qm3.eng.srt

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stanford university

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Alright, well

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A mathematical interlude we'll begin with

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Mathematical interludes again about linear algebra about vector spaces

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but the idea of operators

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But before we do I want to let it before we get operators

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I want say a few more things about vectors

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A few more bits about the mathematics of vectors

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Most of these bits have to do not so much

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with deep mathematics as with good notation

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Good notation is worth than offer a lot

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when you can just manipulate symbols in ways

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uh... that are sort of prearranged

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and uh, do it easily and comfortably

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there's an enormous benefit in that

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when doing abstract mathematics certainly

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but also the abstractions a mathematical abstractions of physics

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The abstractions we are going to talk about

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were largely due to Dirac, Paul Dirac

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who really was the one

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who saw into the way quantum mechanics really fits together

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So let's talk about that

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We have a space of states, the space of states 

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which will come back to is as I said a linear vector space

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meaning to say you can multiply states by numbers

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to get new states is abstract vectors

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abstract vectors

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maybe I uh, threw maybe I threw too much in

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when I say space of states, just the abstract vectors

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you can multiply them by numbers in our case complex numbers 

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and you can add them

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and there are two kinds of vectors, they are bra vectors and ket vectors

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They are roughly speaking related by complex conjugation

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uh, and a vector space has a dimensionality

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The dimensionality is the maximum number of orthogonal vectors

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that you can find in that space

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And once you know what the dimension of spaces is

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you can look for a basis of vectors

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The basis of vectors is a mutually orthogonal collection of normalized

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get the terminology down

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orthonormal basis

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Ortho means that they are all orthogonal to each other

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normal means normalized

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And normalized simply means that the vectors are of unit length

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A collection of them that is maximal

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in other words that you can't find any more

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because there are no more directions left

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That defines a basis and the number of them is equal to the

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uh, the dimensionality of space

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Given such a basis, you can make

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you can write any vector in the space as some kind of sum

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over those basis vectors

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So we can write for any vector A, 

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that is equal to sum 

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over i, i here stands for the basis vectors

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Of course a basis is not a unique thing

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just as there are many sets of mutually orthogonal vectors

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in three-dimensional space

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There are many sets of mutually orthogonal vectors 

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in an abstract vector space, but we pick one

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We pick a set, set and we pick that set and we label i 

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we can write any vector 

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as a sum alpha of i exactly these alphas which uh,

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refer a moment to go

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Alpha of i is a set of complex numbers is a sort of complex coefficients

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times the ith basis vector

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The ith base vector uh

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are labeling i

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This is a ket representation

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But now all we can do with this 

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is we can take the inner product with it

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I want to try to calculate these alphas of i's 

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in terms of some quantities involving the vectors themselves

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Alright so what we do

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is we take the inner product of both sides of the equation

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with the basis vector j, the ket vector j

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On the left hand side we get the inner product 

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of the vector A which is the thing we're trying to express

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and that's equal to the sum over i, 

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alphas of i, times the inner product of the vector j

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with the vector i

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But the inner product of vector j with this vector i 

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is either one or zero

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It's one if i and j are the same vector 

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and zero otherwise

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Why? Because they are orthogonal if i is not equal to j 

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and then normalized

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namely of unit length if i does equal j

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So this bracket over here, this product bras and kets

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is just a Kronecker symbol delta j k

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sorry, delta j i

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zero or one

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and when you do sum

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and it just picks out one and only one contribution

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the contribution in which i is equal to j

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so on the right hand side, you just get alpha j

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No sum

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Sum is collapsed to a one term

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and that's now equal to j A

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So what we learned?

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we've learned that the coefficient here

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in the expansion of any vectors is

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just the inner product of the jth vector

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with  the target vector, the vector we're trying to describe

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uh, the fact that I wrote a j there is irrelevant

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It could also be alpha i

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It doesn't matter which i or which j

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ith coefficient is in the product

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now I can use that

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to rewrite the sum up here

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I now know what the alphas of i are

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for these guys over here

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so let's rewrite this

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sum over i

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I'm going to start with alpha i

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sorry with i

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and then write alphas of i

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I'm going to write alphas of i next to it over here

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in the opposite order

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That's, that's allowed

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nothing uh... or nothing

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non commuted at this stage

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now I want to write alphas of i

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so I write i A

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that's a basic formula, that any vector

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can be rewritten in terms of its

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coefficient or in terms of its inner product of the basis vectors

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times the basis vectors themselves, summed over i

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It's kind of a pretty formula

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and it comes back over and over again

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what it says is whenever you see this summation

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of i times i in that form

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you can sort of throw it away

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It's just A equals A

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But it's an expression for a vector in terms of its components

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I saw that so one simple fact

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the same thing is true for bra vectors

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exactly the same thing is true for bra vectors

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and you would write it in the following form

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a bra vector, there's also a sum over i

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of the inner product of bra vector with the basis vectors

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times the basis bra vectors

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The basis bra vectors are just complex conjugates

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if you like their basic basis, ket vectors

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both of these equations are true

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and we're gonna find out that they were enormously useful

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they were enormously useful

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and powerful

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even though they were very very simple

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all right now we come to the notion of linear operators

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states in quantum mechanics are vectors

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We thought a little bit about how that works

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in the case of a single spin

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observables, observables mean the things that we measure

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the things that we measure

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the objects that we measured the quantity that we measure

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the observables are related to linear operators

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in the space

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So what is a linear operator

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and it'll take us the rest of the evening

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to understand in the detail what has to do with the observables

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So put that out of your head for the moment

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we just doing a mathematical interlude now

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what is the notion of a linear operator

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A linear operator is a process if you like

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which you apply to vectors

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Somebody or other I can't remember

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who might've been John Wheeler

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likes to describe these things as machines

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You would say a linear operator

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is a machine with two little ports in it

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into one port you put in an input vector

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and then you turn the wheel and outcomes an output vector

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So it's a thing which acts on an input vector

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to give an output vector

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so let's uh... call M a linear operator

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Linear will come to, so far we're just talking about operator

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a machine in goes one in outcomes another one

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and we'll talk about what means to be linear in a minute

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so M is some operator and that means

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that given any vector when M acts on it

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It's simply gives another vector, unique

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Given any A, it gives a unique B

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That does not mean

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the converse may not be converse

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opposite may not be the true, may not be

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Uh, that given a B on this side

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that there's only one unique A

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that gives rise to that B

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But it is true that M applies to any vector A

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gives an unique reflection of A

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which is the process M acting on A

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Now what does it mean for M to be a linear operator

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Well, first of all, M can act on Z times A

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With Z times A is a vector

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That's right, that's right

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M can act on Z times A

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Z times A is itself a vector

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Because you're allowed to multiply vectors

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by complex numbers

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Z is a complex number

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The rule about linear operators is

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if they apply on constants times vector

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they just give back the constant

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times what they would have given on A

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I thought it just means that Z, a complex number

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a complex number can be brought through M

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M acting on twice A gives twice

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what M would give on A

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M acting on three times A gives three times

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what M gives on A

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M acting on i times on A, a complex number, i

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just gives i times M would give on A

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That's the first rule about linearity

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And the second rule about linearity is

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that if M acts on the sum of two vectors, A plus B

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then what you get is just the sum

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of what the machine spit out for A

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and what the machine spit out for B

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What the machine spits out when you put in A plus B

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is just the sum of what comes out

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M times A plus M times B

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and that's it, that's the notion of a linear operator

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No more to it than that

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At least that's the full set of rules about linear operators

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Now that's even more can be concrete about them

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You're going to manipulate with them

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Let's suppose that M times A equals B

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Here we reduce B as a piece of the input

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But now B is the output

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just letters I can use in any way I want

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M times A equals B

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I am interested in the component of B along the axis i

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but that I mean

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these objects over here the alphas

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They are the components of the vector A

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along the various basis vector directions

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So alphas of i's of, the components of the vectors

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and I'm interested in the components

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of the output vector

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I would like to know in a concrete way

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what are the components of B

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To find out, I just project them onto the direction i

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A M i

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All I've done is in both sides of the equation project to

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project this, put the inner product with the vector i

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This by definition is beta i

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Another words if I were to expand B

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in the same way I expanded A

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I would use the coefficient beta i

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That's what we learned up on the top board

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This is beta i

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How about here?

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What can I do with this rewrite things in terms of components

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I want to rewrite everything in terms of components

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Actually become Eric Mearre? Operations that you can do

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Well, here is where we use this trick

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Whenever you see a vector A, you can substitute for it

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A sum

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Let's do that right in here

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where A is substituted i

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I haven't substituted it yet

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Now, the summation variables, not i in this case

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i have used as the external non summation variable here

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So let's sum over j

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It doesn't matter what we call the summation variable

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j, A

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I've stuck in for A, the sum over components

269
00:15:22,368 --> 00:15:23,976
times the jth vector

270
00:15:24,125 --> 00:15:27,108
times the jth basis vector

271
00:15:27,249 --> 00:15:33,158
But now, A times j goes to the top again

272
00:15:33,245 --> 00:15:37,900
That's just alpha j

273
00:15:38,047 --> 00:15:40,924
So we now have a formula

274
00:15:41,067 --> 00:15:43,490
completely in terms of components

275
00:15:43,635 --> 00:15:47,688
This is i M j

276
00:15:47,792 --> 00:15:50,301
I'll get to this in a minute but it's a number

277
00:15:50,373 --> 00:15:52,762
Inner products are numbers

278
00:15:52,870 --> 00:15:54,747
M times j is a vector

279
00:15:54,853 --> 00:15:57,424
and you can take it to the inner product to the i

280
00:15:57,613 --> 00:15:59,630
M times j is a vector

281
00:15:59,739 --> 00:16:00,599
and vectors

282
00:16:00,744 --> 00:16:04,528
ket vectors have inner products with basis vectors

283
00:16:04,684 --> 00:16:13,753
times j A, or times alpha j

284
00:16:13,861 --> 00:16:19,927
That's equal to beta i

285
00:16:20,067 --> 00:16:24,898
So all we have to do is give this object a name

286
00:16:25,038 --> 00:16:26,798
What is this object?

