

591
00:40:49,464 --> 00:40:58,683
Ya, that would, that would also say that any attempt to

592
00:40:59,967 --> 00:41:04,799
think about the logics and experiments are different in quantum mechanics

593
00:41:09,418 --> 00:41:16,287
>> in other words you measure A and B and certain

594
00:41:16,589 --> 00:41:18,271
No, no, no, no, no, let's go back

595
00:41:18,989 --> 00:41:20,512
I said the ghost in the world 

596
00:41:21,269 --> 00:41:24,870
makes the spin pointing up along the z axis

597
00:41:25,527 --> 00:41:28,117
That's, we don't know that, but that's true

598
00:41:31,332 --> 00:41:33,675
Then if we measure sigma z first 

599
00:41:34,342 --> 00:41:37,732
Under the circumstances, we will certainly find that A or B is true

600
00:41:39,183 --> 00:41:42,471
But we measure under the same circumstances

601
00:41:42,991 --> 00:41:45,777
The same degree of lack of presence of knowledge

602
00:41:46,615 --> 00:41:50,984
We measured B first, we find out that there is 25% chance that

603
00:41:51,021 --> 00:41:55,581
A or B, this composite statement here is false

604
00:41:56,224 --> 00:41:59,938
>> So I have a question, the operation A or B result is now going to be 

605
00:42:00,042 --> 00:42:02,264
>> a possibility of 25%

606
00:42:03,563 --> 00:42:10,098
Well, not only that, but you have to specify which order you add A and B

607
00:42:10,138 --> 00:42:15,040
So, alright, that's in itself an elementary something new that 

608
00:42:15,096 --> 00:42:22,203
the OR statement does not unify the statement that were in symmetrical way

609
00:42:23,059 --> 00:42:31,771
>> Question? I confused when we, the thing we are prepared to it

610
00:42:36,179 --> 00:42:38,828
that doesn't matter, just to simplify the story,

611
00:42:38,892 --> 00:42:40,346
It wouldn't matter

612
00:42:40,427 --> 00:42:44,018
>> but we enter the experiment with new B that's 

613
00:42:44,058 --> 00:42:46,378
>> (inaudible) the apparatus on the side

614
00:42:46,418 --> 00:42:49,922
>> and I thought what you are saying is

615
00:42:49,962 --> 00:42:53,132
>> now do this multiple time would get an average

616
00:42:53,169 --> 00:42:55,914
Yeah, but we are not doing it, we are trying to find out

617
00:42:55,995 --> 00:42:58,451
>> but you are saying, what you say is

618
00:42:58,509 --> 00:43:03,794
>> when we do that measurement on the side we are preparing the spin again?

619
00:43:03,851 --> 00:43:08,212
>> and that would say that from then on we should always get plus one

620
00:43:09,323 --> 00:43:11,899
Plus one for sigma x, but then if we go for sigma z

621
00:43:11,939 --> 00:43:16,771
we would get an random answer, measuring one and measuring

622
00:43:16,797 --> 00:43:21,142
>> we are gonna get a random answer for the first inspection?

623
00:43:25,461 --> 00:43:30,789
Think about it, I can't stop the class for

624
00:43:32,348 --> 00:43:36,129
>> Question, if you have a measurement device (Ó¡¶ÈÈË£¿)

625
00:43:36,211 --> 00:43:40,650
>> to measure two different particles

626
00:43:43,599 --> 00:43:45,834
Why have you jumped to two particles?

627
00:43:46,699 --> 00:43:52,026
>> I've been wondering the measurement does it spin vectors

628
00:43:52,830 --> 00:43:55,204
What? I don't know

629
00:43:56,901 --> 00:44:05,503
Okay, that was all review, the next thing which I was going to review

630
00:44:05,543 --> 00:44:09,679
But I'm not going to because it take too long, it's vector spaces

631
00:44:10,751 --> 00:44:14,956
I will just remind you real quickly, the vector space are such elements

632
00:44:15,020 --> 00:44:20,989
You can add elements, this is not something you would do with sets

633
00:44:21,029 --> 00:44:25,373
You can add two sets of course to make a bigger set

634
00:44:25,414 --> 00:44:32,211
But there is no natural single element which is the sum of two elements

635
00:44:34,597 --> 00:44:41,058
So the elements of the set, namely the basic individual objects are not

636
00:44:41,099 --> 00:44:43,179
things that you can add and make new elements

637
00:44:46,514 --> 00:44:51,227
The space of state of a quantum system is a linear vector space

638
00:44:51,266 --> 00:44:56,810
and that means given any two states you can add them in some sense

639
00:44:56,850 --> 00:45:00,539
and make another state, that's what we are going to explore

640
00:45:02,201 --> 00:45:06,443
This logic, this peculiar logic, this peculiar distortion of logic

641
00:45:06,499 --> 00:45:09,107
turns out to be the logic of vector spaces

642
00:45:09,171 --> 00:45:12,813
So logic which is embedded into structure and spaces

643
00:45:12,877 --> 00:45:15,540
in a very neat and tidy way

644
00:45:15,581 --> 00:45:18,645
And that's ultimately why vector space is important

645
00:45:19,728 --> 00:45:24,049
Okay, so let's remember the vector of things that you can add 

646
00:45:27,209 --> 00:45:31,203
If there are two vectors A and B, you can add them

647
00:45:31,283 --> 00:45:34,225
You can multiply them by complex numbers

648
00:45:36,930 --> 00:45:42,274
Every vector has a dual vector, which is kind of its complex conjugate

649
00:45:43,346 --> 00:45:49,002
If we call this object ket vectors because they are the second half of a bracket

650
00:45:49,970 --> 00:45:56,126
Then the ket of bra vector which are in one to one correspondent

651
00:45:56,153 --> 00:46:01,715
which are for every ket vector there are a bra vector

652
00:46:01,756 --> 00:46:05,234
But it starts as if that it's a complex conjugate

653
00:46:05,276 --> 00:46:07,977
So there are two dual vector spaces

654
00:46:09,212 --> 00:46:11,578
And you can take in the product of a vector

655
00:46:11,606 --> 00:46:14,058
A bra vector and a ket vector

656
00:46:14,827 --> 00:46:19,180
So the bra vector and the ket vector can be combined together

657
00:46:19,245 --> 00:46:23,355
It gives a complex number, the inner product as a complex number

658
00:46:26,052 --> 00:46:31,393
And it satisfy there's equal to the complex conjugate 

659
00:46:31,433 --> 00:46:36,738
The complex conjugate of the opposite order

660
00:46:38,001 --> 00:46:42,074
So if you take a complex vector and a complex vector space

661
00:46:42,129 --> 00:46:47,179
You take the inner product interchange the two vectors

662
00:46:47,249 --> 00:46:50,276
Then it's equivalent to it's complex conjugation

663
00:46:50,332 --> 00:46:52,563
That's pretty much what you need to know

664
00:46:53,957 --> 00:46:57,879
That allows you to define given the idea of a inner product

665
00:46:57,930 --> 00:47:01,733
That allows you to define the notion of a length of the vector

666
00:47:01,773 --> 00:47:05,109
We don't usually think of the length of the vector

667
00:47:05,160 --> 00:47:07,122
We usually think of the square of the length of the vector

668
00:47:07,162 --> 00:47:09,429
And the square of the length of the vector

669
00:47:09,511 --> 00:47:15,462
Let's say the letter is A, is the inner product of it's own conjugate

670
00:47:15,502 --> 00:47:20,788
Or it's own dual, usually we say it's the inner product with itself

671
00:47:20,844 --> 00:47:23,262
But that's slightly abuse of terminology

672
00:47:23,852 --> 00:47:25,075
And this is always real

673
00:47:25,692 --> 00:47:31,499
The reason it's always real because if we interchange A and A 

674
00:47:31,579 --> 00:47:33,172
We just get back to the same thing

675
00:47:33,252 --> 00:47:38,213
But this tells us the inner product of A and A is it's own complex conjugate

676
00:47:38,276 --> 00:47:46,163
It's own complex conjugate, it's real, it's also positive, that's easy to prove

677
00:47:48,068 --> 00:47:50,973
And it can be thought of the square root of the length of the vector

678
00:47:51,965 --> 00:47:56,604
It's analogous to, closely analogous to a dot product 

679
00:47:56,685 --> 00:47:58,629
with a three-vector with itself 

680
00:47:59,338 --> 00:48:03,972
If A and B were three-vectors then we wouldn't distinguish the bra and the ket

681
00:48:04,827 --> 00:48:11,308
But we would write that A dot B is the dot product between the two vectors

682
00:48:11,388 --> 00:48:12,345
Analogous to this

683
00:48:13,098 --> 00:48:18,989
And we would also write A dot A was the square of the length of A 

684
00:48:20,789 --> 00:48:23,190
The inner product of a vector with itself 

685
00:48:23,253 --> 00:48:25,386
can be thought of as square of the length of the vector

686
00:48:25,972 --> 00:48:29,380
And finally there is the notion of orthogonality 

687
00:48:30,218 --> 00:48:33,692
Two vectors are orthogonal when their inner product is zero

688
00:48:49,034 --> 00:48:52,378
Same definition for dot product

689
00:48:52,442 --> 00:48:55,533
When the dot product between two vectors is zero

690
00:48:55,611 --> 00:48:57,003
Those vectors are orthogonal 

691
00:48:57,874 --> 00:49:02,166
Those are the definition that we need for a vector space

692
00:49:03,414 --> 00:49:08,796
The other definition is the definition of the dimensionality of the vector space

693
00:49:08,835 --> 00:49:11,652
How many dimensions does the vector space have?

694
00:49:12,482 --> 00:49:13,941
And the way to decide that is 

695
00:49:14,246 --> 00:49:18,226
by asking the maximum numbers of orthogonal vectors

696
00:49:18,268 --> 00:49:19,993
You can find in the vector space

697
00:49:20,036 --> 00:49:26,458
For ordinary vectors, let's say in two three four five dimensions

698
00:49:26,515 --> 00:49:27,947
this is obvious

699
00:49:28,906 --> 00:49:33,773
In two dimensions I can take a vector, just pick any vector 

700
00:49:34,293 --> 00:49:38,045
make a unit vector just by shrinking it's length to unit length

701
00:49:38,145 --> 00:49:42,333
Then I look around and see if we can find another orthogonal vector to it

702
00:49:43,105 --> 00:49:45,981
Yeah, in fact I can find two, one point to this way one point to that way

703
00:49:46,082 --> 00:49:51,291
But that's not a important distinction, we find one other direction

704
00:49:51,697 --> 00:49:56,401
Or one other vector that apart from the numerical multiple

705
00:49:56,502 --> 00:49:58,489
Apart from the numerical multiple 

706
00:49:58,605 --> 00:50:00,793
there is one other vector which is orthogonal to it

707
00:50:00,894 --> 00:50:03,345
So there are two mutually, 

708
00:50:03,446 --> 00:50:07,209
the maximal number of mutually orthogonal vector you can find is two

709
00:50:08,161 --> 00:50:11,217
If you go to three direction you can find a third one, but no more

710
00:50:11,318 --> 00:50:14,017
If you find a fourth one, you are out of directions

711
00:50:14,882 --> 00:50:18,407
So three-dimensional space has three orthogonal vectors

712
00:50:19,143 --> 00:50:21,727
Four dimensional has four and so forth

713
00:50:21,828 --> 00:50:24,799
And same thing is true for complex vector spaces

714
00:50:25,735 --> 00:50:29,638
We describe the version of complex vector spaces

715
00:50:29,738 --> 00:50:35,785
Which is just column vector, alpha one alpha two alpha three...