287
00:16:26,943 --> 00:16:28,460
just uh... tell you again

288
00:16:28,602 --> 00:16:31,580
You have some operator with abstract entity

289
00:16:31,687 --> 00:16:34,022
You have a collection of basis vectors

290
00:16:34,130 --> 00:16:39,492
You apply the abstract operator to the vector j

291
00:16:39,601 --> 00:16:41,128
and take its  inner product with i

292
00:16:41,239 --> 00:16:44,522
and call this thing m i j

293
00:16:44,666 --> 00:16:45,743
It's a number

294
00:16:45,846 --> 00:16:48,309
and maybe a complex number

295
00:16:48,466 --> 00:16:50,329
But it's a number

296
00:16:50,435 --> 00:16:55,777
Another words, it's possible to characterize linear operators

297
00:16:55,882 --> 00:17:00,753
by what we called them matrix elements

298
00:17:00,861 --> 00:17:06,322
These are called matrix elements of the linear operator M

299
00:17:06,464 --> 00:17:10,041
This times alpha js, summed over j

300
00:17:10,184 --> 00:17:12,605
I left out the sum over j here

301
00:17:12,758 --> 00:17:14,594
Let's put back in

302
00:17:14,747 --> 00:17:17,334
sum over j is equal to beta i

303
00:17:17,477 --> 00:17:22,946
If somebody hands you the collection of them i js

304
00:17:23,061 --> 00:17:27,062
and somebody hands you a vector in the form of its components

305
00:17:27,166 --> 00:17:30,749
you can then start sweeping through it

306
00:17:30,817 --> 00:17:34,012
and calculating what the components of the output are

307
00:17:34,085 --> 00:17:37,267
in terms of the input

308
00:17:37,375 --> 00:17:40,333
m i j is a matrix

309
00:17:40,439 --> 00:17:42,189
It has two indices

310
00:17:42,260 --> 00:17:44,365
It's a square matrix

311
00:17:44,436 --> 00:17:47,255
If the dimensionality of the spaces n

312
00:17:47,364 --> 00:17:50,259
then m is an n way

313
00:17:50,365 --> 00:17:55,384
m is an n by n matrix, the square matrix

314
00:17:55,538 --> 00:17:56,967
It's a square matrix

315
00:17:57,105 --> 00:18:03,653
Alpha can be regarded as a column

316
00:18:03,794 --> 00:18:07,416
and the output is also a column

317
00:18:07,521 --> 00:18:09,865
It's a column because A was a ket vector

318
00:18:10,009 --> 00:18:11,722
and B was a ket vector

319
00:18:11,829 --> 00:18:14,914
I saw another way to write the same thing

320
00:18:14,985 --> 00:18:17,118
This is one way to write

321
00:18:17,189 --> 00:18:20,172
M times A equals B

322
00:18:20,314 --> 00:18:22,656
Well here's one way to write it

323
00:18:22,764 --> 00:18:25,644
Abstract is another way to write it as a concrete sum

324
00:18:25,753 --> 00:18:27,823
and the third ways to recognize that

325
00:18:27,902 --> 00:18:30,214
this is just the multiplication of a matrix

326
00:18:30,317 --> 00:18:32,737
times a column vector

327
00:18:32,880 --> 00:18:36,446
You're writing the matrix M

328
00:18:36,516 --> 00:18:40,027
by displaying all of its components

329
00:18:40,134 --> 00:18:46,173
m one one, m one two, m one three, dot dot dot

330
00:18:46,247 --> 00:18:52,268
m two one, m two two, dot dot dot

331
00:18:52,338 --> 00:18:54,524
This is just an array that exhibits

332
00:18:54,668 --> 00:18:56,893
the matrix elements will not learn the quality

333
00:18:57,040 --> 00:18:59,725
the matrix elements of m

334
00:18:59,868 --> 00:19:03,083
times the column vector A

335
00:19:03,189 --> 00:19:09,121
alpha one alpha two dot dot dot, alpha N

336
00:19:09,227 --> 00:19:13,790
and what is the rule this this of course is supposed to result

337
00:19:13,908 --> 00:19:20,241
an output, which is beta one beta two, dot dot dot, beta N

338
00:19:20,380 --> 00:19:23,471
What's the rule that's implied by this equation over here?

339
00:19:23,578 --> 00:19:24,996
It's very simple

340
00:19:25,109 --> 00:19:29,732
If you want, the first entry over here

341
00:19:29,839 --> 00:19:31,703
You take the first role

342
00:19:31,810 --> 00:19:35,058
and you take its inner product with a column here

343
00:19:35,163 --> 00:19:36,446
and m, not the inner product

344
00:19:36,518 --> 00:19:37,700
you can called it the inner product

345
00:19:37,807 --> 00:19:39,452
m one one alpha one

346
00:19:39,635 --> 00:19:41,360
plus m one two alpha two

347
00:19:41,503 --> 00:19:43,215
plus m one three alpha three

348
00:19:43,360 --> 00:19:46,120
right down to the bottom

349
00:19:46,235 --> 00:19:50,700
That's exactly this formula, m one two

350
00:19:50,808 --> 00:19:55,156
Sorry m one one alpha one plus m one two alpha two

351
00:19:55,295 --> 00:19:56,787
and so forth

352
00:19:56,896 --> 00:19:59,157
is exactly the standard rule

353
00:19:59,306 --> 00:20:03,396
for multiplying a matrix by a column vector

354
00:20:03,539 --> 00:20:05,855
So now we have a third way

355
00:20:05,995 --> 00:20:11,338
to represent the action of a matrix on a vector

356
00:20:11,450 --> 00:20:14,414
This is less abstract

357
00:20:14,532 --> 00:20:16,472
It's concrete if you know the matrix elements

358
00:20:16,594 --> 00:20:18,602
you know exactly what to do with them

359
00:20:18,743 --> 00:20:21,516
The only thing about it is it does depend on

360
00:20:21,655 --> 00:20:25,184
a particular choice of your basis vectors

361
00:20:25,373 --> 00:20:29,423
The specifics of the matrix elements here

362
00:20:29,565 --> 00:20:32,738
and the column vector will depend on your choice of

363
00:20:32,807 --> 00:20:36,852
Once you pick a basis, you stick with it

364
00:20:36,996 --> 00:20:39,284
But you could have chosen a different basis

365
00:20:39,418 --> 00:20:42,452
in which case these components would have been different

366
00:20:42,565 --> 00:20:45,153
The matrix elements would have been different

367
00:20:45,297 --> 00:20:48,213
After all the matrix elements are themselves

368
00:20:48,340 --> 00:20:51,142
related to the basis vectors

369
00:20:51,283 --> 00:20:54,136
and so you lose something and you gain something

370
00:20:54,242 --> 00:20:56,748
by going to a matrix type of presentation

371
00:20:56,857 --> 00:21:03,433
What you lose is the geometric idea of a relationship

372
00:21:03,543 --> 00:21:04,625
between two vectors

373
00:21:04,805 --> 00:21:07,642
That's independent of the basis choice

374
00:21:07,770 --> 00:21:09,764
What you gain is

375
00:21:09,945 --> 00:21:15,458
a computational actual mechanical algorithm

376
00:21:15,635 --> 00:21:18,644
for producing, for the machine itself

377
00:21:18,841 --> 00:21:19,571
Here is the machine

378
00:21:19,719 --> 00:21:21,122
The machine is a matrix

379
00:21:21,226 --> 00:21:23,054
and what it does is a grinning through

380
00:21:23,163 --> 00:21:25,350
m one one alpha one blah blah blah blah

381
00:21:25,450 --> 00:21:27,762
That goes to the next row does the thing again

382
00:21:27,837 --> 00:21:30,631
goes to the next row, does the thing again

383
00:21:30,741 --> 00:21:39,037
and just spits out the uh, final answer

384
00:21:39,177 --> 00:21:41,675
These things of course also true for ordinary vectors

385
00:21:41,818 --> 00:21:43,203
in three-dimensional space

386
00:21:43,309 --> 00:21:44,925
They are not specific uh,

387
00:21:45,071 --> 00:21:45,929
We tend not to use them

388
00:21:46,033 --> 00:21:49,072
as much as three dimensional ordinary vectors

389
00:21:49,178 --> 00:21:54,122
But they're quite, uh, essential in quantum mechanics

390
00:21:54,266 --> 00:21:54,519
Yes?

391
00:21:54,520 --> 00:21:56,513
Student: It's a little getting ahead

392
00:21:56,620 --> 00:22:00,427
Student: could be choice of basis that you make a different to

393
00:22:00,534 --> 00:22:03,060
Student: whether or not you can see the solution to a problem?

394
00:22:03,217 --> 00:22:08,886
Well, precisely you asked whether you can make a difference

395
00:22:08,994 --> 00:22:11,535
to the way you see through a problem

396
00:22:11,639 --> 00:22:13,486
Certainly a wrong choice of basis

397
00:22:13,629 --> 00:22:16,192
can leave you completely confused

398
00:22:16,299 --> 00:22:17,451
I think what you really meant is

399
00:22:17,557 --> 00:22:20,156
can different choices of basis lead to

400
00:22:20,274 --> 00:22:21,876
different physical answers

401
00:22:22,019 --> 00:22:23,749
and that had better not be

402
00:22:23,888 --> 00:22:25,790
Then you're doing something wrong

403
00:22:25,899 --> 00:22:27,690
Student: yeah, smart, (mumble)

404
00:22:27,870 --> 00:22:30,058
Yeah yeah it's often the case that uh

405
00:22:30,200 --> 00:22:31,733
that a convenient choice a basis will

406
00:22:31,877 --> 00:22:34,947
make your computations especially simple

407
00:22:35,093 --> 00:22:39,219
at a bad choice a basis might make them horribly complicated

408
00:22:39,363 --> 00:22:41,301
That's true but uh

409
00:22:41,479 --> 00:22:43,525
but at the end of the day

410
00:22:43,632 --> 00:22:46,327
the answers should not depend on it

411
00:22:46,472 --> 00:22:48,684
the result of experiment

412
00:22:48,829 --> 00:22:54,585
Okay so now we have, we have the idea of a linear operator

413
00:22:54,695 --> 00:23:02,053
Operating on several different versions of it

414
00:23:02,161 --> 00:23:03,000
Student: Question?

415
00:23:03,000 --> 00:23:03,084
Yeah

416
00:23:03,090 --> 00:23:06,140
Student: Is that general, it seems that

417
00:23:06,259 --> 00:23:11,089
Student: within a definite operator that gives you a ket from bra vector

418
00:23:11,237 --> 00:23:13,931
No, no, no

419
00:23:14,072 --> 00:23:17,683
Linear vector act on ket vector to give ket vectors

420
00:23:17,769 --> 00:23:19,614
Student: That's not the complex conjugation, not allowed

421
00:23:19,729 --> 00:23:20,067
That's right

422
00:23:20,176 --> 00:23:25,711
Complex, very good, let's, yeah, uh

423
00:23:25,781 --> 00:23:30,557
The simplest kind of linear operators

424
00:23:30,666 --> 00:23:33,657
is just multiplication by a complex number

425
00:23:33,771 --> 00:23:38,239
I'll multiply A by a complex number and get another vector

426
00:23:38,332 --> 00:23:39,770
Basically the same vector

427
00:23:39,880 --> 00:23:42,071
but twice as long or something

428
00:23:42,176 --> 00:23:44,361
That is a linear operator

429
00:23:44,471 --> 00:23:49,282
You might ask whether complex conjugation is a linear operator

430
00:23:49,398 --> 00:23:53,424
I take, I take the components of a vector

431
00:23:53,566 --> 00:23:55,837
and replace them by their complex conjugates

432
00:23:56,018 --> 00:23:58,583
The answer is no, is not

433
00:23:58,799 --> 00:24:02,107
Uh, and you can check the definitions

434
00:24:02,247 --> 00:24:04,239
multiplying the components of a vector,

435
00:24:04,350 --> 00:24:07,620
or not multiplying by complex conjugating is not

436
00:24:07,761 --> 00:24:11,055
What it does do is it takes a vector

437
00:24:11,199 --> 00:24:15,445
and it's a different kind of machine complex conjugation

438
00:24:15,633 --> 00:24:18,782
It's a machine which takes a bra vector to a ket vector

439
00:24:18,926 --> 00:24:20,870
or ket vector to a bra vector

440
00:24:20,978 --> 00:24:22,356
and that's a,

441
00:24:22,497 --> 00:24:25,708
strictly it's called anti-linear operator

442
00:24:25,804 --> 00:24:31,471
It's not anti-linear, it's just it's not the opposite of linear

443
00:24:31,595 --> 00:24:34,058
It's just a word

444
00:24:34,164 --> 00:24:36,460
Well, the reason to worry

445
00:24:36,461 --> 00:24:37,436
the reason to call it anti-linear is

446
00:24:37,543 --> 00:24:39,675
because the process of complex conjugation

447
00:24:39,785 --> 00:24:42,045
is in some deep way related to anti particles

448
00:24:42,148 --> 00:24:45,356
But we're not there yet

449
00:24:45,464 --> 00:24:52,836
Yeah, all right, now let's talk, let's talk about ket vector

450
00:24:52,949 --> 00:24:55,497
sorry bra vectors

451
00:24:55,601 --> 00:24:59,961
So far we have defined a linear operator

452
00:25:00,103 --> 00:25:03,658
to act only on ket vectors

453
00:25:03,797 --> 00:25:08,187
But every linear operator can also be given a definition

454
00:25:08,301 --> 00:25:10,973
We need a definition of course we haven't found it yet

455
00:25:11,117 --> 00:25:12,410
We need a definition

456
00:25:12,517 --> 00:25:16,497
But there is also a definition of any given linear operator

457
00:25:16,641 --> 00:25:18,976
acting on a bra vector

458
00:25:19,081 --> 00:25:24,756
Okay, let's see if we can concoct a definition

459
00:25:24,900 --> 00:25:29,451
and then we'll stick with that definition

460
00:25:29,523 --> 00:25:34,119
What might be the rules?

461
00:25:34,225 --> 00:25:37,735
But the action on that same linear operator

462
00:25:37,837 --> 00:25:41,347
when it acts on a bra vector

463
00:25:41,435 --> 00:25:43,175
First of all, the standard notation

464
00:25:43,278 --> 00:25:45,873
It's very very good to keep track of notation

465
00:25:46,097 --> 00:25:48,480
and to use the same notation of rules

466
00:25:48,596 --> 00:25:52,983
Now we need, but keeps things in order

467
00:25:53,162 --> 00:25:56,313
When you act on a bra vector

468
00:25:56,449 --> 00:25:58,967
put the corresponding operator on the right side of it

469
00:25:59,074 --> 00:26:01,758
Alright it gives something

470
00:26:01,896 --> 00:26:04,638
Take that something and take its inner product

471
00:26:04,746 --> 00:26:07,209
The bracket around indicate

472
00:26:07,351 --> 00:26:12,204
that M acting on A is itself an entities

473
00:26:12,345 --> 00:26:20,399
and now let's take its inner product with another vector B

474
00:26:20,545 --> 00:26:26,826
Well, if I removed the bracket here

475
00:26:26,967 --> 00:26:29,068
I would have a symbol

476
00:26:29,188 --> 00:26:30,681
which would look like this

477
00:26:30,682 --> 00:26:34,591
A M B, and I really wouldn't know

478
00:26:34,703 --> 00:26:38,450
if I am supposed to do is act with M on B

479
00:26:38,559 --> 00:26:41,593
and then take the inner product with A

480
00:26:41,738 --> 00:26:47,067
or act with M on A and then take the inner product with B

481
00:26:47,247 --> 00:26:51,035
The answer is with the standard definition

482
00:26:51,151 --> 00:26:57,322
of how linear operators act to the left on bra vectors

483
00:26:57,428 --> 00:26:59,510
It doesn't matter

484
00:26:59,651 --> 00:27:04,855
In another words, we define the action of an M on A to the left

485
00:27:04,927 --> 00:27:08,244
such that and I will show you we can always do this

486
00:27:08,348 --> 00:27:16,103
such that it doesn't matter

487
00:27:16,243 --> 00:27:19,447
which way we do it

488
00:27:19,625 --> 00:27:24,277
A M B is the same as A M B

489
00:27:24,384 --> 00:27:28,468
uh, that you could remove the brackets

490
00:27:28,610 --> 00:27:31,112
Ok, let's see that we can do this

491
00:27:31,285 --> 00:27:34,440
In order to say it, let's start with the definition this way

492
00:27:34,614 --> 00:27:37,125
Let's start with definition this way

493
00:27:37,272 --> 00:27:42,855
Ok, so with definition this way

494
00:27:42,994 --> 00:27:47,300
now I'm going to use that trick up on the top

495
00:27:47,444 --> 00:27:51,728
When you write a vector by inserting

496
00:27:51,833 --> 00:27:54,095
This is a name for this operation here

497
00:27:54,242 --> 00:27:57,808
It's called inserting a complete set of states

498
00:27:57,948 --> 00:28:02,704
complete simply means it's a basis

499
00:28:02,849 --> 00:28:05,339
It's called inserting a complete set of states

500
00:28:05,448 --> 00:28:06,178
It did nothing

501
00:28:06,315 --> 00:28:08,483
It took A, and rewrote A

502
00:28:08,592 --> 00:28:11,203
but rewrote A in terms of its components

503
00:28:11,342 --> 00:28:15,322
so let's do that over, on B over here

504
00:28:15,416 --> 00:28:16,477
We can do the same thing over here

505
00:28:16,575 --> 00:28:24,347
We write this is equal to B, j, j

506
00:28:24,490 --> 00:28:25,299
Summed over j

507
00:28:25,404 --> 00:28:27,308
I am not going to write the summations signs

508
00:28:27,417 --> 00:28:30,518
maybe I'll put them at the end out here

509
00:28:30,624 --> 00:28:32,801
Summation i and j

510
00:28:32,944 --> 00:28:37,082
So it's the same vector with B

511
00:28:37,223 --> 00:28:39,579
Then put M there

512
00:28:39,835 --> 00:28:42,293
bracket around it, put M there

513
00:28:42,401 --> 00:28:45,701
and I'll do exactly the same thing on the ket vector

514
00:28:45,795 --> 00:28:48,704
Remember the ket vector can also be expanded

515
00:28:48,848 --> 00:28:50,452
in the same way

516
00:28:50,561 --> 00:29:02,616
So here we go, A, i, i

517
00:29:02,718 --> 00:29:07,722
What we see here? We see, this is alpha star i

518
00:29:07,829 --> 00:29:12,684
Why that I put the star there?

519
00:29:12,765 --> 00:29:15,766
Because the bra vectors are complex conjugates

520
00:29:15,909 --> 00:29:21,752
alphas of i is an M i j

521
00:29:21,899 --> 00:29:25,799
i, inner product of i and times j

522
00:29:25,903 --> 00:29:31,154
and then there is beta j

523
00:29:31,261 --> 00:29:33,115
That's all we have here

524
00:29:33,229 --> 00:29:36,913
This is summed on i and j

525
00:29:37,129 --> 00:29:42,114
M with i on the left and j on the right that's a M i j

526
00:29:42,223 --> 00:29:44,844
j beta, alpha i

527
00:29:44,911 --> 00:29:46,088
and you can see from the way

528
00:29:46,161 --> 00:29:48,568
that we've written this that it doesn't matter

529
00:29:48,673 --> 00:29:51,786
if I first sum over j

530
00:29:51,928 --> 00:29:56,022
and think of M is acting on the ket vector beta

531
00:29:56,168 --> 00:29:57,983
or I sum over i

532
00:29:58,087 --> 00:30:01,969
and think of M as first acting to the left on the alpha

533
00:30:02,077 --> 00:30:03,559
you get the same answer

534
00:30:03,675 --> 00:30:06,419
It's just a double sum of i and j

535
00:30:06,539 --> 00:30:08,588
You have the same answer

536
00:30:08,692 --> 00:30:12,623
and so if you define

537
00:30:12,805 --> 00:30:19,114
Another words, you can define M in such a way

538
00:30:19,215 --> 00:30:23,289
that it doesn't matter which order or the way you put the brackets

539
00:30:23,392 --> 00:30:25,314
That easies your life

540
00:30:25,419 --> 00:30:26,585
You don't have to remember whether you meant

541
00:30:26,694 --> 00:30:30,511
M acts on B and then you take an inner product with A

542
00:30:30,655 --> 00:30:33,351
or M acts on A and you take the inner product with B

543
00:30:33,458 --> 00:30:35,074
Same thing

544
00:30:35,218 --> 00:30:43,029
Ok, but there is a something, there is, let's

545
00:30:43,135 --> 00:30:45,527
It's the power of a nice notation

546
00:30:45,670 --> 00:30:48,490
But now let's consider another question

547
00:30:48,607 --> 00:30:59,839
Supposing that a linear operator M on A gives B