716
00:50:36,769 --> 00:50:39,521
And their dual, which is a row vector

717
00:50:40,236 --> 00:50:43,969
Which would make up out of the complex conjugate of the variable

718
00:50:46,609 --> 00:50:51,970
So once we have a vector, we can start looking for vectors orthogonal to it

719
00:50:52,071 --> 00:50:54,994
Vectors which are perpendicular in this sense

720
00:50:58,857 --> 00:51:00,828
Find the other one that look like third one

721
00:51:00,928 --> 00:51:05,444
look for the fourth one, however many you find

722
00:51:05,545 --> 00:51:07,676
You will eventually run out of direction

723
00:51:07,776 --> 00:51:09,308
That's the dimensionality of space

724
00:51:09,751 --> 00:51:14,072
To a special case in which there are only two entries here

725
00:51:15,407 --> 00:51:17,551
It's a two dimensional vectors space 

726
00:51:17,651 --> 00:51:20,975
you can only find two mutually orthogonal vectors

727
00:51:22,183 --> 00:51:25,191
You can find many pairs of mutually orthogonal vectors  

728
00:51:25,291 --> 00:51:28,856
In real space, they are just these, these, these

729
00:51:29,752 --> 00:51:31,040
Pointing in different directions

730
00:51:31,156 --> 00:51:33,488
But once you find two there won't be any other one

731
00:51:34,610 --> 00:51:38,568
Okay, so we have the idea of mutually orthogonal vectors

732
00:51:39,570 --> 00:51:43,511
Now we have to state what is the connection between the vector space

733
00:51:43,611 --> 00:51:47,391
and the space of possible states of a quantum system

734
00:51:51,185 --> 00:51:52,921
The quantum system that we are gonna deal with 

735
00:51:53,021 --> 00:51:56,425
is still very primitive and simple, a single spin

736
00:51:56,527 --> 00:52:00,289
Until we get the hand on that, we won't move to more complicated things

737
00:52:02,577 --> 00:52:04,281
>> Is that in the illustration that 

738
00:52:05,633 --> 00:52:09,465
>> the orthogonal vectors have to be perpendicular to each other  

739
00:52:10,065 --> 00:52:11,369
That's what orthogonal means

740
00:52:12,121 --> 00:52:16,881
>> so suppose you have a qubit that tilt to a hologram

741
00:52:16,982 --> 00:52:21,569
>> it's still three-dimensional but the system is not perpendicular

742
00:52:22,265 --> 00:52:25,833
Perpendicular and orthogonal are synonymic terms

743
00:52:28,473 --> 00:52:31,457
Perpendicular and orthogonal both mean right angle

744
00:52:34,281 --> 00:52:36,969
You are confused by something but I don't know what it is 

745
00:52:39,979 --> 00:52:45,409
Ya, ya, I know you are, but

746
00:52:47,497 --> 00:52:53,801
Perpendicular orthogonal, right angles are, they mean all the same thing

747
00:52:54,569 --> 00:52:58,433
>> I think when you say linearly independent it's orthogonal

748
00:52:58,537 --> 00:53:02,553
>> but if you (inaudible)

749
00:53:03,177 --> 00:53:08,281
That's true, let's not confuse it, this is right

750
00:53:09,331 --> 00:53:12,553
Linearly independent that is different than orthogonal, 

751
00:53:12,654 --> 00:53:15,081
orthogonal is linearly independent, 

752
00:53:15,186 --> 00:53:18,049
but linearly independent is less strong than orthogonal

753
00:53:18,854 --> 00:53:22,134
These two vectors like that they are linearly independent

754
00:53:22,238 --> 00:53:28,218
But they are not orthogonal ,both orthgonal, okay, let's come back

755
00:53:30,650 --> 00:53:36,362
The state of a quantum system is a postulate forming a vector space

756
00:53:36,986 --> 00:53:39,074
Form a complex vector space 

757
00:53:39,175 --> 00:53:43,095
In fact the space of a symble system we are discussing 

758
00:53:43,196 --> 00:53:45,599
form a two dimensional vector space

759
00:53:48,695 --> 00:53:50,810
Let's write down some vectors, we know some states

760
00:53:50,910 --> 00:53:55,015
We know that for example that is a state that can be prepared 

761
00:53:55,116 --> 00:54:00,218
in which the spin is pointing upward or downward along the z axis

762
00:54:01,647 --> 00:54:07,287
We characterize the state by the experiment that we did together in that state

763
00:54:08,951 --> 00:54:15,471
That experiment as the apparatus vertical upward

764
00:54:16,127 --> 00:54:20,743
and it created a state of the system which either could have been down or up

765
00:54:20,843 --> 00:54:22,519
But let's say it's up

766
00:54:23,665 --> 00:54:25,926
We can label that state in a number of ways

767
00:54:26,027 --> 00:54:27,470
We can call it up, we can call it down

768
00:54:27,571 --> 00:54:32,558
We can call it plus... up and down is the same as plus and minus 

769
00:54:32,672 --> 00:54:35,286
We can draw them with an arrow that is up and down

770
00:54:35,387 --> 00:54:40,670
What other ways could we, ah we can write it as sigma z equal to plus one

771
00:54:40,771 --> 00:54:42,870
And sigma z equal to minus one

772
00:54:42,971 --> 00:54:47,990
They all mean the same thing, up down, plus minus

773
00:54:48,091 --> 00:54:51,126
Sigma z equals plus one, sigma z equals minus one

774
00:54:51,227 --> 00:54:53,582
All the same thing, alright?

775
00:54:53,683 --> 00:54:57,902
So there are two states that we know about that can access

776
00:54:59,913 --> 00:55:04,928
Or we can prepare it by turning the apparatus upward along the z axis

777
00:55:05,007 --> 00:55:09,692
Let's give them names, these state of vectors in vector space 

778
00:55:09,862 --> 00:55:15,677
according to hypothesis and it can be called, let's say up and down 

779
00:55:23,014 --> 00:55:28,414
If this is true the vector space is simple, just two-dimensional vector space

780
00:55:29,131 --> 00:55:32,358
And that means that any vector in that space 

781
00:55:32,500 --> 00:55:36,446
can be written as some linear combination of up and down

782
00:55:36,580 --> 00:55:41,086
Another words, every vector in two-dimensional space 

783
00:55:41,239 --> 00:55:46,333
can be written in terms of two basic vectors up and down

784
00:55:48,149 --> 00:55:52,381
So enough to write any state whatever, up and down

785
00:55:54,046 --> 00:56:02,622
Okay, so what about those states by tilting our apparatus along the x axis

786
00:56:04,118 --> 00:56:06,590
We get something different; it's not the same thing

787
00:56:06,722 --> 00:56:11,038
It's not up, because up means up along the z axis

788
00:56:11,158 --> 00:56:14,686
It's not down, because down means down along the z axis

789
00:56:14,811 --> 00:56:16,870
And we know they are different, 

790
00:56:16,989 --> 00:56:19,582
because we know the result of the experiment 

791
00:56:19,718 --> 00:56:22,670
on these two different kinds of states will generally be different

792
00:56:23,758 --> 00:56:29,726
So we can call those states technically (?) left and right 

793
00:56:29,876 --> 00:56:33,054
It's up and down, it's right and left

794
00:56:34,294 --> 00:56:37,974
There are two other states, we can draw them this way if you like

795
00:56:38,087 --> 00:56:44,463
It also correspond to sigma x equals one and sigma x equals minus one

796
00:56:44,625 --> 00:56:53,527
And we could label them if you like, label them right and left

797
00:56:53,659 --> 00:56:58,708
Finally we know that if we could turn on apparatus 

798
00:56:58,848 --> 00:57:03,953
so the apparatus was pointing along the y axis

799
00:57:04,069 --> 00:57:10,706
We can access to more states, I think we can call those in or out 

800
00:57:10,845 --> 00:57:12,907
Into the blackboard and out of the blackboard

801
00:57:13,031 --> 00:57:21,522
So there is in and out, into the blackboard and out of the blackboard

802
00:57:21,522 --> 00:57:25,742
And they correspond to sigma y equals plus or minus one

803
00:57:25,742 --> 00:57:30,967
We could label them in and out

804
00:57:33,615 --> 00:57:38,752
All of the logic that I describe to you and all of the experimental results

805
00:57:38,894 --> 00:57:43,286
As we will see, we will not see it immediately

806
00:57:43,450 --> 00:57:46,195
Took some time to understand all of these

807
00:57:46,351 --> 00:57:51,283
It is all summarized by a few statements 

808
00:57:51,412 --> 00:57:55,667
but the first of which is that space of all states 

809
00:57:55,806 --> 00:57:58,662
of the simple systems is two-dimensional 

810
00:57:58,798 --> 00:58:04,795
has only two independent states which means that these other things

811
00:58:04,927 --> 00:58:07,189
can be written in terms of up and down

812
00:58:08,645 --> 00:58:14,320
So let's go to the general linear combination of up and down 

813
00:58:14,464 --> 00:58:15,743
and ask what they mean

814
00:58:15,876 --> 00:58:21,327
Let's take a vector, I guess we can call it a A

815
00:58:21,458 --> 00:58:26,927
which is a form, the coefficient, the complex coefficient alpha one

816
00:58:28,250 --> 00:58:30,960
times up plus another complex 

817
00:58:31,099 --> 00:58:38,557
I suppose we should call this alpha up, alpha down times down

818
00:58:38,697 --> 00:58:46,033
The alphas are complex numbers, the complex vector space we multiply

819
00:58:46,149 --> 00:58:49,726
It could be real numbers, but in general they will be complex numbers

820
00:58:49,858 --> 00:58:55,260
Let us suppose we knew what that meant

821
00:58:55,424 --> 00:58:58,237
What kind of state is that, where it came from

822
00:58:58,390 --> 00:59:03,166
Some new beasts, it's not up, it's not down somewhere in between

823
00:59:05,535 --> 00:59:10,848
Next postulate, the next postulate is that if we were to measure up and down

824
00:59:10,979 --> 00:59:14,559
If we were, if somebody prepare us with a electron in this state

825
00:59:14,729 --> 00:59:19,967
And we measure up and down, we will find probability up and down 

826
00:59:20,099 --> 00:59:24,863
And those probabilities would be probability for up, 

827
00:59:30,415 --> 00:59:35,679
It's the square of the magnitude of these coefficient 

828
00:59:35,797 --> 00:59:41,792
And the probability for down is equal to alpha star down and alpha down

829
00:59:41,925 --> 00:59:46,368
Another words, it's basically the square of the coefficient

830
00:59:46,459 --> 00:59:54,049
Or the square of the component of A in the direction up and down

831
00:59:54,159 --> 00:59:55,919
It's alpha, 

832
00:59:56,049 --> 01:00:00,549
it's essentially the components of the general vector along the two axis

833
01:00:00,699 --> 01:00:02,455
Up and down

834
01:00:06,262 --> 01:00:09,034
There is something that I failed to tell you

835
01:00:09,160 --> 01:00:11,074
It's also a postulate, there are many many postulates 

836
01:00:11,210 --> 01:00:13,451
But you can see they fit together in a neat package

837
01:00:13,590 --> 01:00:18,027
There are probably a few postulates which I'm saying 

838
01:00:22,430 --> 01:00:24,556
I bet we can reduce it down to four postulates, four axioms

839
01:00:24,679 --> 01:00:27,660
But rather than try an axiom and applies it

840
01:00:31,636 --> 01:00:33,140
Do a little more informally

841
01:00:33,269 --> 01:00:37,968
What's the relation between up and down?