548
00:30:59,978 --> 00:31:01,706
Incidentally amusing A and B

549
00:31:01,813 --> 00:31:05,349
all over the place in different ways

550
00:31:05,493 --> 00:31:07,309
There was just two vectors

551
00:31:07,524 --> 00:31:11,298
Here, A is the input vector, B is the output vector

552
00:31:11,415 --> 00:31:14,786
So let's suppose that the machine M

553
00:31:14,895 --> 00:31:20,592
when you stick A into it, out spits B

554
00:31:20,737 --> 00:31:27,165
We could ask what's the corresponding bra vector relationship

555
00:31:27,282 --> 00:31:30,817
Is there a linear operator which has the property

556
00:31:30,962 --> 00:31:36,853
that wanted acts on

557
00:31:36,997 --> 00:31:39,451
I'm not going to call it M yet

558
00:31:39,596 --> 00:31:41,188
It's not M

559
00:31:41,295 --> 00:31:44,123
Let's call it scripting M

560
00:31:44,232 --> 00:31:48,999
Is there for every M which acts on A to give B

561
00:31:49,109 --> 00:31:51,680
Is there corresponding scripting M

562
00:31:51,821 --> 00:32:01,665
which acts on the corresponding bra vector B

563
00:32:01,806 --> 00:32:05,714
to give the corresponding bra vector over here

564
00:32:05,857 --> 00:32:06,537
Well, yes, there is

565
00:32:06,645 --> 00:32:07,617
We are going to figure out

566
00:32:07,730 --> 00:32:10,422
how to figure out what it is soon enough

567
00:32:10,535 --> 00:32:14,338
But it is not the original M

568
00:32:14,477 --> 00:32:17,053
One example would be what if

569
00:32:17,193 --> 00:32:21,414
M was just multiplication by a complex number

570
00:32:21,551 --> 00:32:24,913
then what is scripting M B

571
00:32:25,059 --> 00:32:27,709
Multiplication by the complex conjugate number

572
00:32:27,853 --> 00:32:30,553
So whatever M is

573
00:32:30,694 --> 00:32:33,763
It's not necessarily the same as scripting M

574
00:32:33,910 --> 00:32:35,656
and it would be the same as scripting M?

575
00:32:35,801 --> 00:32:40,014
If it was multiplication by a real number

576
00:32:40,192 --> 00:32:42,525
but it was multiplication by a complex number

577
00:32:42,665 --> 00:32:44,329
then in general

578
00:32:44,472 --> 00:32:48,148
generally it will not give the same relationship

579
00:32:48,294 --> 00:32:51,272
We would normally just put a complex conjugate sign

580
00:32:51,411 --> 00:32:53,935
if we want to operate in the other direction

581
00:32:54,055 --> 00:32:58,555
Alright the notation that is standard,

582
00:32:58,669 --> 00:33:05,603
this to say that the operator which one acting to the left

583
00:33:05,744 --> 00:33:07,694
is the reflection if you like

584
00:33:07,801 --> 00:33:11,844
it's the dual of the operator M acting to the right

585
00:33:11,989 --> 00:33:14,311
does not called M

586
00:33:14,456 --> 00:33:18,660
It is called M dagger

587
00:33:18,771 --> 00:33:26,636
The dagger is an operator version of complex conjugation

588
00:33:26,743 --> 00:33:30,457
It's an operator version of complex conjugation

589
00:33:30,563 --> 00:33:33,131
or matrix version of complex conjugation

590
00:33:33,319 --> 00:33:35,308
It's called Hermitian conjugation about

591
00:33:35,486 --> 00:33:38,757
We'll get to these fancy words soon enough

592
00:33:38,935 --> 00:33:41,591
Well, ok, this is abstract

593
00:33:41,771 --> 00:33:45,998
but let's see if we can get to a concrete notion

594
00:33:46,105 --> 00:33:50,080
M i j, the abstract operator

595
00:33:50,192 --> 00:33:52,975
Is equivalent to a concrete matrix

596
00:33:53,083 --> 00:33:55,540
if I give you the concrete matrix

597
00:33:55,681 --> 00:33:57,319
that I'm giving you M

598
00:33:57,426 --> 00:34:00,275
The question is in terms of that concrete matrix

599
00:34:00,382 --> 00:34:03,610
is there a concrete matrix for M dagger?

600
00:34:03,751 --> 00:34:04,946
The answer is yes

601
00:34:05,058 --> 00:34:05,972
How do you find it?

602
00:34:06,116 --> 00:34:09,023
We find it just by proceeding dead ahead

603
00:34:09,131 --> 00:34:16,777
I would like to know, what the matrix elements

604
00:34:16,958 --> 00:34:27,060
let's see which way I did it

605
00:34:27,165 --> 00:34:35,327
Yeah, this is what I would call

606
00:34:35,518 --> 00:34:38,464
the i j th matrix element of M dagger

607
00:34:38,569 --> 00:34:39,691
I don't know what M dagger is yet

608
00:34:39,800 --> 00:34:40,258
I have a suspicion

609
00:34:40,259 --> 00:34:43,305
that something to do with complex conjugates

610
00:34:43,444 --> 00:34:46,225
But this is its matrix elements

611
00:34:46,365 --> 00:34:48,375
concrete set of numbers

612
00:34:48,520 --> 00:34:52,421
Can we find out what they are?

613
00:34:52,566 --> 00:34:56,002
Alright, yeah I think I have to step back for a minute

614
00:34:56,139 --> 00:35:02,032
Let me suppose that M acts on the basis vector

615
00:35:02,141 --> 00:35:05,868
to give i prime

616
00:35:06,012 --> 00:35:09,095
i prime is just the result of sticking the basis vector

617
00:35:09,206 --> 00:35:12,519
into the machine, into the M machine

618
00:35:12,628 --> 00:35:13,724
and this is what it comes out

619
00:35:13,836 --> 00:35:16,218
let's suppose that's the result

620
00:35:16,364 --> 00:35:23,334
Now let's take j M i

621
00:35:23,477 --> 00:35:27,932
That's equal to j i prime

622
00:35:28,028 --> 00:35:31,321
All I've done is taking inner product with j

623
00:35:31,389 --> 00:35:35,820
Here's the matrix element of the matrix M

624
00:35:35,896 --> 00:35:37,490
and it's just this inner product that

625
00:35:37,597 --> 00:35:38,977
that was a trivial operation

626
00:35:39,124 --> 00:35:41,168
I didn't do anything, Ok?

627
00:35:41,310 --> 00:35:45,744
I didn't do anything very significant

628
00:35:45,885 --> 00:35:51,973
If this is true then by definition

629
00:35:52,110 --> 00:35:59,277
then by definition, M dagger acting to the left

630
00:35:59,421 --> 00:36:06,007
What does M dagger acting to the left give on i?

631
00:36:06,153 --> 00:36:07,766
Ok, look at this equation here

632
00:36:07,874 --> 00:36:14,107
M dagger acts to the left gives the bra vector B

633
00:36:14,252 --> 00:36:17,202
So I thought that all the dagger does is

634
00:36:17,380 --> 00:36:19,362
that it turns equation over

635
00:36:19,501 --> 00:36:23,011
So what does M dagger acting on i give?

636
00:36:23,120 --> 00:36:26,165
It gives the bra vector i prime

637
00:36:26,307 --> 00:36:29,363
If M dagger acts on the bra i

638
00:36:29,514 --> 00:36:32,410
it must give the bra vector i prime

639
00:36:32,558 --> 00:36:35,593
It just flips the equation from bras to kets

640
00:36:35,735 --> 00:36:43,919
So we have over here then is i prime j

641
00:36:44,067 --> 00:36:45,288
That's the matrix element

642
00:36:45,397 --> 00:36:55,818
that's this matrix element over here

643
00:36:55,962 --> 00:36:58,647
Okay, so let's look at what we have

644
00:36:58,795 --> 00:37:03,813
We have a matrix element of M is the inner product of j with i prime

645
00:37:03,955 --> 00:37:07,914
The matrix element of M dagger

646
00:37:08,020 --> 00:37:09,629
is the inner product of i prime with j

647
00:37:09,736 --> 00:37:13,284
What's the relationship between these two?

648
00:37:13,384 --> 00:37:18,816
Student: (mumble) complex conjugate

649
00:37:18,897 --> 00:37:20,726
just complex conjugate

650
00:37:20,866 --> 00:37:26,768
If you take two vectors, A and B

651
00:37:26,877 --> 00:37:30,583
and you think of them in the opposite order

652
00:37:30,652 --> 00:37:34,392
The bra A times the ket B versus the bra B times the ket A

653
00:37:34,473 --> 00:37:38,325
the relationship between them is just complex conjugation

654
00:37:38,432 --> 00:37:43,653
So this here is just a complex conjugate of this

655
00:37:43,792 --> 00:37:48,410
So what we've derived is that the matrix element

656
00:37:48,552 --> 00:37:51,885
the i j th matrix element of M dagger

657
00:37:51,994 --> 00:38:07,582
is the complex conjugate of the j i th matrix element of M

658
00:38:07,687 --> 00:38:11,506
So if I know the matrix elements of M

659
00:38:11,648 --> 00:38:15,117
What do I do to get the matrix elements of M dagger

660
00:38:15,221 --> 00:38:16,791
It's very simple

661
00:38:17,013 --> 00:38:20,411
You complex conjugate but that's not quite enough

662
00:38:20,515 --> 00:38:27,282
you complex conjugate and you interchange i s and j s

663
00:38:27,390 --> 00:38:31,112
you interchange i and j, ok?

664
00:38:31,260 --> 00:38:36,591
Another way to write it is that M dagger i j

665
00:38:36,700 --> 00:38:44,751
is equal to M j i star

666
00:38:44,890 --> 00:38:45,998
So now we know a lot

667
00:38:46,139 --> 00:38:52,349
We know how to construct operators which act to the left

668
00:38:52,451 --> 00:38:55,813
to do the same mapping

669
00:38:55,916 --> 00:38:58,560
that they did when I went to the right

670
00:38:58,743 --> 00:39:02,426
The bra A to the bra B

671
00:39:02,568 --> 00:39:07,555
if the original operator do the same thing to the kets

672
00:39:07,697 --> 00:39:11,334
Alright that's the notion of Hermitian conjugation

673
00:39:11,441 --> 00:39:15,367
This is called Hermitian conjugation, The word Hermitian

674
00:39:15,471 --> 00:39:18,584
is named after some mathematician

675
00:39:18,696 --> 00:39:22,362
whose name was Hermite

676
00:39:22,470 --> 00:39:28,860
and Hermitian conjugation is the process of

677
00:39:28,946 --> 00:39:32,456
interchanging i and j and taking its complex conjugate

678
00:39:32,599 --> 00:39:35,635
in terms of matrices

679
00:39:35,710 --> 00:39:38,223
If I have a matrix

680
00:39:38,327 --> 00:39:40,320
this is an easy way to remember it

681
00:39:40,428 --> 00:39:43,105
M one one, M one two, dot dot dot

682
00:39:43,211 --> 00:39:48,668
M two one, M two two, dot dot dot

683
00:39:48,811 --> 00:39:53,730
If that's the matrix representing M

684
00:39:53,801 --> 00:39:56,111
then the matrix representing its conjugated

685
00:39:56,217 --> 00:39:58,057
its Hermitian conjugate is

686
00:39:58,152 --> 00:40:01,666
to simply interchange rows and columns

687
00:40:01,738 --> 00:40:06,163
That means reflecting it about the diagonal

688
00:40:06,270 --> 00:40:09,586
and then complex conjugating

689
00:40:09,693 --> 00:40:13,442
interchanging i and j

690
00:40:13,517 --> 00:40:17,335
let's interchange m one two with M two one

691
00:40:17,446 --> 00:40:20,660
you just flip it about the diagonal

692
00:40:20,769 --> 00:40:23,976
and you complex conjugated

693
00:40:24,045 --> 00:40:29,189
So if I want the matrix elements of the Hermitian conjugate

694
00:40:29,270 --> 00:40:31,312
Well, I can't say any more

695
00:40:31,417 --> 00:40:32,991
you just do what I said

696
00:40:33,059 --> 00:40:35,493
You interchange rows and columns

697
00:40:35,566 --> 00:40:37,126
and you complex conjugate

698
00:40:37,240 --> 00:40:39,442
Interchanging rows and columns is the same thing

699
00:40:39,550 --> 00:40:43,759
is reflecting about the diagonal

700
00:40:43,802 --> 00:40:46,689
You reflect about the diagonal

701
00:40:46,800 --> 00:40:47,994
Let's do it

702
00:40:48,104 --> 00:40:50,510
and then you complex conjugate

703
00:40:50,617 --> 00:40:52,406
Let me give an example

704
00:40:52,508 --> 00:40:55,544
Supposing I have the matrix

705
00:40:55,656 --> 00:41:02,333
Two, six plus i times seven

706
00:41:02,439 --> 00:41:07,838
Four minus i and nine, nine plus i

707
00:41:07,948 --> 00:41:09,363
ok, nine plus i

708
00:41:09,508 --> 00:41:12,264
What's the Hermitian conjugation of that

709
00:41:12,373 --> 00:41:14,724
Well, you first flip it

710
00:41:14,829 --> 00:41:17,594
flipping it leaves the diagonal elements in place

711
00:41:17,698 --> 00:41:21,311
interchanges the off diagonal elements

712
00:41:21,452 --> 00:41:22,457
So you flip it

713
00:41:22,565 --> 00:41:25,389
Two, nine plus i

714
00:41:25,532 --> 00:41:32,823
six plus i times seven,four minus i

715
00:41:32,937 --> 00:41:36,161
What is that process called of uh, flipping like that

716
00:41:36,303 --> 00:41:38,805
Transpose, that's called the transpose

717
00:41:38,895 --> 00:41:40,067
So flipping is the transpose

718
00:41:40,174 --> 00:41:42,897
but then you have the complex conjugated

719
00:41:43,004 --> 00:41:44,495
Complex conjugating just means

720
00:41:44,563 --> 00:41:47,990
wherever you see i put minus i

721
00:41:48,095 --> 00:41:50,776
So this becomes four plus i

722
00:41:50,915 --> 00:41:55,935
one minus seven i and nine minus i

723
00:41:56,077 --> 00:41:57,347
and that's the matrix

724
00:41:57,501 --> 00:42:01,399
that represents Hermitian conjugate of the original matrix

725
00:42:01,544 --> 00:42:05,509
So it's easy, it's easy, it's concrete

726
00:42:05,615 --> 00:42:08,911
and it's also abstract it has an abstract side

727
00:42:09,019 --> 00:42:10,969
The Hermitian conjugate just allows you

728
00:42:11,049 --> 00:42:15,511
to turn in equation from a bra vector to a ket vector

729
00:42:15,615 --> 00:42:19,034
and in terms of concrete matrix elements

730
00:42:19,142 --> 00:42:21,288
You flip i and j

731
00:42:21,396 --> 00:42:32,787
and you complex conjugate not defines M dagger

732
00:42:32,885 --> 00:42:35,667
Now is that in generally matrices

733
00:42:35,812 --> 00:42:39,674
a sort of as complex objects

734
00:42:39,855 --> 00:42:43,674
and the notion of complex conjugation

735
00:42:43,781 --> 00:42:46,787
is replaced by this Hermitian conjugation

736
00:42:46,898 --> 00:42:50,988
simultaneous flipping and uh,

737
00:42:51,077 --> 00:42:52,982
Now what is a real number?

738
00:42:53,090 --> 00:42:56,861
A real number is one which is its own complex conjugate

739
00:42:56,967 --> 00:42:59,527
If I write z is equal to z star

740
00:42:59,635 --> 00:43:01,017
you jump up and you say

741
00:43:01,106 --> 00:43:03,861
that is a real number

742
00:43:04,002 --> 00:43:08,385
There's a corresponding notion for operators

743
00:43:08,492 --> 00:43:10,219
It's the notion

744
00:43:10,359 --> 00:43:16,442
and operator which is its own Hermitian conjugate

745
00:43:16,529 --> 00:43:21,281
An operator which is its own Hermitian conjugate

746
00:43:21,388 --> 00:43:24,284
That means you interchange and complex conjugate

747
00:43:24,361 --> 00:43:26,217
you get back to the same thing

748
00:43:26,360 --> 00:43:29,125
that is called Hermitian

749
00:43:29,232 --> 00:43:31,743
Hermitian is the analog for matrices

750
00:43:31,826 --> 00:43:36,496
or the analog for operators of real

751
00:43:36,566 --> 00:43:39,256
Alright so what it says a Hermitian matrix

752
00:43:39,358 --> 00:43:45,286
Now let's specialize to the case of Hermitian matrices

753
00:43:45,356 --> 00:43:52,843
I'll just spell out Hermitian for you

754
00:43:52,923 --> 00:43:58,944
Hermitian for matrices is the analog of real for numbers

755
00:43:59,079 --> 00:44:04,870
A Hermitian operator or matrix is one which is equal

756
00:44:04,941 --> 00:44:12,612
to its own Hermitian conjugate in terms of matrices

757
00:44:12,716 --> 00:44:20,483
It's M i j, is equal to M j i star

758
00:44:20,588 --> 00:44:23,306
Let's try to see if we can construct Hermitian matrix

759
00:44:23,485 --> 00:44:29,313
Let's see we can construct a Hermitian matrix

760
00:44:29,420 --> 00:44:31,705
The first thing about Hermitian matrices

761
00:44:31,815 --> 00:44:34,937
is that diagonal elements are always real

762
00:44:35,042 --> 00:44:36,622
take a diagonal element

763
00:44:36,731 --> 00:44:39,496
for diagonal element i and j are equal

764
00:44:39,603 --> 00:44:43,874
i j is the same as j i if on the diagonal

765
00:44:43,982 --> 00:44:46,465
So that says that for the diagonal elements

766
00:44:46,572 --> 00:44:48,383
they're always real

767
00:44:48,469 --> 00:44:53,668
so Hermitian matrix first of all has real diagonal elements

768
00:44:53,774 --> 00:45:00,784
Uh, one, seven, I like seven, minus two point six

769
00:45:00,851 --> 00:45:02,596
on the diagonal

770
00:45:02,701 --> 00:45:04,622
Off the diagonal

771
00:45:04,729 --> 00:45:08,137
they have the property that when you reflect them

772
00:45:08,284 --> 00:45:10,333
They simply complex conjugate

773
00:45:10,482 --> 00:45:12,702
so we would put i over here

774
00:45:12,814 --> 00:45:16,211
We would put minus i over here for Hermitian matrix

775
00:45:16,355 --> 00:45:20,406
If you are would put four minus i over here

776
00:45:20,512 --> 00:45:22,581
you would put four plus i over here

777
00:45:22,684 --> 00:45:24,403
and if you have five over here

778
00:45:24,516 --> 00:45:28,020
you put five over here

779
00:45:28,130 --> 00:45:30,154
So Hermitian matrix has the property

780
00:45:30,264 --> 00:45:32,961
first of all diagonal elements are real

781
00:45:33,068 --> 00:45:37,718
and the off diagonal elements, not symmetric

782
00:45:37,824 --> 00:45:40,400
But a different kind of symmetry

783
00:45:40,509 --> 00:45:44,609
A symmetry when you reflect them the complex conjugate

784
00:45:44,717 --> 00:45:47,370
That's called a Hermitian matrix

785
00:45:47,449 --> 00:45:50,991
A Hermitian matrix has an enormous importance

786
00:45:51,073 --> 00:45:53,966
and Hermitian matrices have enormous importance

787
00:45:54,069 --> 00:45:56,643
in quantum mechanics

788
00:45:56,748 --> 00:45:59,373
They represent the observable quantities

789
00:45:59,454 --> 00:46:00,670
The measurable quantities

790
00:46:00,814 --> 00:46:03,466
The things that you can measure

791
00:46:03,576 --> 00:46:04,362
We're gonna come to that

792
00:46:04,470 --> 00:46:06,767
We're going to do a bit of that

793
00:46:06,839 --> 00:46:11,070
especially for the single spin I hope tonight

794
00:46:11,147 --> 00:46:15,498
But before we do we have to know a little more about Hermitian matrices

795
00:46:15,607 --> 00:46:17,892
In particular we need a concept

796
00:46:17,998 --> 00:46:21,237
a concept of eigenvector and eigenvalue

797
00:46:21,377 --> 00:46:25,366
I know I'm going fast and I know I'm quite aware that if you don't know