842
01:00:39,620 --> 01:00:42,972
are they negative to each other?

843
01:00:43,112 --> 01:00:47,080
No, they are orthogonal, now that's a postulate

844
01:00:47,220 --> 01:00:52,064
The basic postulate is that the thing is measurably different 

845
01:00:52,185 --> 01:00:57,193
the two states are such you can cleanly tell the difference 

846
01:00:57,327 --> 01:00:59,159
between them by experiment

847
01:00:59,159 --> 01:01:00,558
They are orthogonal, 

848
01:01:00,686 --> 01:01:07,423
orthogonality literally means they are sufficiently different

849
01:01:07,565 --> 01:01:12,585
That you can, with a single experiment or probably or maybe a few experiment

850
01:01:12,709 --> 01:01:14,035
to tell the difference between them uniquely

851
01:01:14,185 --> 01:01:18,723
I will tell you what it correspond to a set

852
01:01:21,348 --> 01:01:26,574
You might say that two sets are orthogonal if they don't share any elements

853
01:01:26,684 --> 01:01:29,734
This is not a standard terminology, but you might say it anyway

854
01:01:29,867 --> 01:01:31,992
They are orthogonal they do not share any elements

855
01:01:32,806 --> 01:01:34,839
That mean they are distinctively different

856
01:01:34,970 --> 01:01:38,552
There is no chance that an experiment would confuse two of them

857
01:01:38,686 --> 01:01:43,575
Experiment that a nice experiment to distinguish them 

858
01:01:43,710 --> 01:01:45,815
will not lead to any ambiguity

859
01:01:45,979 --> 01:01:47,711
The same is true for up and down

860
01:01:47,842 --> 01:01:49,680
There is distinctly different of up and down

861
01:01:49,827 --> 01:01:51,543
If you measure up, you will measure up again and again and again

862
01:01:51,684 --> 01:01:54,943
And the version of that in quantum mechanics

863
01:01:55,069 --> 01:01:59,511
Or the implication of that in quantum mechanics

864
01:01:59,662 --> 01:02:03,455
Or the vectors is the thing that correspond to 

865
01:02:03,569 --> 01:02:08,823
physically unambiguously different situation of orthogonal to each other

866
01:02:08,953 --> 01:02:12,247
So up and down, or orthogonal to each other

867
01:02:14,039 --> 01:02:18,135
What if we have a combination of up and down was some coefficient

868
01:02:18,265 --> 01:02:22,359
Then it's the probability of be up and probability to be down

869
01:02:22,463 --> 01:02:28,471
And the probability are given by alpha star alpha up

870
01:02:28,618 --> 01:02:34,729
And alpha star alpha down again, this is a postulate

871
01:02:34,839 --> 01:02:40,058
We will see in a moment what it correspond to

872
01:02:40,058 --> 01:02:42,366
Well, what's the relationship between 

873
01:02:42,461 --> 01:02:44,157
the probability for up and probability for down

874
01:02:44,283 --> 01:02:46,943
The answer is if we add them we get one

875
01:02:47,078 --> 01:02:53,878
And therefore alpha star alpha plus beta star beta is equal to one

876
01:02:55,637 --> 01:02:59,361
Any reasonable state of a system which has a total probability one for

877
01:02:59,492 --> 01:03:01,496
All the different things that can happen

878
01:03:02,296 --> 01:03:04,537
should satisfy the sum of the probabilities 

879
01:03:04,714 --> 01:03:07,307
or the distinct probabilities should add up to one 

880
01:03:07,422 --> 01:03:16,875
So that said, alpha star alpha plus beta, not beta

881
01:03:17,006 --> 01:03:21,923
Alpha up star up, alpha down star alpha down

882
01:03:22,048 --> 01:03:26,204
Did I use beta before? Should equal to one

883
01:03:26,345 --> 01:03:33,428
But that's equivalent to the statement of inner product A with itself

884
01:03:33,538 --> 01:03:36,879
is equal to one

885
01:03:39,608 --> 01:03:40,636
Only to think about it

886
01:03:44,843 --> 01:03:46,485
Prove this

887
01:03:51,392 --> 01:03:56,491
Prove that alpha star alpha summed over up and down 

888
01:03:56,618 --> 01:03:59,354
is equal to the inner product of A with itself

889
01:03:59,478 --> 01:04:01,338
It does follow from the postulate

890
01:04:04,682 --> 01:04:08,970
And the assumption that u and d are perpendicular unit vectors

891
01:04:09,095 --> 01:04:14,506
u and d are perpendicular unit vector like that assumption 

892
01:04:14,610 --> 01:04:15,962
that is orthogonal to each other

893
01:04:16,095 --> 01:04:18,466
That unit vectors, if they are not unit vectors 

894
01:04:18,620 --> 01:04:21,258
you can change them into unit vectors

895
01:04:21,387 --> 01:04:26,058
And then you can prove that the inner product of a vector with itself 

896
01:04:26,188 --> 01:04:30,972
is just the sum of the squares of the conjugate times itself

897
01:04:31,110 --> 01:04:34,735
That's a little thing to prove 

898
01:04:34,868 --> 01:04:40,987
But what does that say, that says the physical state correspond to

899
01:04:41,116 --> 01:04:44,804
a vector in a vector space with a unit length, a unit vector

900
01:04:45,560 --> 01:04:49,788
A unit standing for the fact that the total probability is one

901
01:04:50,885 --> 01:05:00,716
So states are only equal to vector in a vector space but they are unit vectors

902
01:05:02,415 --> 01:05:12,479
The other term for unit vector is normalized vectors, normalized vectors

903
01:05:13,639 --> 01:05:16,103
Normalized is another word for unit

904
01:05:17,023 --> 01:05:22,879
So all physical states, all states or what you can do 

905
01:05:23,016 --> 01:05:25,855
to that electron spin or anything else

906
01:05:26,991 --> 01:05:29,167
Correspond to unit vectors

907
01:05:34,023 --> 01:05:42,111
And given any such unit vectors the probability associate with it are

908
01:05:42,249 --> 01:05:44,615
alpha star alpha up, alpha star alpha down

909
01:05:44,761 --> 01:05:48,927
So that's a bit of information 

910
01:05:53,816 --> 01:05:58,910
But wait a minute, what happen to these other vectors right and left?

911
01:05:58,910 --> 01:06:01,525
What happen to right and left?

912
01:06:06,275 --> 01:06:06,405
Right and left seem like different state 

913
01:06:06,405 --> 01:06:08,828
But according to the postulate, 

914
01:06:10,145 --> 01:06:14,157
right and left must be linearly combination of up and down

915
01:06:14,299 --> 01:06:19,485
It's not too surprising because if you prepare right or left

916
01:06:19,622 --> 01:06:23,189
then measure up or down, you get plus one or minus one

917
01:06:24,060 --> 01:06:26,252
That's all it is, plus one or minus one

918
01:06:26,393 --> 01:06:29,685
So it's not too surprising whatever left or right is

919
01:06:29,828 --> 01:06:32,864
It's somehow a combination of up and down

920
01:06:34,012 --> 01:06:43,120
And that statement is buried in the properties of vectors

921
01:06:43,120 --> 01:06:47,942
So let's look at the state left and also the state right

922
01:06:48,117 --> 01:06:50,160
What do we know about it?

923
01:06:50,306 --> 01:06:59,270
Well, we know that it must be a combination alpha up up plus alpha down down

924
01:06:59,410 --> 01:07:05,380
We can also write that in the form of column vector alpha up and alpha down

925
01:07:07,080 --> 01:07:09,020
Number one, 

926
01:07:09,150 --> 01:07:15,460
number two, supposing electron is created pointing along the right direction

927
01:07:16,610 --> 01:07:20,870
The x axis, what do the probability for it to be up and down? Remember?

928
01:07:23,450 --> 01:07:31,260
Half, half, that means alpha up star alpha up must equal to half

929
01:07:31,400 --> 01:07:36,870
And alpha star down alpha down must equal to half

930
01:07:36,990 --> 01:07:43,420
Let's make a guess, let's make a guess, the simplest guess would be to put

931
01:07:47,170 --> 01:07:50,570
one over square root of two and one over square of two here

932
01:07:51,740 --> 01:07:56,623
Alpha up is one over square of two, if alpha up is one over square of two

933
01:07:56,738 --> 01:08:00,982
What is alpha star alpha? One half

934
01:08:02,591 --> 01:08:03,894
This one is also one half

935
01:08:04,027 --> 01:08:06,545
Now we could put, we could put some other things

936
01:08:06,673 --> 01:08:08,526
There is one ambiguity is there

937
01:08:08,672 --> 01:08:10,417
We could put a minus sign here

938
01:08:11,286 --> 01:08:14,631
Or we can put a complex coefficient in front of here

939
01:08:14,762 --> 01:08:17,646
Without changing the magnitude of it

940
01:08:17,777 --> 01:08:20,446
We will come back to that, we will come back to what that means

941
01:08:21,518 --> 01:08:23,926
We will come back to what that ambiguity means

942
01:08:25,134 --> 01:08:30,823
But let's take this as a working hypothesis for what the state right is

943
01:08:30,975 --> 01:08:34,140
That lead us to the state left

944
01:08:34,140 --> 01:08:39,664
What do we know about left?

945
01:08:39,806 --> 01:08:46,130
Well, right or left is physically distinguishable by a single experiment

946
01:08:46,264 --> 01:08:51,538
You can orient your apparatus along the x direction 

947
01:08:51,671 --> 01:08:58,602
Measure sigma x, and determine unambiguously whether it correspond to right or left

948
01:08:58,745 --> 01:09:02,434
What does it mean about the vector right and left?

949
01:09:02,565 --> 01:09:04,572
It means they are orthogonal 

950
01:09:05,356 --> 01:09:10,023
Orthogonal or physically measurable distinct are the same things

951
01:09:10,153 --> 01:09:15,548
So that tells us that left right should be equal to zero

952
01:09:15,665 --> 01:09:23,008
Incidentally we have assumed that up down is equal to zero

953
01:09:23,918 --> 01:09:28,156
Up down, orthogonal, by symmetrically 

954
01:09:28,292 --> 01:09:31,860
it's nothing different between the x axis and z axis, 

955
01:09:32,004 --> 01:09:35,826
so if the two vector up and down are orthogonal stands for reason that

956
01:09:35,826 --> 01:09:38,116
two vectors left and right must be orthogonal

957
01:09:38,253 --> 01:09:43,734
here's right, is there a vector orthogonal to it, ya?

958
01:09:43,870 --> 01:09:48,312
There is a vector which is orthogonal to it, what is it?

959
01:09:48,434 --> 01:09:56,044
One over square of two up, minus one over square of two down

960
01:09:57,377 --> 01:10:01,412
Now it's the ambiguous to put the minus sign here or here, 

961
01:10:01,542 --> 01:10:03,340
Either way would work

962
01:10:03,481 --> 01:10:08,332
We will find out there is no physical difference between those two possibilities

963
01:10:08,475 --> 01:10:16,526
We will find out later there is no physics, and the overall sign or a vector

964
01:10:16,526 --> 01:10:19,386
So here is one which is orthogonal

965
01:10:19,514 --> 01:10:21,633
How do we check this is orthogonal to that?

966
01:10:21,765 --> 01:10:29,955
Well, let's write the row, here's a column vector describing right

967
01:10:30,097 --> 01:10:33,915
Let's write down the row vector on it,

968
01:10:34,022 --> 01:10:35,723
That's not the column vector, here is the column vector

969
01:10:35,855 --> 01:10:39,011
The column vector describing right is one over square of two

970
01:10:39,163 --> 01:10:43,395
One over square of two, that's these two

971
01:10:43,540 --> 01:10:46,579
What is the row vector describing left

972
01:10:46,707 --> 01:10:48,155
If we want to take the inner product

973
01:10:48,294 --> 01:10:51,690
We want to combine the row vector with the column vector

974
01:10:51,840 --> 01:10:55,234
So the row vector describing this right over here

975
01:10:55,371 --> 01:11:00,420
is one over square root of two minus one over square of two

976
01:11:04,915 --> 01:11:07,515
And to take the inner product between them

977
01:11:07,644 --> 01:11:11,715
You take the product of first coefficient times the coefficient

978
01:11:11,836 --> 01:11:14,147
Plus the product of the second one times the second one

979
01:11:14,261 --> 01:11:17,259
Standard rules, by taking the inner product 

980
01:11:17,380 --> 01:11:19,332
between a column vector and a row vector

981
01:11:19,483 --> 01:11:23,067
You match the first component with the first component multiply them, 

982
01:11:23,209 --> 01:11:26,003
second component with the second component, multiply them

983
01:11:26,130 --> 01:11:30,179
and go right down to the, to all the components and add them up

984
01:11:31,884 --> 01:11:33,067
okay, so what's the answer?

985
01:11:33,229 --> 01:11:36,772
One half, sorry, one over square root of two

986
01:11:36,904 --> 01:11:40,707
One over square root of two is half minus, one over square of two

987
01:11:42,339 --> 01:11:43,643
This inner product is zero

988
01:11:43,759 --> 01:11:48,612
So we found a second vector which correspond to right and left

989
01:11:48,750 --> 01:11:51,035
A candidate if you like for right and left

990
01:11:51,200 --> 01:11:56,515
And what's more you will notice that this vector also has the property

991
01:11:56,647 --> 01:12:03,496
That the probability for it to be up and down is equal to half for each one

992
01:12:03,650 --> 01:12:08,035
Why? Because the square of this one half, 

993
01:12:08,151 --> 01:12:11,875
and the square of the magnitude of this one is one half

994
01:12:12,007 --> 01:12:16,937
So we have a nice symmetrical relationship between left right and up and down

995
01:12:17,063 --> 01:12:19,771
Incidentally it's kind of interesting 

996
01:12:19,898 --> 01:12:23,013
Supposing them, where is my eraser?

997
01:12:30,834 --> 01:12:34,594
Alright, it's kind of interesting, I've written right and left in terms of up and down

998
01:12:34,733 --> 01:12:38,648
I could of course to write up and down in terms of right and left

999
01:12:38,778 --> 01:12:40,058
Let's see what we get

1000
01:12:40,204 --> 01:12:44,690
If we add these two equations the downs were canceled

1001
01:12:44,813 --> 01:12:46,442
And we will just get up

1002
01:12:46,600 --> 01:12:51,154
We will get two over square, we will get square two that's two over the square

1003
01:12:51,298 --> 01:12:55,794
two times up equals to right plus left

1004
01:12:55,927 --> 01:12:57,890
And then divided by square of two

1005
01:12:59,874 --> 01:13:08,586
And we will find that down is equal to right minus left over square of two

1006
01:13:08,716 --> 01:13:12,166
So you see it's completely reciprocal relationship between them

1007
01:13:13,366 --> 01:13:21,386
If we began our experiments with the apparatus along the x direction

1008
01:13:21,526 --> 01:13:26,376
We might have started, we might have started with left and right

1009
01:13:26,516 --> 01:13:30,856
And come to the conclusion that up and down are combination of left and right

1010
01:13:32,046 --> 01:13:35,386
Okay, that leaves in and out, what about in and out?

1011
01:13:36,506 --> 01:13:41,546
Is there a third pair of vectors, in and out?

1012
01:13:41,676 --> 01:13:49,356
Which stand in the same relationship to right and left and to up and down

1013
01:13:50,976 --> 01:13:54,816
As they stand to each other, there is a third combination which has

1014
01:13:56,046 --> 01:13:58,856
Which is symmetrically defined

1015
01:13:58,976 --> 01:14:00,536
Let's put it up here

1016
01:14:01,546 --> 01:14:07,536
We have up and down, and they are orthogonal to each other

1017
01:14:08,646 --> 01:14:13,576
We have right and left, right which is equal to one over square of two

1018
01:14:13,696 --> 01:14:16,876
We can write it as up plus down, and left

1019
01:14:21,176 --> 01:14:29,705
And same thing one over square of two times up minus down

1020
01:14:29,848 --> 01:14:34,374
We also have right left equal to zero

1021
01:14:34,515 --> 01:14:40,114
More over we have something that we are not going to write

1022
01:14:40,225 --> 01:14:42,988
But I will say it anyway

1023
01:14:43,103 --> 01:14:46,540
If you start in right and you say what's the probability for up and down?

1024
01:14:46,764 --> 01:14:48,612
It's one half

1025
01:14:48,744 --> 01:14:51,892
If you start left and you ask the probability for up and down

1026
01:14:51,997 --> 01:14:53,164
It's one half

1027
01:14:53,270 --> 01:14:55,468
What happen if you start with up and down

1028
01:14:55,602 --> 01:14:57,132
What's the probability for right and left?

1029
01:15:01,180 --> 01:15:03,780
Well, it's also one half

1030
01:15:03,897 --> 01:15:06,469
We know it's one half from the physics of the situation

1031
01:15:06,578 --> 01:15:11,275
But it's also true if we rewrite up and down in terms of right and left

1032
01:15:11,410 --> 01:15:14,540
The coefficients are also one over square of two

1033
01:15:14,681 --> 01:15:18,076
So they are nicely symmetrical to each other

1034
01:15:18,206 --> 01:15:22,084
Alright so that leaves us with the last possibility

1035
01:15:22,210 --> 01:15:25,925
Or the last perpendicular direction in and out

1036
01:15:32,081 --> 01:15:36,865
Okay so let's try to guess, it's gonna have an up

1037
01:15:37,021 --> 01:15:43,010
And it's gonna have plus and down

1038
01:15:44,474 --> 01:15:46,586
But it's gonna be some coefficient here

1039
01:15:46,701 --> 01:15:48,578
And we don't know what those coefficients are yet

1040
01:15:50,082 --> 01:15:52,108
What do we know about the coefficients?

1041
01:15:53,076 --> 01:15:56,860
The complex in general, in general they are complex 

1042
01:15:58,226 --> 01:16:01,530
>> there's not just be, the square would be one half

1043
01:16:01,671 --> 01:16:04,954
The square would just, or the magnitude of the square has to be one half

1044
01:16:06,985 --> 01:16:10,439
We also know that the coefficient here have to be such

1045
01:16:10,572 --> 01:16:13,537
That in and out are perpendicular to each other

1046
01:16:16,918 --> 01:16:22,574
And we also should find that if we rewrite in and out

1047
01:16:23,693 --> 01:16:29,946
in terms of left and right, the coefficient should also have modulus one half

1048
01:16:30,123 --> 01:16:31,726
just by symmetry

1049
01:16:31,726 --> 01:16:35,106
Turns out that enough to determine them completely

1050
01:16:35,106 --> 01:16:37,097
I'm not gonna prove it that's a homework assignment 

1051
01:16:37,234 --> 01:16:38,831
but that's nice homework assignment

1052
01:16:38,970 --> 01:16:40,771
Let's see what the requirements are

1053
01:16:40,928 --> 01:16:45,104
The requirements are that in out equals to zero

1054
01:16:45,218 --> 01:16:52,104
That both in or out when written in terms of up or down

1055
01:16:54,629 --> 01:16:59,898
Should have coefficients whose magnitude is a half

1056
01:17:03,110 --> 01:17:05,052
Square magnitude is a half

1057
01:17:05,202 --> 01:17:08,539
When also when written in terms of left and right

1058
01:17:11,268 --> 01:17:15,084
The same corresponding coefficients here

1059
01:17:15,341 --> 01:17:18,443
Should have magnitude plus or minus a half

1060
01:17:18,570 --> 01:17:24,620
That is enough to determine the coefficient in terms of up and down

1061
01:17:24,762 --> 01:17:28,301
With a small ambiguity which is not interesting 

1062
01:17:28,421 --> 01:17:30,357
I will tell you later what that ambiguity is

1063
01:17:30,503 --> 01:17:32,061
There's no physics in it

1064
01:17:32,893 --> 01:17:36,749
First of all is the overall sign of the vector hat has no physics

1065
01:17:36,876 --> 01:17:38,254
We will see and find out

1066
01:17:38,370 --> 01:17:42,897
Alright what is the solution? The solution is the one here

1067
01:17:43,005 --> 01:17:47,505
And the i here, with a square root of two you have to put

1068
01:17:47,619 --> 01:17:50,598
one over square of two, and one over square of two

1069
01:17:53,527 --> 01:18:04,693
And out is the same thing, up, minus i over square of two, down

1070
01:18:06,677 --> 01:18:09,535
Let's write them in terms of column and row vectors 

1071
01:18:09,671 --> 01:18:13,719
>> so that will be the i over square of two on the in vector?

1072
01:18:14,839 --> 01:18:21,335
Well i one yeah, is that what you are asking?

1073
01:18:22,447 --> 01:18:25,463
Actually I had it right, it was one times i

1074
01:18:31,338 --> 01:18:33,932
We can write these in terms of column vectors

1075
01:18:34,067 --> 01:18:39,116
Let's write the various column vectors, up, what would you write for up?

1076
01:18:40,403 --> 01:18:50,700
Well, we can write one zero, up is just up with no down component

1077
01:18:50,816 --> 01:18:59,724
Down is just zero one, a one in the down place and a zero in the up place

1078
01:18:59,841 --> 01:19:07,500
What about right? It's equal to one over square root of two

1079
01:19:07,660 --> 01:19:09,460
One over square root of two

1080
01:19:09,601 --> 01:19:18,636
Left, that's equal to one over square root of two 

1081
01:19:18,767 --> 01:19:21,492
minus one over square root of two

1082
01:19:22,940 --> 01:19:31,372
Now let's put in, in is one over square root of two,

1083
01:19:31,488 --> 01:19:39,149
i over square root of two and right, sorry

1084
01:19:39,270 --> 01:19:44,332
Out is equal to one over square root of two minus i over square root of two

1085
01:19:47,636 --> 01:19:55,564
First question, is this orthogonal to this? How do we check?