798
00:46:25,444 --> 00:46:26,878
these ideas you're going

799
00:46:26,983 --> 00:46:30,940
you have to struggle tonight to keep up

800
00:46:31,078 --> 00:46:33,073
but they're standard ideas

801
00:46:33,186 --> 00:46:35,632
There're lots of textbooks on linear algebra

802
00:46:35,740 --> 00:46:38,058
You can find it and you can sit down quietly

803
00:46:38,165 --> 00:46:41,035
and then learn these ideas in about two hours

804
00:46:41,138 --> 00:46:44,277
I would say two three hours

805
00:46:44,384 --> 00:46:49,424
Ok, so let's talk about Hermitian

806
00:46:49,532 --> 00:46:55,770
Let's first talk about eigenvectors and eigenvalues

807
00:46:55,875 --> 00:47:08,101
(erase the whiteboard)

808
00:47:08,172 --> 00:47:09,963
We can even think about these things incidentally

809
00:47:10,039 --> 00:47:12,095
in ordinary three-dimensional space

810
00:47:12,201 --> 00:47:12,672
within a vector space

811
00:47:12,783 --> 00:47:17,052
just the space of ordinary three vectors

812
00:47:17,161 --> 00:47:22,263
Uh, there are operators, the represented by matrices

813
00:47:22,343 --> 00:47:26,105
and those operators, operators on vectors to give other vectors

814
00:47:26,250 --> 00:47:29,025
and typically if a matrix is all interesting

815
00:47:29,139 --> 00:47:32,462
if operator is all interesting whatever mean to be interesting

816
00:47:32,567 --> 00:47:36,587
Well, it's a little bit complex

817
00:47:36,665 --> 00:47:38,286
It would generally have the property

818
00:47:38,393 --> 00:47:41,258
that if you put a vector into the machine

819
00:47:41,364 --> 00:47:42,160
the vector comes out

820
00:47:42,268 --> 00:47:44,384
will be pointing in a different direction

821
00:47:44,485 --> 00:47:48,278
That would be generally true, generically true

822
00:47:48,419 --> 00:47:51,528
Ah, so directionality is not preserved

823
00:47:51,635 --> 00:47:54,343
by linear operators as a rule

824
00:47:54,450 --> 00:47:57,356
On the other hand there may be particular vectors

825
00:47:57,469 --> 00:48:00,290
and typically there are particular vectors

826
00:48:00,396 --> 00:48:04,844
uh, that when you apply in matrix

827
00:48:04,989 --> 00:48:06,398
Those vectors would depend on

828
00:48:06,491 --> 00:48:09,605
which matrix you talking about or an operator

829
00:48:09,716 --> 00:48:12,413
There may be specific vectors associated with

830
00:48:12,519 --> 00:48:14,154
the specific operator

831
00:48:14,263 --> 00:48:18,172
that do simply get multiplied by numbers

832
00:48:18,272 --> 00:48:20,803
Another words that come out in the same direction

833
00:48:20,904 --> 00:48:26,113
An example, an example of an operational

834
00:48:26,184 --> 00:48:28,809
a linear operational on a vector is a rotation about

835
00:48:28,883 --> 00:48:32,479
an axis that can be represented by a matrix

836
00:48:32,583 --> 00:48:35,116
Uh, in three dimensions

837
00:48:35,186 --> 00:48:39,094
if you rotate the vector space about some axis

838
00:48:39,201 --> 00:48:40,182
Then generally speaking

839
00:48:40,297 --> 00:48:43,906
an arbitrary vector will wind up in a different direction

840
00:48:44,047 --> 00:48:46,586
However there is one particular direction

841
00:48:46,678 --> 00:48:48,033
for which that doesn't happen

842
00:48:48,107 --> 00:48:49,852
namely along the axis of rotation

843
00:48:49,923 --> 00:48:52,342
if you rotate about the axis of rotation

844
00:48:52,438 --> 00:48:55,450
The vector comes out pointing in the same direction

845
00:48:55,556 --> 00:48:57,410
It comes out multiple of itself

846
00:48:57,518 --> 00:49:01,039
Now in this case, simply equal to itself

847
00:49:01,180 --> 00:49:05,699
Alright, that's the notion of the eigenvectors of an operator

848
00:49:05,809 --> 00:49:09,133
They are the vectors if there are any

849
00:49:09,242 --> 00:49:12,020
They're not necessarily any

850
00:49:12,128 --> 00:49:15,746
For example in rotations about two and two dimensions

851
00:49:15,854 --> 00:49:17,048
There are no eigenvectors

852
00:49:17,049 --> 00:49:18,612
There are no directions which come out

853
00:49:18,613 --> 00:49:21,634
No directions which come out unaffected

854
00:49:21,778 --> 00:49:29,209
But, so in general may or may not be the, eigenvectors

855
00:49:29,319 --> 00:49:32,563
But if there are eigenvectors defined in the following way

856
00:49:32,667 --> 00:49:36,393
Again I'm gonna use M for the generic operator

857
00:49:36,498 --> 00:49:40,514
M acts on an eigenvector

858
00:49:40,619 --> 00:49:44,036
Let's call the eigenvector I

859
00:49:44,187 --> 00:49:48,958
I for eigen and now of course I is not for eigen

860
00:49:49,069 --> 00:49:51,223
Eigen becomes an E

861
00:49:51,367 --> 00:49:54,538
But I am going to call it I anyway

862
00:49:54,626 --> 00:49:59,379
If M acts on one of its eigenvectors

863
00:49:59,483 --> 00:50:00,734
Eigenvectors are not things

864
00:50:00,841 --> 00:50:02,530
which are independent of the matrices

865
00:50:02,637 --> 00:50:06,327
Given a matrix or operator

866
00:50:06,471 --> 00:50:11,150
it has certain eigen directions or certain eigenvectors

867
00:50:11,249 --> 00:50:15,569
and the rule is, or the definition

868
00:50:15,714 --> 00:50:17,444
is the set of vectors

869
00:50:17,584 --> 00:50:21,058
which just get multiplied by numbers

870
00:50:21,164 --> 00:50:24,754
Let's call for the ith eigenvector here

871
00:50:24,866 --> 00:50:27,885
The eigenvalue is called lambda I

872
00:50:27,992 --> 00:50:31,004
and simply multiplies the input

873
00:50:31,145 --> 00:50:35,600
Now different eigenvectors

874
00:50:35,705 --> 00:50:38,649
of the same matrix will have different eigenvalues

875
00:50:38,755 --> 00:50:40,779
There may be and in general

876
00:50:40,853 --> 00:50:43,087
that's the generic situation

877
00:50:43,194 --> 00:50:46,301
If there are several eigenvectors

878
00:50:46,403 --> 00:50:48,042
they will have different eigenvalues

879
00:50:48,148 --> 00:50:53,426
However the eigenvectors are not completely unambiguous

880
00:50:53,568 --> 00:50:57,456
If you take an eigenvector and multiply it by a number

881
00:50:57,560 --> 00:51:01,970
the result is still an eigenvector with the same eigenvalue

882
00:51:02,086 --> 00:51:05,809
If I multiply this equation by a complex numbers z

883
00:51:05,889 --> 00:51:08,286
I can bring the complex number z through

884
00:51:08,392 --> 00:51:11,491
and I'll find out that any numerical

885
00:51:11,566 --> 00:51:14,704
even complex numerical, multiple over

886
00:51:14,813 --> 00:51:18,061
an eigenvector is also an eigenvector

887
00:51:18,168 --> 00:51:24,498
Uh, so strictly speaking you should talk about eigen directions

888
00:51:24,573 --> 00:51:27,392
the set of vectors, uh, is uh,

889
00:51:27,459 --> 00:51:29,160
uh, including

890
00:51:29,232 --> 00:51:33,212
or not worrying about their length so speak?

891
00:51:33,365 --> 00:51:35,140
But in any case this is the direction of an

892
00:51:35,250 --> 00:51:40,442
this is the definition of an eigenvector

893
00:51:40,548 --> 00:51:43,063
Alright, some matrices have eigenvectors

894
00:51:43,157 --> 00:51:44,984
Other matrices don't have eigenvectors

895
00:51:45,124 --> 00:51:48,845
There's something important about Hermitian matrices

896
00:51:48,947 --> 00:51:50,325
Or Hermitian operators

897
00:51:50,433 --> 00:51:55,142
Hermitian operators is not only have eigenvectors

898
00:51:55,221 --> 00:51:58,202
They have a complete set of eigenvectors

899
00:51:58,310 --> 00:51:59,677
Complete means

900
00:51:59,749 --> 00:52:01,763
that there are enough of them to expand any vector

901
00:52:01,870 --> 00:52:03,969
and even better

902
00:52:04,117 --> 00:52:10,671
they have orthornormal family of eigenvectors

903
00:52:10,775 --> 00:52:12,153
In general different eigenvectors

904
00:52:12,271 --> 00:52:14,588
have different eigenvalues as I said

905
00:52:14,697 --> 00:52:16,790
But it will be true that matrix Hermitian

906
00:52:16,941 --> 00:52:22,953
it has n, the dimensionality of the spaces n

907
00:52:23,038 --> 00:52:27,921
there will be n mutually orthogonal eigenvectors

908
00:52:28,064 --> 00:52:32,123
Another words, the eigenvectors of Hermitian matrices

909
00:52:32,267 --> 00:52:35,503
defined a basis

910
00:52:35,609 --> 00:52:39,149
I'm going to let you approve part of that theorem

911
00:52:39,256 --> 00:52:40,807
It's a very easy theorem to prove

912
00:52:40,917 --> 00:52:42,764
I'm gonna prove the other half of it

913
00:52:42,910 --> 00:52:46,320
I'll prove the half, that says that

914
00:52:46,407 --> 00:52:50,783
if you have two eigenvector with different eigenvalues

915
00:52:50,868 --> 00:52:52,271
and that's the generic situation

916
00:52:52,383 --> 00:52:54,862
The eigenvalues are all different

917
00:52:54,974 --> 00:52:58,856
If you have two different eigenvectors with different eigenvalues

918
00:52:58,968 --> 00:53:02,722
we'll prove that they're orthogonal to each other

919
00:53:02,829 --> 00:53:04,901
Your job is just to prove

920
00:53:05,006 --> 00:53:07,649
that you won't run out of eigenvectors

921
00:53:07,757 --> 00:53:11,857
until you have n of them

922
00:53:11,964 --> 00:53:12,796
That's not hard

923
00:53:12,905 --> 00:53:13,815
It's not hard

924
00:53:13,924 --> 00:53:15,751
In any case it's true

925
00:53:15,830 --> 00:53:17,576
Alright so let's prove

926
00:53:17,662 --> 00:53:20,095
that if we have two eigenvectors

927
00:53:20,204 --> 00:53:22,143
Let's, M is a Hermitian, right?