1086
01:19:55,716 --> 01:19:59,677
We take this times the complex conjugate of this

1087
01:19:59,800 --> 01:20:03,204
And take this times the complex conjugate of this

1088
01:20:03,317 --> 01:20:06,103
That's the rule of caluculating inner product

1089
01:20:06,103 --> 01:20:09,304
We could constructing the row vector, we don't need to

1090
01:20:09,431 --> 01:20:12,047
No let's, let's, let's, we don't need to

1091
01:20:12,169 --> 01:20:17,273
Remember the row vector has complex conjugate entries

1092
01:20:18,851 --> 01:20:21,132
But it's easy enough to do the inner product

1093
01:20:21,274 --> 01:20:26,174
The inner product is the sum of the product of the entries

1094
01:20:27,287 --> 01:20:30,575
times their complex conjugate

1095
01:20:31,487 --> 01:20:34,583
Alright so one of these has to be the complex conjugate

1096
01:20:34,583 --> 01:20:37,533
What's the product of these two complex conjugates?

1097
01:20:40,609 --> 01:20:43,314
One half, it's just one half

1098
01:20:44,202 --> 01:20:47,003
So this one times this gives you a one half

1099
01:20:47,003 --> 01:20:51,288
And how about this times the complex conjugate of this?

1100
01:20:51,427 --> 01:20:55,618
Minus one half, minus one half

1101
01:20:55,773 --> 01:21:00,013
Minus sign over two, what is the complex conjugate of i over square root of two

1102
01:21:01,670 --> 01:21:03,404
Minus sign over square root of two, 

1103
01:21:03,530 --> 01:21:05,494
so we have to multiply minus sign over square root of two

1104
01:21:06,186 --> 01:21:07,378
times another minus sign over the square root of two

1105
01:21:07,764 --> 01:21:12,520
which gives us minus a half, i times i is minus one

1106
01:21:12,520 --> 01:21:17,711
So that gives us minus a half, if we add them up we get zero

1107
01:21:17,862 --> 01:21:21,315
So yes, these two are orthogonal to each other

1108
01:21:23,609 --> 01:21:27,177
The magnitude of the coefficients here

1109
01:21:27,310 --> 01:21:31,352
the square of the magnitude of the coefficient here is equal to a half

1110
01:21:31,520 --> 01:21:35,765
which proves the in states or the out state 

1111
01:21:35,910 --> 01:21:39,365
the probability of being up and down is one half

1112
01:21:40,789 --> 01:21:44,727
How about the probability of being left or right?

1113
01:21:44,843 --> 01:21:47,071
How might you figure that out?

1114
01:21:51,271 --> 01:21:51,858
Say it again?

1115
01:21:55,055 --> 01:21:59,360
Definition of i is the square root of minus one

1116
01:21:59,360 --> 01:22:03,559
So i times i is always minus one

1117
01:22:06,434 --> 01:22:10,075
i is not, oh sorry, this is

1118
01:22:10,207 --> 01:22:12,524
This i is not in

1119
01:22:13,947 --> 01:22:30,411
Okay, I and Out, Right and Left, Up and that's the way it is, Down

1120
01:22:30,540 --> 01:22:32,947
And go back through out all the pages and change it

1121
01:22:36,722 --> 01:22:40,378
Actually we can could have change it more easily, just change this i to capital I

1122
01:22:40,493 --> 01:22:43,047
But that would be, that would be bad form

1123
01:22:43,176 --> 01:22:47,588
>> Is there anything about this coordinate system 

1124
01:22:47,690 --> 01:22:51,450
>> that implying a left or right hand coordinate system?

1125
01:22:54,496 --> 01:22:57,136
Very very good question and the answer is yes

1126
01:22:57,259 --> 01:23:00,836
But we will come to it, yes that's a very very good question

1127
01:23:00,977 --> 01:23:09,134
What is this in here is telling us that it's a right hand coordinate system

1128
01:23:09,280 --> 01:23:10,926
That or a left hand coordinate system

1129
01:23:10,926 --> 01:23:14,187
And I didn't want to go there yet, but we will come to it

1130
01:23:14,332 --> 01:23:18,132
We will come to it when we explore the Pauli matrices 

1131
01:23:19,080 --> 01:23:24,256
For the moment, we just, we are doing, we are sort of guessing away

1132
01:23:24,256 --> 01:23:27,447
Bumbling through, trying to find the right vectors 

1133
01:23:27,585 --> 01:23:31,931
to describe the various states of the system

1134
01:23:34,564 --> 01:23:37,539
Okay, this is my proposal if you like

1135
01:23:37,667 --> 01:23:41,746
My best guess for the moment for what the states up down

1136
01:23:41,746 --> 01:23:47,169
Those are basic, we start with those, in terms of those right and left R 

1137
01:23:47,170 --> 01:23:48,656
and not in and out R

1138
01:23:48,656 --> 01:23:51,395
Now what you could do, 

1139
01:23:52,592 --> 01:23:58,565
we saw that you could rewrite that up and down in terms of left and right

1140
01:23:58,678 --> 01:24:01,394
I've shown you how to do that 

1141
01:24:01,534 --> 01:24:06,536
And that means you could write i and o in terms of left and right

1142
01:24:08,264 --> 01:24:11,684
You could check what the coefficients are

1143
01:24:11,805 --> 01:24:15,224
I hesitate to do it because I'm sure of bumble it

1144
01:24:15,356 --> 01:24:20,874
In the other hand I tended to do it because I'm not sure to get it right

1145
01:24:22,081 --> 01:24:25,836
I will take one shot at it, one shot and if it doesn't work out

1146
01:24:31,875 --> 01:24:36,954
Okay, up is left plus right, down is left minus right

1147
01:24:37,096 --> 01:24:38,529
With a square root of two

1148
01:24:38,672 --> 01:24:41,238
So let's see if we can figure out what in is first

1149
01:24:41,349 --> 01:24:46,957
Let's just do in, in is one over square root of two times up

1150
01:24:47,077 --> 01:24:54,309
What did I say up was? Up I said was left plus

1151
01:25:01,029 --> 01:25:07,750
is left plus right, left plus right

1152
01:25:11,308 --> 01:25:13,452
But with a one over square root of two

1153
01:25:15,012 --> 01:25:17,180
That gives me one half

1154
01:25:17,313 --> 01:25:26,980
It's left over two, plus right over two, that's scary

1155
01:25:28,296 --> 01:25:37,167
And then plus i over square root of two times down

1156
01:25:37,282 --> 01:25:38,773
Anybody remember what down was?

1157
01:25:39,866 --> 01:25:43,093
Left minus right over here, let's

1158
01:25:43,234 --> 01:25:55,927
That will give me left, right over another square root of two

1159
01:25:56,068 --> 01:26:00,255
minus left over another square root of two

1160
01:26:03,839 --> 01:26:05,767
So the whole thing look like

1161
01:26:05,919 --> 01:26:09,275
>>Is that two one halves, 

1162
01:26:11,987 --> 01:26:16,345
>>No, the second term, no, let's go back, right there

1163
01:26:16,478 --> 01:26:18,783
>>You get a half and then you got a two

1164
01:26:22,744 --> 01:26:31,869
And this could just be one half, the square of two right

1165
01:26:32,913 --> 01:26:34,139
Did I have it right?

1166
01:26:35,709 --> 01:26:39,910
Okay so what do we get, we get the coefficient times the left

1167
01:26:40,055 --> 01:26:58,918
So here we are, I is equal to one half, one minus i over two

1168
01:26:59,066 --> 01:27:09,672
Was that right? One minus i over two times left plus

1169
01:27:09,808 --> 01:27:14,726
I think it's one plus i over two times right

1170
01:27:17,322 --> 01:27:21,103
>>I think you would write O instead of I?

1171
01:27:21,221 --> 01:27:23,855
Here? I think we are doing I

1172
01:27:23,975 --> 01:27:28,967
But I'm not sure, doesn't matter

1173
01:27:32,855 --> 01:27:35,344
Uh, I think we were doing this one

1174
01:27:35,468 --> 01:27:40,463
Up is left plus right, and down is left minus right

1175
01:27:40,592 --> 01:27:42,223
The same i

1176
01:27:44,135 --> 01:27:50,383
Okay, now, what's the magnitude of the two coefficients here?

1177
01:27:50,492 --> 01:27:53,671
Then times it's complex conjugate

1178
01:27:53,806 --> 01:27:56,999
Let's check, I don't see any square root of two in these things, 

1179
01:27:56,999 --> 01:27:58,373
Do you see any square root of two here?

1180
01:27:58,490 --> 01:28:01,981
You might have expected some square root of two huh?

1181
01:28:02,155 --> 01:28:08,326
Nevertheless, I maintain you take this kind of complex conjugate 

1182
01:28:08,440 --> 01:28:09,982
I hope that was true

1183
01:28:10,119 --> 01:28:14,558
I maintain that if you take this kind of complex conjugate

1184
01:28:14,702 --> 01:28:17,641
You will get one half, let's check it

1185
01:28:17,780 --> 01:28:26,521
It's one minus i, times one plus i over, two times two is four

1186
01:28:27,713 --> 01:28:32,909
One minus i plus one plus i, where is the law?

1187
01:28:33,026 --> 01:28:38,307
It's two, it's two over four, which equal to one half

1188
01:28:39,908 --> 01:28:41,979
Another word, this coefficient here, 

1189
01:28:42,083 --> 01:28:44,849
although it does not contain any explicit square root of two

1190
01:28:45,894 --> 01:28:49,429
It times it's complex conjugate is equal to one over two

1191
01:28:49,429 --> 01:28:53,424
So the magnitude of this complex number here

1192
01:28:53,533 --> 01:28:55,886
is one over the square root of two

1193
01:28:56,015 --> 01:28:59,699
It's written in the form that you don't see any explicit one over square root of two

1194
01:28:59,842 --> 01:29:03,289
But the magnitude of this one over square root of two

1195
01:29:03,289 --> 01:29:05,587
And so is the magnitude of this

1196
01:29:05,725 --> 01:29:14,286
The implication is that if you started in the state I, into the blackboard

1197
01:29:15,502 --> 01:29:17,807
And instead of measuring up or down

1198
01:29:18,649 --> 01:29:24,591
You measure left or right, the probability would be a half, alright?

1199
01:29:24,721 --> 01:29:32,083
So we found three mutually relate, not, orthogonal to each other

1200
01:29:32,251 --> 01:29:35,280
Up and down is not orthogonal to left and right

1201
01:29:35,415 --> 01:29:38,632
They are orthogonal to each other 

1202
01:29:38,754 --> 01:29:41,496
But they are not orthogonal in the vertical direction

1203
01:29:41,630 --> 01:29:48,099
But each one, if expanded in any of the others

1204
01:29:48,099 --> 01:29:53,091
would have coefficient whose magnitude was one over the square root of two

1205
01:29:54,623 --> 01:29:57,613
So they are symmetrically related in that sense

1206
01:30:00,045 --> 01:30:04,957
And what does it means?