928
00:53:22,233 --> 00:53:25,000
Let's, M is Hermitian

929
00:53:25,085 --> 00:53:30,119
M equals a Hermitian matrix

930
00:53:30,225 --> 00:53:32,805
I use matrix as an operator anonymously

931
00:53:32,914 --> 00:53:39,555
which means that M equals M dagger

932
00:53:39,673 --> 00:53:42,283
Okay, let's suppose that M is Hermitian

933
00:53:42,358 --> 00:53:44,799
and it has an eigenvalue lambda I

934
00:53:44,937 --> 00:53:46,633
The first observation

935
00:53:46,742 --> 00:53:51,195
is that the eigenvalues of Hermitian matrices are real

936
00:53:51,297 --> 00:53:53,274
That even comes before when I said previously

937
00:53:53,417 --> 00:53:57,416
If there is an eigenvalue is real, Ok?

938
00:53:57,522 --> 00:54:00,573
How do we prove that it's real?

939
00:54:00,685 --> 00:54:03,824
Let's prove that it's real

940
00:54:03,929 --> 00:54:08,647
Well, let's take this equation and flip it over

941
00:54:08,719 --> 00:54:11,135
into a bra equation

942
00:54:11,241 --> 00:54:14,521
To flip it over into a bra equation

943
00:54:14,594 --> 00:54:21,383
I flipped I, I take and then replace it by M dagger

944
00:54:21,495 --> 00:54:22,776
Well that's not going to do much

945
00:54:22,842 --> 00:54:24,984
because M is equal to M dagger

946
00:54:25,130 --> 00:54:29,891
And on the right hand side, I flipped I

947
00:54:29,974 --> 00:54:34,494
and what will I do with lambda in general?

948
00:54:34,593 --> 00:54:38,637
Complex conjugate, complex conjugate lambda

949
00:54:38,746 --> 00:54:43,961
So far we haven't proved that it's real

950
00:54:44,064 --> 00:54:46,578
Every time you flip from bras to kets

951
00:54:46,686 --> 00:54:48,823
you have to complex conjugate all the numbers

952
00:54:48,941 --> 00:54:50,221
that are there

953
00:54:50,331 --> 00:54:53,982
Alright, but now, assumption, M is Hermitian

954
00:54:54,090 --> 00:54:58,402
So it's equal to its own Hermitian conjugate

955
00:54:58,513 --> 00:55:02,728
And now we have this equation

956
00:55:02,835 --> 00:55:07,042
Now let's take the inner product

957
00:55:07,147 --> 00:55:10,985
of both of these equations with I

958
00:55:11,126 --> 00:55:12,769
and the upper equation

959
00:55:12,866 --> 00:55:19,476
we're gonna take the inner product

960
00:55:19,543 --> 00:55:21,154
on the left with I

961
00:55:21,293 --> 00:55:24,294
and what does that give us that gives us I I

962
00:55:24,402 --> 00:55:28,740
times lambda I

963
00:55:28,845 --> 00:55:32,157
Let's do the same thing over here

964
00:55:32,301 --> 00:55:41,402
except now multiply on the right

965
00:55:41,497 --> 00:55:44,638
This and this are obviously the same

966
00:55:44,783 --> 00:55:46,037
M with I on the left

967
00:55:46,148 --> 00:55:49,834
and I on the right is the same as this

968
00:55:49,920 --> 00:55:52,793
But this is not obviously the same

969
00:55:52,869 --> 00:55:58,847
not unless lambda is equal to lambda star

970
00:55:58,917 --> 00:56:01,780
I I here are the same as I I here

971
00:56:01,890 --> 00:56:03,227
That cancels out

972
00:56:03,371 --> 00:56:06,254
and therefore we come to the conclusion

973
00:56:06,361 --> 00:56:09,737
that if a matrix is Hermitian

974
00:56:09,836 --> 00:56:12,040
if it's equal to its own Hermitian conjugate

975
00:56:12,116 --> 00:56:13,340
Then its eigenvalues

976
00:56:13,417 --> 00:56:16,232
are equal to their own complex conjugates

977
00:56:16,303 --> 00:56:17,724
and they are real

978
00:56:17,814 --> 00:56:21,198
Eigenvalues of Hermitian matrices are real

979
00:56:21,285 --> 00:56:23,657
That's the first step

980
00:56:23,765 --> 00:56:24,968
Now let's prove

981
00:56:25,078 --> 00:56:27,793
that if we have two different eigenvalues

982
00:56:27,898 --> 00:56:30,916
the corresponding eigenvectors must be orthogonal

983
00:56:31,028 --> 00:56:32,999
Now remember what orthogonal means

984
00:56:33,070 --> 00:56:34,753
from a physical point of view

985
00:56:34,857 --> 00:56:38,118
It means physically distinct

986
00:56:38,226 --> 00:56:39,734
that is an experiment

987
00:56:39,843 --> 00:56:43,230
that you can do to distinguish the two of them unambiguously

988
00:56:43,338 --> 00:56:44,734
That's what orthogonal means

989
00:56:44,845 --> 00:56:46,586
from a physical point of view

990
00:56:46,689 --> 00:56:49,444
Let's prove that the eigenvectors

991
00:56:49,588 --> 00:56:54,932
of M with different eigenvalue are orthogonal

992
00:56:54,997 --> 00:56:58,043
Different eigenvalues physically descent distinguishable

993
00:56:58,147 --> 00:56:59,994
You should keep that in your head

994
00:57:00,081 --> 00:57:05,235
Okay, so here we are, let's, let's start with M on I

995
00:57:05,315 --> 00:57:09,056
equals lambda I I

996
00:57:09,142 --> 00:57:14,637
And M on J equals lambda J J

997
00:57:14,721 --> 00:57:18,700
two different eigenvectors, I and J

998
00:57:18,803 --> 00:57:20,688
First we'll do

999
00:57:20,775 --> 00:57:22,189
is turned this one over

1000
00:57:22,262 --> 00:57:25,982
and turn it into a ket equation, uh, a bra, it is a ket equation

1001
00:57:26,090 --> 00:57:29,127
We take this one and turn it into a bra equation

1002
00:57:29,221 --> 00:57:30,933
so that looks like this

1003
00:57:31,056 --> 00:57:37,415
J, M dagger, but M dagger is the same as M

1004
00:57:37,522 --> 00:57:41,433
equals the real number lambda J

1005
00:57:41,513 --> 00:57:49,321
eigenvector, eigenvalues of Hermitian matrices are real, times J

1006
00:57:49,474 --> 00:57:54,580
Now here we have, okay, so we, we have

1007
00:57:54,687 --> 00:57:59,653
two equations

1008
00:57:59,732 --> 00:58:01,920
what can you do to them

1009
00:58:02,028 --> 00:58:05,342
so that we can relate them

1010
00:58:05,452 --> 00:58:07,581
What we can do with them to relate them

1011
00:58:07,725 --> 00:58:10,105
is to multiply this one by J on the left

|1012
00:58:10,225 --> 00:58:12,184
and this one by I on the right

1013
00:58:12,293 --> 00:58:13,531
and we will get the same expression

1014
00:58:13,611 --> 00:58:15,825
on the left hand side of the equation

1015
00:58:15,939 --> 00:58:16,528
So let's do that

1016
00:58:16,639 --> 00:58:28,865
J, J, I, I

1017
00:58:28,944 --> 00:58:32,078
What we have on the left is the same

1018
00:58:32,186 --> 00:58:33,858
in the upper and lower equation

1019
00:58:33,999 --> 00:58:37,673
What we have on the right is not necessarily the same

1020
00:58:37,784 --> 00:58:40,821
so that requires that it's the same

1021
00:58:40,918 --> 00:58:44,717
Requiring that it's the same says that lambda I

1022
00:58:44,823 --> 00:58:49,906
minus lambda J

1023
00:58:50,013 --> 00:58:56,328
times the inner product of J with I must equal zero

1024
00:58:56,436 --> 00:58:58,217
I've just written a fence

1025
00:58:58,292 --> 00:59:01,139
I've just written that this is equal to this

1026
00:59:01,250 --> 00:59:03,187
and then transposed

1027
00:59:03,293 --> 00:59:05,464
That's lambda I minus lambda J

1028
00:59:05,535 --> 00:59:09,664
times the inner product of J and I equal zero

1029
00:59:09,772 --> 00:59:14,726
Well, if the product of two things and I have two things

1030
00:59:14,832 --> 00:59:18,476
lambda I minus lambda J  and the inner product of J and I

1031
00:59:18,561 --> 00:59:20,803
The product of two things are zero

1032
00:59:20,906 --> 00:59:25,722
it says unambiguously at least one of them is zero

1033
00:59:25,803 --> 00:59:28,673
It usually says one of them is zero, Ok?

1034
00:59:28,814 --> 00:59:33,183
The inner product of I and J may or may not be zero

1035
00:59:33,290 --> 00:59:35,953
But let's take the case what we know

1036
00:59:36,060 --> 00:59:38,022
that the eigenvectors correspond to

1037
00:59:38,131 --> 00:59:39,688
different values of the eigen

1038
00:59:39,774 --> 00:59:42,852
sorry, the eigenvalues are different

1039
00:59:42,960 --> 00:59:43,954
That's what I'm interested

1040
00:59:44,060 --> 00:59:45,833
I'm interested in eigenvectors

1041
00:59:45,944 --> 00:59:50,037
corresponding to different values of the eigenvalue

1042
00:59:50,180 --> 00:59:54,993
So by assumption lambda I is not equal to lambda J

1043
00:59:55,097 --> 00:59:58,623
The conclusion is that the inner product of I with J

1044
00:59:58,725 --> 01:00:01,074
must be equal to zero

1045
01:00:01,182 --> 01:00:02,540
So what does that mean?