1207
01:30:05,100 --> 01:30:08,379
In effect, if you start with anyone one of them

1208
01:30:08,514 --> 01:30:12,069
And measure the other component, if you start with up or down

1209
01:30:12,188 --> 01:30:17,789
And measure sigma x and sigma y, you will get probability a half

1210
01:30:18,893 --> 01:30:23,619
No matter what you start with, any one of these three combination

1211
01:30:23,619 --> 01:30:28,832
Any one of these six combination and measure a perpendicular combination 

1212
01:30:28,943 --> 01:30:31,873
you always get probability plus or minus a half

1213
01:30:32,031 --> 01:30:37,796
Now I've put in here some postulates about what probabilities are

1214
01:30:37,937 --> 01:30:42,786
without telling you, but I think we were coming along pretty well

1215
01:30:42,915 --> 01:30:45,586
We have some mathematics here which is telling us

1216
01:30:45,718 --> 01:30:51,170
That these three collections, or pairs of vectors

1217
01:30:51,354 --> 01:30:55,298
are nicely symmetrically related to each other 

1218
01:31:08,830 --> 01:31:13,155
That's the most important fact, yes,

1219
01:31:23,660 --> 01:31:28,323
You know we don't have to, this is orthogonal to this

1220
01:31:28,432 --> 01:31:39,260
Oh sorry, three bases, three bases at angles with respect to each other

1221
01:31:40,761 --> 01:31:45,138
That's a good question, the question is could we find, excellent question

1222
01:31:45,256 --> 01:31:50,283
Could we find another double 

1223
01:31:51,067 --> 01:31:57,652
Let's call it big and little, no, we are already used the l

1224
01:31:57,814 --> 01:32:02,710
Uh, fat and skinny, fat and skinny

1225
01:32:02,710 --> 01:32:10,170
Can we find another pair such that fact is orthogonal to skinny

1226
01:32:11,631 --> 01:32:16,313
And the inner product that's what it actually came down to

1227
01:32:16,479 --> 01:32:23,961
Or that the expansion coefficients of each one in terms of the other kind of bases

1228
01:32:24,122 --> 01:32:26,769
are always equal to one over square root of two 

1229
01:32:26,907 --> 01:32:29,065
or always one half in magnitude?

1230
01:32:31,193 --> 01:32:35,673
The answer is no, there is no third, there is no fourth combination

1231
01:32:35,789 --> 01:32:39,609
that's in the same relationship, that's not obvious, that's not obvious

1232
01:32:41,642 --> 01:32:43,873
It is a fact that we live in a three-dimensional world

1233
01:32:44,008 --> 01:32:48,825
It's exactly the fact that we live in a three-dimensional world right?

1234
01:32:50,321 --> 01:32:53,177
It's exactly the fact that the statement of 

1235
01:32:53,178 --> 01:32:55,970
three-dimension of space can be represented

1236
01:32:55,970 --> 01:33:02,726
in two-dimensional, in two-dimensional vector space terms

1237
01:33:02,870 --> 01:33:05,570
Alright let me tell you something else, which is true

1238
01:33:07,098 --> 01:33:12,490
If I give you a column vector, let's say a column vector 

1239
01:33:13,578 --> 01:33:15,964
Alpha one, alpha two

1240
01:33:16,100 --> 01:33:21,515
Or it's just the coefficient of the expansion alpha up or alpha down

1241
01:33:21,643 --> 01:33:22,506
If you like

1242
01:33:24,656 --> 01:33:29,802
The question is how much freedom is there in the alphas 

1243
01:33:29,932 --> 01:33:35,498
such that it doesn't change the physical nature of the state? How much freedom?

1244
01:33:35,641 --> 01:33:40,139
Does every combination of alpha up and alpha down 

1245
01:33:40,270 --> 01:33:41,905
correspond to a different state?

1246
01:33:42,032 --> 01:33:42,903
The answer is no

1247
01:33:43,039 --> 01:33:49,160
First of all, the total probability has to add up to one

1248
01:33:49,330 --> 01:33:51,296
So that's one constraint right there 

1249
01:33:51,447 --> 01:33:59,337
that alpha up star alpha up plus alpha down star alpha down is equal to one

1250
01:34:00,665 --> 01:34:03,172
That's another fact that we have not got to yet

1251
01:34:03,304 --> 01:34:08,713
But we will find that every physically observable quantity 

1252
01:34:08,840 --> 01:34:12,137
always associate with a state

1253
01:34:12,258 --> 01:34:17,583
Always involve an alpha times the complex conjugate alpha

1254
01:34:17,721 --> 01:34:22,342
You will find that, it's, we won't put it in

1255
01:34:22,487 --> 01:34:25,862
It will emerge just as the consequences of the postulate

1256
01:34:25,996 --> 01:34:33,852
But what it says is that if you multiply both of the alphas here

1257
01:34:34,010 --> 01:34:38,997
By the same phase factor, does anybody know what are the phase factors?

1258
01:34:39,124 --> 01:34:44,530
The phase factor was a complex number of the form e to the i theta

1259
01:34:45,802 --> 01:34:51,738
of unit length, doesn't change the magnitude of either coefficient here

1260
01:34:51,870 --> 01:34:57,968
That if we multiply every entries by the same phase factor

1261
01:34:57,968 --> 01:35:01,020
It doesn't affect anything, that means that 

1262
01:35:01,158 --> 01:35:03,635
if I just give you two complex numbers here 

1263
01:35:03,767 --> 01:35:07,959
there are two redundant piece of information 

1264
01:35:08,094 --> 01:35:12,240
that really don't have anything to do with physical character of the states

1265
01:35:12,359 --> 01:35:15,384
First is the over all magnitude, 

1266
01:35:16,594 --> 01:35:20,442
because the over all magnitude should always be such that

1267
01:35:20,587 --> 01:35:22,568
that sum of these square roots add up to one

1268
01:35:23,568 --> 01:35:29,893
So that means one core constraint if you like, and the redundancy

1269
01:35:30,033 --> 01:35:35,794
The redundancy is that if you multiply both entries by the same phase factor here

1270
01:35:35,937 --> 01:35:37,947
It also doesn't change the states

1271
01:35:39,284 --> 01:35:40,253
So then you could ask, 

1272
01:35:40,393 --> 01:35:47,158
how many independent variables are there which characterize the state of a system?

1273
01:35:47,158 --> 01:35:50,200
Of this system, how many of them are there?

1274
01:35:50,358 --> 01:35:55,829
Well, let's count, there are two independent variables 

1275
01:35:55,962 --> 01:35:59,395
If we don't worry about either of this constrain over here

1276
01:35:59,514 --> 01:36:03,118
How many things does it take to characterize two complex numbers?

1277
01:36:04,219 --> 01:36:07,059
Four, four real numbers, right?

1278
01:36:07,949 --> 01:36:14,163
But now, one of them doesn't count, because we have to satisfy the constrain

1279
01:36:14,305 --> 01:36:16,899
That the total length of this must be equal to one

1280
01:36:17,049 --> 01:36:21,876
One of them doesn't count, but also the another one doesn't could because 

1281
01:36:22,004 --> 01:36:25,812
The overall phase if we are to multiply everything by the same phase factor

1282
01:36:25,961 --> 01:36:29,941
Here it also doesn't count, doesn't change the nature of the states

1283
01:36:30,137 --> 01:36:37,688
So two of those four parameters are irrelevant 

1284
01:36:37,816 --> 01:36:39,508
to the physical character of the states

1285
01:36:39,638 --> 01:36:41,888
How many of that leave over? Two

1286
01:36:42,010 --> 01:36:44,276
Two real numbers

1287
01:36:44,421 --> 01:36:50,260
How many numbers does it take to characterize the direction of a unit vector?

1288
01:36:52,404 --> 01:36:53,798
Two 

1289
01:36:53,932 --> 01:36:57,908
What do they call? Longitude and latitude or something like that

1290
01:36:58,042 --> 01:37:00,869
Just the direction of the vector can be characterized by the two angles

1291
01:37:02,144 --> 01:37:05,657
These two pairs of number are the same 

1292
01:37:06,484 --> 01:37:11,444
The fact that it takes two real parameters 

1293
01:37:11,569 --> 01:37:16,461
to describe a two-component vector like this

1294
01:37:16,579 --> 01:37:19,391
is closely related to the fact 

1295
01:37:19,392 --> 01:37:25,560
it takes two real angles to describe the direction of a three-vector

1296
01:37:25,731 --> 01:37:28,267
in ordinary three-dimensional space 

1297
01:37:29,581 --> 01:37:35,866
If we would have added more components here, it would be more complicated

1298
01:37:35,997 --> 01:37:40,547
Yes there is a deep connection between the fact that

1299
01:37:40,697 --> 01:37:43,155
Very deep connection between the facts that 

1300
01:37:43,294 --> 01:37:47,419
we only find three independent combination of this

1301
01:37:48,771 --> 01:37:51,237
Three independent combinations, and the fat and skinny 

1302
01:37:51,362 --> 01:37:56,795
We can't find fat and skinny basically they won't be there

1303
01:37:56,926 --> 01:38:00,275
Not without increasing the dimensionality of the vector space

1304
01:38:00,445 --> 01:38:04,475
That's in fact very deeply connected with three-dimensional space

1305
01:38:05,779 --> 01:38:10,363
Space was nine dimensional or something like that 

1306
01:38:10,509 --> 01:38:13,443
We would not be able to represent 

1307
01:38:13,577 --> 01:38:17,339
the direction of a little arrow in it quantum mechanically

1308
01:38:17,470 --> 01:38:19,243
Like two component vectors 

1309
01:38:20,348 --> 01:38:22,348
We will need more components 

1310
01:38:23,451 --> 01:38:29,051
>> it seems like just physically you would take three because 

1311
01:38:29,158 --> 01:38:31,899
>> two would specify the collection but you could say 

1312
01:38:32,030 --> 01:38:34,028
>> whether it is plus one or minus one

1313
01:38:35,220 --> 01:38:37,603
No no no, the plus one or minus one

1314
01:38:38,603 --> 01:38:40,963
>> inherited from the space right here

1315
01:38:41,096 --> 01:38:42,787
Yeah

1316
01:38:42,928 --> 01:38:48,894
>> How does it stay alive if you have three complex numbers in vector space

1317
01:38:49,089 --> 01:38:51,759
You mean if you want to study some higher dimensinoal space?

1318
01:38:51,907 --> 01:38:53,962
Oh god we have to figure it out that how many questions 

1319
01:38:54,103 --> 01:38:56,108
and how many directions and how many parameters are there to

1320
01:38:57,086 --> 01:39:02,674
That's more complicated, that we are not gonna try 

1321
01:39:04,962 --> 01:39:10,835
I got to tell you I got really scared when I realize 

1322
01:39:10,964 --> 01:39:13,244
we won't get any square root of two by doing this

1323
01:39:14,387 --> 01:39:28,883
>>this map is based on three space? (inaudible)

1324
01:39:29,016 --> 01:39:31,291
Yes it is, all of what you said is true

1325
01:39:31,428 --> 01:39:33,970
>> If you say that there is nine dimensions, 

1326
01:39:34,077 --> 01:39:37,405
>> would this experiment still agrees?

1327
01:39:41,853 --> 01:39:42,012
It would be a version of it, It would be a version of it,

1328
01:39:42,012 --> 01:39:44,254
It would be more complicated and 

1329
01:39:44,389 --> 01:39:50,237
you couldn't describe the spin of a particle by just by two components like this

1330
01:39:50,382 --> 01:39:51,885
It will take more components

1331
01:39:51,885 --> 01:39:53,586
>> What would be the apparatus of it?

1332
01:39:53,695 --> 01:39:59,345
Spinner theory, similar things, similar things

1333
01:40:01,313 --> 01:40:05,489
Don't try to, look, this is enough for now

1334
01:40:05,631 --> 01:40:12,594
Understanding this, if we run to four dimensional space like this

1335
01:40:12,752 --> 01:40:14,545
We would need four component object but ...

1336
01:40:14,680 --> 01:40:19,025
The three to four would be 

1337
01:40:19,151 --> 01:40:23,697
>> You said the states which was corresponded to 

1338
01:40:23,814 --> 01:40:27,569
>> what about the (inaudible)

1339
01:40:31,426 --> 01:40:37,432
But of course if you add two unit vectors you can get a none unit vector, you can

1340
01:40:38,425 --> 01:40:41,597
But then, the rule of the state of the system are divided through 

1341
01:40:41,726 --> 01:40:45,401
divided through by a overall factor until it's normalized

1342
01:40:45,401 --> 01:40:51,126
They have a same significance as 

1343
01:40:51,258 --> 01:40:54,883
probability distribution which don't summed up to one

1344
01:40:56,114 --> 01:40:59,612
You can always go back and say somehow somewhere along the line

1345
01:40:59,746 --> 01:41:04,124
I got some answers for probabilities which add up to something greater than one

1346
01:41:04,273 --> 01:41:06,364
I will just divide through by

1347
01:41:06,365 --> 01:41:11,631
tell the probabilities to find the normalized probabilitie distributions

1348
01:41:11,631 --> 01:41:17,909
So in intermediate steps of calculation might well be 

1349
01:41:18,047 --> 01:41:20,391
presents of vectors which are not normalized

1350
01:41:20,522 --> 01:41:23,607
But at the end of the day, the states of the systems are normalized vectors

1351
01:41:23,736 --> 01:41:31,006
Normalized vectors, and with no memories of of their overall phases is

1352
01:41:31,125 --> 01:41:32,692
There is a mathematical term for that 

1353
01:41:37,388 --> 01:41:38,261
Questions?

1354
01:41:38,408 --> 01:41:41,167
I've gone through a lot of stuffs

1355
01:41:43,598 --> 01:41:46,078
We will make it more precise 

1356
01:41:46,214 --> 01:41:49,977
And the next time when we discuss observables 

1357
01:41:50,076 --> 01:41:54,102
In particular when we discuss what the mathematical nature 

1358
01:41:54,229 --> 01:41:57,341
of these sigma x and sigma y and the sigma z-s are 

1359
01:41:57,341 --> 01:42:02,347
For the moment we just specify them sort of by experiment

1360
01:42:02,482 --> 01:42:06,221
So you hold your apparatus this way and measure something call sigma x

1361
01:42:06,221 --> 01:42:08,994
And you hold it that way and measure something call sigma y

1362
01:42:09,120 --> 01:42:12,718
And we use some symmetries namely 

1363
01:42:12,840 --> 01:42:16,115
the equivalence of different directions of space 

1364
01:42:16,270 --> 01:42:32,597
to guide us into finding expressions for the various physical vectors 

1365
01:42:32,755 --> 01:42:35,014
describing different orientation of the spins

1366
01:42:35,151 --> 01:42:37,598
But we haven't been very rigorous about it

1367
01:42:37,729 --> 01:42:41,158
It's been no, I've made no pretense of really 

1368
01:42:41,289 --> 01:42:43,950
a mathematical clear theorization of these things

1369
01:42:45,118 --> 01:42:48,542
Sort of see to the plan and guessing to a logic stance

1370
01:42:48,710 --> 01:42:52,333
Next time we will make some rigorous postulates 

1371
01:42:53,350 --> 01:42:55,766
about what these things really means

1372
01:42:55,874 --> 01:43:03,157
And their relationship, these things and these things 

1373
01:43:03,289 --> 01:43:05,726
and what the relationship between them is

1374
01:43:05,863 --> 01:43:11,301
The vectors and the things that you measure, these are the states

1375
01:43:11,301 --> 01:43:13,137
But these are the things you measure about

1376
01:43:13,257 --> 01:43:16,212
And the question is what is the connection between them

1377
01:43:16,345 --> 01:43:19,439
Okay, any more question about this?

1378
01:43:19,567 --> 01:43:27,494
>> About these lectures like the sequence on the web

1379
01:43:27,625 --> 01:43:29,283
I imagine they are, but I don't remember, 

1380
01:43:29,424 --> 01:43:33,266
there is not only one way to think about quantum mechanics, so I bet they are

1381
01:43:33,403 --> 01:43:38,097
And you know if you sitting through this for the third time that's your problem

1382
01:43:39,301 --> 01:43:43,785
Believe me, I'm sitting through it for the

1383
01:43:47,904 --> 01:43:51,273
>> was that the phase factor that we could ignore?

1384
01:43:51,388 --> 01:43:52,936
The answer is yes, I think so

1385
01:43:55,040 --> 01:43:57,944
>> was that the phase factor we are gonna be doing anything?

1386
01:44:00,448 --> 01:44:07,655
Well, mostly we would be saying, look, it doesn't matter, it doesn't matter

1387
01:44:09,287 --> 01:44:14,174
The, okay, let's talk about the mathematics of ignoring something

1388
01:44:17,918 --> 01:44:20,496
Ya, that's right

1389
01:44:24,041 --> 01:44:33,315
You know that's true, let's talk about the mathematics and ignoring things from

1390
01:44:38,082 --> 01:44:41,050
Alright the point is zero dimension right?

1391
01:44:43,763 --> 01:44:47,778
A circle is one dimensional, there is one parameter and an angle on it

1392
01:44:47,896 --> 01:44:51,147
But supposing I tell you there is no significance 

1393
01:44:51,260 --> 01:44:52,856
to tell you where you are on that circle

1394
01:44:52,990 --> 01:44:57,310
There is no significance whatever or points on that circle 

1395
01:44:57,429 --> 01:44:59,652
equivalent to every other point

1396
01:44:59,791 --> 01:45:03,739
And you might say, well, for all interesting questions

1397
01:45:03,874 --> 01:45:05,507
A circle is just a point 

1398
01:45:05,647 --> 01:45:09,050
If you can ignore where you are on the circle 

1399
01:45:10,226 --> 01:45:13,019
There is only one point on the circle 

1400
01:45:13,169 --> 01:45:16,227
You are either on the circle, if you are on the circle 

1401
01:45:16,341 --> 01:45:17,875
The circle is one point 

1402
01:45:18,016 --> 01:45:22,987
You can say the same thing about 

1403
01:45:23,103 --> 01:45:25,699
That's another example, what's the other example?

1404
01:45:28,731 --> 01:45:32,971
No, let's take this example 

1405
01:45:33,073 --> 01:45:38,284
If you have, what is that complex number?

1406
01:45:38,418 --> 01:45:42,347
A complex number is a point on the plane

1407
01:45:42,478 --> 01:45:47,571
But for some reason the angle where you are on the plane doesn't matter

1408
01:45:47,767 --> 01:45:49,984
The only thing that's interesting is the distance 

1409
01:45:50,844 --> 01:45:54,174
But now supposing you have two complex numbers 

1410
01:45:54,304 --> 01:45:57,359
And all I tell you is that if you have two complex numbers 

1411
01:45:57,489 --> 01:46:02,252
The thing which doesn't matter is the sum of their phases

1412
01:46:03,346 --> 01:46:06,233
The sum of their phases, the sum of their angles

1413
01:46:06,374 --> 01:46:10,914
Then there is still one angle left over which maybe important

1414
01:46:11,050 --> 01:46:13,434
The difference of the angles 

1415
01:46:13,567 --> 01:46:15,708
So I have two complex numbers 

1416
01:46:15,846 --> 01:46:19,714
And I tell you, for reason that I haven't specify yet 

1417
01:46:20,580 --> 01:46:24,260
That the overall phase which can be thought of the sum of the phase angle

1418
01:46:24,399 --> 01:46:27,691
If the sum of the phase angle for whatever the reason 

1419
01:46:27,807 --> 01:46:30,470
plays no role in the physics of the situation

1420
01:46:30,470 --> 01:46:33,648
There is still the difference of the angle there

1421
01:46:33,800 --> 01:46:37,165
That's exactly what the case is, the sum of the angle

1422
01:46:37,305 --> 01:46:41,876
You know the complex numbers is that going to these entries here

1423
01:46:42,016 --> 01:46:43,949
is of no physical significance

1424
01:46:44,103 --> 01:46:49,816
We counted up four real parameters for two complex numbers

1425
01:46:49,931 --> 01:46:53,956
The overall magnitudes of it doesn't count because we divide that out

1426
01:46:53,956 --> 01:46:56,646
That removes one of them

1427
01:46:56,646 --> 01:47:00,694
And then the sum of the angle doesn't count for whatever reasons

1428
01:47:02,356 --> 01:47:06,014
That removes the other one that leaves you with two real numbers

1429
01:47:06,135 --> 01:47:10,407
And that's in correspondence with the two real number it takes

1430
01:47:10,536 --> 01:47:12,763
to pick the direction in space

1431
01:47:13,856 --> 01:47:18,997
A lot of subtlety, but nothing really difficult

1432
01:47:19,133 --> 01:47:20,549
The mathematic is really easy

1433
01:47:20,674 --> 01:47:23,275
The mathematic for quantum mechanics is easier 

1434
01:47:23,404 --> 01:47:25,502
than the mathematic for classical physics

1435
01:47:26,750 --> 01:47:31,479
Two by two matrices are very simple

1436
01:47:31,625 --> 01:47:34,698
Mathematic is easier, it's unfamiliar 

1437
01:47:39,919 --> 01:47:44,278
>> there's something that always been lately unsettling

1438
01:47:44,405 --> 01:47:48,891
>> when we get to the heart of the idea of up and down

1439
01:47:49,009 --> 01:47:52,342
>> are along a single axis, one pointing this way one pointing that way

1440
01:47:52,467 --> 01:47:54,217
>> Why they would be orthogonal now?

1441
01:47:56,219 --> 01:47:59,332
Orthogonal doesn't mean perpendicular in three-dimensional space

1442
01:47:59,451 --> 01:48:01,179
It means physically distinguishable

1443
01:48:01,179 --> 01:48:05,851
It means unambiguously distinguishable 

1444
01:48:05,969 --> 01:48:11,028
Left and right are not distinctly distinguishable uniquely from up and down

1445
01:48:11,164 --> 01:48:18,183
Why? Because if you have a thing oriented along left and you measure up and down

1446
01:48:18,290 --> 01:48:19,475
You might get up

1447
01:48:19,596 --> 01:48:27,127
>> So you think of two orthogonal axis being two independent variables

1448
01:48:27,243 --> 01:48:28,729
>> you could vary independantly?

1449
01:48:28,729 --> 01:48:35,518
Yes, that's right, you could vary independently the probability of up and down

1450
01:48:35,638 --> 01:48:39,255
Of course they have to add up to one, alright?

1451
01:48:39,385 --> 01:48:41,325
They have to add up to one, 

1452
01:48:41,453 --> 01:48:44,755
but ignoring the question of whether they add up to one

1453
01:48:44,920 --> 01:48:51,225
You have the freedom to independently vary the coefficient in terms of u

1454
01:48:51,325 --> 01:48:53,439
The coefficient in term of the d

1455
01:48:53,439 --> 01:48:59,809
And in fact they have to add up to one, it is a fact

1456
01:49:01,152 --> 01:49:05,939
Don't confuse orthogonality in space of states

1457
01:49:05,939 --> 01:49:13,023
with orthogonality in three-dimensional space

1458
01:49:14,039 --> 01:49:23,753
The analog idea in phase space or in configuration space of classical system

1459
01:49:24,962 --> 01:49:30,133
Orthogonality of two probability distribution

1460
01:49:31,456 --> 01:49:35,135
Let's think of, suppose you have phase space

1461
01:49:35,259 --> 01:49:39,065
Or configuration space of some classical system, the phase space

1462
01:49:40,280 --> 01:49:42,824
And I give you a probability distribution

1463
01:49:42,957 --> 01:49:47,659
A simple probability distribution might be zero everywhere except for some patch

1464
01:49:47,659 --> 01:49:51,761
Where the probability is uniform

1465
01:49:51,902 --> 01:50:01,664
Okay, the analog of orthogonality between two distinct probability distribution

1466
01:50:01,831 --> 01:50:06,720
is that they don't overlap, they are non-overlapping 

1467
01:50:09,823 --> 01:50:12,111
There is no ambiguity

1468
01:50:14,479 --> 01:50:18,847
That's the analog of orthogonality

1469
01:50:19,001 --> 01:50:21,919
Up and down don't relap

1470
01:50:22,050 --> 01:50:24,031
If you are up, not down, if you are down, not up

1471
01:50:25,828 --> 01:50:30,431
Now if you think about, there is a situation 

1472
01:50:30,572 --> 01:50:37,167
in which orthogonality is the same in space and in the space of states

1473
01:50:37,306 --> 01:50:38,981
It's the polarization of the photons

1474
01:50:40,177 --> 01:50:43,035
Polarization of photons is a little bit different 

1475
01:50:43,180 --> 01:50:46,660
We will get to it because it's a little more complicated

1476
01:50:48,492 --> 01:50:57,462
A little better if we have different word for perpendicularity in space

1477
01:50:57,599 --> 01:51:02,004
Or orthogonality in space of state, different concepts

1478
01:51:06,909 --> 01:51:11,348
I mean, it's very much what's going on here that 

1479
01:51:11,497 --> 01:51:25,588
Being oriented along right, can I do an experiment

1480
01:51:25,716 --> 01:51:31,689
which can distinguish between a spin pointing up and a spin pointing down?

1481
01:51:32,897 --> 01:51:36,491
Yes, you can bring your apparatus over to it, detect it

1482
01:51:36,645 --> 01:51:41,381
If it's up, it will point one, if it's down, it will point minus one

1483
01:51:41,484 --> 01:51:42,837
It will give you a minus one

1484
01:51:43,008 --> 01:51:47,349
Can you use that apparatus to tell you uniquely the difference 

1485
01:51:47,349 --> 01:51:53,449
between a spin pointing in the x axis or the spin pointing along the z axis?

1486
01:51:53,606 --> 01:51:56,515
What would you do to tell the difference?

1487
01:51:56,649 --> 01:51:58,573
What would you measure?

1488
01:51:58,717 --> 01:52:03,229
No matter what you measure, it's not gonna uniquely tell you the difference

1489
01:52:03,229 --> 01:52:08,086
If you measure sigma z, you will get plus one, 

1490
01:52:08,213 --> 01:52:10,668
but it doesn't tell you anything about sigma x

1491
01:52:10,804 --> 01:52:16,252
It doesn't, you could have gotten sigma z equals to plus one because

1492
01:52:16,383 --> 01:52:21,500
it was pointing along the z axis and you got a certainty one

1493
01:52:21,655 --> 01:52:26,759
Or it was pointing along the x axis with a probability of a half of pointing up

1494
01:52:26,759 --> 01:52:30,245
There is no way to tell

1495
01:52:30,377 --> 01:52:33,223
There is no way to tell by a single, well-defined experiment

1496
01:52:33,377 --> 01:52:38,199
Where the spin was pointing, let's say, up, or out

1497
01:52:39,654 --> 01:52:45,264
>>What if you measure the photon can tell you of that?

1498
01:52:45,264 --> 01:52:50,648
With one experiment of photon either will it come on and off

1499
01:52:50,804 --> 01:52:53,020
From that you can not tell uniquely

1500
01:52:53,187 --> 01:52:59,745
No you can not tell

1501
01:52:59,896 --> 01:53:02,727
That's exactly the same as measuring the spin

1502
01:53:02,877 --> 01:53:05,600
If you get up in your measurement 

1503
01:53:05,750 --> 01:53:09,304
If does not tell you whether it's initially up 

1504
01:53:09,304 --> 01:53:13,471
Or was initially side way with a probability half up

1505
01:53:13,609 --> 01:53:20,672
There is no unique single experiment you can do to distinguish this from this

1506
01:53:20,778 --> 01:53:24,694
There is a experiment you can do to distinguish this from this

1507
01:53:24,694 --> 01:53:28,763
That's what makes up and down orthogonal 

1508
01:53:28,923 --> 01:53:31,267
But down and left not orthogonal

1509
01:53:31,400 --> 01:53:34,205
They overlapped, they overlapped

1510
01:53:34,333 --> 01:53:38,061
And in fact the probability that if you were in left

1511
01:53:38,192 --> 01:53:40,701
That you measure down when you measure sigma z

1512
01:53:40,829 --> 01:53:44,096
That's a measure of the overlap between them

1513
01:53:46,380 --> 01:53:50,529
They are overlapped which means in technical language they are not orthogonal

1514
01:53:53,354 --> 01:54:02,875
>> You originally said you didn't know it was perpendicular to upper z, 

1515
01:54:02,876 --> 01:54:06,314
>>and we measured that out

1516
01:54:06,314 --> 01:54:11,756
What we are trying to do is confirm the proposition which has been up 

1517
01:54:11,892 --> 01:54:18,535
Or the z component of the spin is up or the x component of the spin is up

1518
01:54:19,731 --> 01:54:22,142
That's a single proposition, it's not two proposition

1519
01:54:22,287 --> 01:54:30,356
It's been up, or it's been up along z, or it's up along x

1520
01:54:30,463 --> 01:54:34,753
It's a single proposition if the first detection 

1521
01:54:34,870 --> 01:54:37,165
measured the spin up along the z axis, we are finished

1522
01:54:46,446 --> 01:54:51,708
And since we set up before hand, we know in that case we would did

1523
01:54:51,838 --> 01:54:53,649
At the right hand side

1524
01:54:54,790 --> 01:54:55,455
Okay

1525
01:54:55,592 --> 01:55:02,956
>> Would it be right to summarize that propositions are not commutative?

1526
01:55:04,814 --> 01:55:15,529
Uh, yeah, propositions technically correspond to projection operators

1527
01:55:15,652 --> 01:55:17,433
Projection operators are not commutative

1528
01:55:19,689 --> 01:55:28,009
It would be proper to summarize that, it's part of the point you know

1529
01:55:28,139 --> 01:55:30,987
Depends on which order you

1530
01:55:31,953 --> 01:55:34,321
>> So you caution us of trying to visualize 

1531
01:55:36,481 --> 01:55:42,025
>> So is the sigma along the z axis or on x axis

1532
01:55:42,162 --> 01:55:43,969
>> Isn't that a visualization?

1533
01:55:44,094 --> 01:55:47,849
It is, it is, it is, it is ,it is

1534
01:55:52,889 --> 01:56:00,553
Well ,you know, yeah, you know it is , it is a visualization

1535
01:56:01,649 --> 01:56:08,137
Do keep in mind though this interesting fact that on the average

1536
01:56:09,097 --> 01:56:13,865
The spins, if you measure many of them, many many of them,

1537
01:56:14,000 --> 01:56:21,129
The average value do behave like a component of a little vector

1538
01:56:21,254 --> 01:56:26,537
In particular if you take a large number of these things

1539
01:56:26,670 --> 01:56:29,929
And you make sure if they all pointing upward

1540
01:56:30,064 --> 01:56:35,177
And then you measure some oblique component of them

1541
01:56:35,300 --> 01:56:42,249
with your apparatus being tilted, then the average that you will get

1542
01:56:42,386 --> 01:56:49,385
for the oblique component would just be the projection of upright vector

1543
01:56:49,536 --> 01:56:51,256
on to the oblique direction

1544
01:56:51,256 --> 01:56:53,665
And exactly as if you are measuring the component of a vector

1545
01:56:53,784 --> 01:57:00,881
So the upshot is, if you have a large number of them

1546
01:57:00,998 --> 01:57:10,156
In many cases you can visualize with classical physics, after all

1547
01:57:10,156 --> 01:57:14,295
Classical physics is still to large extend the approximation of systems

1548
01:57:14,450 --> 01:57:18,220
which consists of many many parts which all behave 

1549
01:57:19,348 --> 01:57:21,896
simultaneously the same way

1550
01:57:21,896 --> 01:57:34,727
>> More question, is there any properties which describe this unique 

1551
01:57:34,870 --> 01:57:39,859
>>relationship between that (inaudible)

1552
01:57:40,072 --> 01:57:43,156
well, I suppose, I suppose there is

1553
01:57:43,286 --> 01:57:45,715
You better be more specific then that

1554
01:57:50,081 --> 01:57:54,663
Eigen invariants does have to do with phase factor here

1555
01:57:54,771 --> 01:57:57,060
Well it does have to do with them

1556
01:57:57,195 --> 01:57:58,906
But it does have to do with them

1557
01:57:59,036 --> 01:58:05,128
But we can't get into that right now

1558
01:58:06,424 --> 01:58:10,392
>> Things which get me confused about 

1559
01:58:10,526 --> 01:58:13,944
>> calling the measurement of the (inaudible), in classical sense that

1560
01:58:14,084 --> 01:58:25,132
>> but it's not like, you can't really say what value before we do the measure

1561
01:58:25,264 --> 01:58:27,486
>> we can only say what it is after you do the experiment

1562
01:58:27,604 --> 01:58:32,590
but then, once we measure it, we can measure it again

1563
01:58:32,738 --> 01:58:35,366
And the same thing again, that will be consistency

1564
01:58:35,507 --> 01:58:40,483
>>Put it in a certain state that we will

1565
01:58:40,615 --> 01:58:44,674
>> If we get the same exact preparation over and over again

1566
01:58:44,811 --> 01:58:46,266
>> it will give you a distribution

1567
01:58:47,658 --> 01:58:53,193
That's right, you can't always think that a system has a property

1568
01:58:53,329 --> 01:58:55,322
It's just your job to find out what it is

1569
01:59:00,018 --> 01:59:03,498
For more, please visit us at Stanford.edu