1046
01:00:02,648 --> 01:00:06,061
That means that I and J are orthogonal

1047
01:00:06,164 --> 01:00:08,698
So now we have a theorem

1048
01:00:08,804 --> 01:00:14,236
the eigenvectors of Hermitian matrices

1049
01:00:14,342 --> 01:00:19,617
with different eigenvalues are orthogonal

1050
01:00:19,756 --> 01:00:22,687
Uh, let's just take the generic case

1051
01:00:22,763 --> 01:00:24,775
where all of the eigenvalues are different

1052
01:00:24,845 --> 01:00:27,300
We will come back to what happens when some of them may be the same

1053
01:00:27,442 --> 01:00:30,473
It's not important right now

1054
01:00:30,614 --> 01:00:33,815
if all the eigenvalues have to be different to each other

1055
01:00:33,891 --> 01:00:35,140
Then all the eigenvectors

1056
01:00:35,246 --> 01:00:39,182
are mutually orthogonal to each other

1057
01:00:39,292 --> 01:00:40,663
If there are enough of them

1058
01:00:40,773 --> 01:00:42,998
and your job is to prove enough of them

1059
01:00:43,107 --> 01:00:44,989
they form a basis

1060
01:00:45,065 --> 01:00:48,397
Another words this collection of things called I and J here

1061
01:00:48,476 --> 01:00:51,219
is a special case of the kind of basis

1062
01:00:51,288 --> 01:00:53,940
which we labeled with small I and J

1063
01:00:54,004 --> 01:00:57,174
But it's the basis of eigenvectors

1064
01:00:57,279 --> 01:01:01,539
that's the basis of eigenvectors of the Hermitian operator

1065
01:01:01,615 --> 01:01:03,103
as I said Hermitian operators

1066
01:01:03,215 --> 01:01:06,709
will be identified with observable quantities

1067
01:01:06,783 --> 01:01:09,539
and to say that eigenvectors are orthogonal

1068
01:01:09,646 --> 01:01:13,390
has something other to do with physical differences

1069
01:01:13,528 --> 01:01:15,900
that are measurable unambiguously

1070
01:01:16,046 --> 01:01:21,886
between those states of the system

1071
01:01:21,964 --> 01:01:25,043
Ok, we are now prepared to state

1072
01:01:25,149 --> 01:01:28,560
the principles of quantum mechanics

1073
01:01:28,667 --> 01:01:32,852
We have enough mathematical information

1074
01:01:32,928 --> 01:01:36,218
or enough mathematical formalism

1075
01:01:36,292 --> 01:01:40,170
to state the entire of show

1076
01:01:40,251 --> 01:01:42,351
we can put on the blackboard now

1077
01:01:42,407 --> 01:01:44,661
what quantum mechanics is and what it says

1078
01:01:44,799 --> 01:01:48,799
and uh, what the rules are

1079
01:01:48,890 --> 01:01:53,371
or what the correspondences are

1080
01:01:53,481 --> 01:01:54,563
what the mathematical

1081
01:01:54,671 --> 01:01:56,032
sorry what the, uh,

1082
01:01:56,136 --> 01:01:59,989
what the physical meanings of these various things are

1083
01:02:00,096 --> 01:02:01,051
You won't get there

1084
01:02:01,160 --> 01:02:03,068
You won't understand this the first time around

1085
01:02:03,138 --> 01:02:03,884
It'll be foreign

1086
01:02:03,957 --> 01:02:06,008
it'll be to kill your what is it talking about

1087
01:02:06,099 --> 01:02:08,579
But of course we will work it out in an example

1088
01:02:08,667 --> 01:02:11,028
so we'll see how the ideas work

1089
01:02:11,141 --> 01:02:13,212
so don't get disturbed

1090
01:02:13,320 --> 01:02:18,605
if what I throw at you now sounds a little bit meaningless

1091
01:02:18,721 --> 01:02:21,965
That's ok. By the time we'll finish some examples

1092
01:02:22,047 --> 01:02:25,287
it should be clear how we use this

1093
01:02:25,359 --> 01:02:28,969
Alright, first of all there are measurable quantities

1094
01:02:29,070 --> 01:02:32,454
The result of a measurement is always a real number

1095
01:02:32,564 --> 01:02:34,458
It takes at least two measurements

1096
01:02:34,565 --> 01:02:36,468
and to somehow to compatible measurements

1097
01:02:36,541 --> 01:02:39,933
two measurements that can both be performed simultaneously

1098
01:02:40,073 --> 01:02:42,964
to measure a complex number

1099
01:02:43,072 --> 01:02:47,536
The basic ingredients and basic measurables are real numbers

1100
01:02:47,598 --> 01:02:49,591
that's what's comes out of your detector

1101
01:02:49,664 --> 01:02:51,311
You can measure two of them

1102
01:02:51,380 --> 01:02:52,703
if you can measure two of them

1103
01:02:52,774 --> 01:02:55,544
If it is possible to simultaneously measure two things

1104
01:02:55,649 --> 01:02:58,011
and put them together into a complex number

1105
01:02:58,081 --> 01:03:00,342
but really is just two measurements

1106
01:03:00,449 --> 01:03:03,676
So each measurement, each thing that you measure

1107
01:03:03,775 --> 01:03:09,260
has a real result associated with it

1108
01:03:09,370 --> 01:03:13,616
The physical measurables, the called observables

1109
01:03:13,686 --> 01:03:16,139
quantum mechanics called observables

1110
01:03:16,245 --> 01:03:17,421
the things that you can measure

1111
01:03:17,524 --> 01:03:23,206
The observables are identified with Hermitian matrices

1112
01:03:23,312 --> 01:03:25,265
or Hermitian operators

1113
01:03:25,351 --> 01:03:28,682
Observables, it's not a q, what like a q from

1114
01:03:28,683 --> 01:03:38,417
(hand writing)

1115
01:03:38,531 --> 01:03:39,761
I want to write equals

1116
01:03:39,866 --> 01:03:42,270
are represented by, I'll just put an arrow

1117
01:03:42,375 --> 01:03:56,296
are represented by Hermitian linear, of course, operators

1118
01:03:56,436 --> 01:04:02,167
First thing

1119
01:04:02,247 --> 01:04:05,023
Let's just say, in the study the spin

1120
01:04:05,129 --> 01:04:09,544
in the uh, in the uh, in a Rough Lose way

1121
01:04:09,691 --> 01:04:12,419
we saw them identified some observables

1122
01:04:12,564 --> 01:04:14,349
The observables that we identified

1123
01:04:14,461 --> 01:04:17,195
with the components of the spin

1124
01:04:17,301 --> 01:04:19,281
The observables that we could identify

1125
01:04:19,352 --> 01:04:23,566
were associated with our apparatus point apparatus in some direction

1126
01:04:23,672 --> 01:04:26,340
and measure the spin along those directions

1127
01:04:26,410 --> 01:04:28,497
we said, we called if we measured the spin

1128
01:04:28,583 --> 01:04:32,199
along the z axis, we called it sigma z

1129
01:04:32,307 --> 01:04:34,508
We can measure it

1130
01:04:34,613 --> 01:04:37,064
or we could turn our apparatus on its side

1131
01:04:37,171 --> 01:04:40,073
and measure sigma x

1132
01:04:40,182 --> 01:04:43,432
Or we can turn on our apparatus soon

1133
01:04:43,537 --> 01:04:48,362
from the x axis to the y axis and measures sigma y

1134
01:04:48,469 --> 01:04:50,780
We talked about those things

1135
01:04:50,920 --> 01:04:55,117
We didn't talk about sigmas being Hermitian operators

1136
01:04:55,226 --> 01:04:57,731
We haven't talked about them being operators at all

1137
01:04:57,836 --> 01:05:01,167
But we did talk about measuring them

1138
01:05:01,243 --> 01:05:04,918
these will become Hermitian operators in time, Ok?

1139
01:05:05,019 --> 01:05:08,762
Next, the possible values

1140
01:05:08,868 --> 01:05:11,891
that a given observable can take on

1141
01:05:11,998 --> 01:05:14,525
What are they for sigma z?

1142
01:05:14,664 --> 01:05:18,587
One of the possible, plus or minus one, right?

1143
01:05:18,691 --> 01:05:28,656
The possible observable values

1144
01:05:28,763 --> 01:05:30,398
measured in an experiment

1145
01:05:30,540 --> 01:05:38,231
those observable values are the eigenvalues

1146
01:05:38,336 --> 01:05:48,260
They're real because the matrices are Hermitian

1147
01:05:48,405 --> 01:05:51,102
The eigenvectors, what about eigenvectors?

1148
01:05:51,256 --> 01:05:53,644
The eigenvectors correspond to states

1149
01:05:53,752 --> 01:05:56,675
States, vectors, states, vectors

1150
01:05:56,784 --> 01:05:59,877
The eigenvectors are the states

1151
01:05:59,982 --> 01:06:04,332
in which the corresponding quantities are definite

1152
01:06:04,437 --> 01:06:06,547
Another words in which the measurement of them

1153
01:06:06,657 --> 01:06:08,247
is unambiguous

1154
01:06:08,353 --> 01:06:12,812
Let's, let's follow through on this here a little bit

1155
01:06:12,917 --> 01:06:15,648
For all three of these observables

1156
01:06:15,756 --> 01:06:19,546
the eigenvalues had better be plus or minus one

1157
01:06:19,692 --> 01:06:23,002
Okay now let's come to the eigenvectors

1158
01:06:23,109 --> 01:06:26,287
The eigenvectors, for example

1159
01:06:26,393 --> 01:06:29,208
the eigenvector with a given eigenvalue

1160
01:06:29,316 --> 01:06:30,509
is a state of the system

1161
01:06:30,618 --> 01:06:33,552
where if you measure that particular observable

1162
01:06:33,657 --> 01:06:36,961
you will be definitely get the eigenvalue

1163
01:06:37,073 --> 01:06:39,526
associated with that eigenvector

1164
01:06:39,634 --> 01:06:50,605
So the eigenvectors, those are

1165
01:06:50,714 --> 01:06:51,087
We should have a backwards

1166
01:06:51,195 --> 01:06:56,621
we should put eigenvectors on the side

1167
01:06:56,727 --> 01:06:59,802
and on the left side the physical meaning of them

1168
01:06:59,946 --> 01:07:11,070
the physical meaning is the states in which

1169
01:07:11,179 --> 01:07:29,749
observable has unambiguous result

1170
01:07:29,895 --> 01:07:33,867
Example, we talk, for example

1171
01:07:33,968 --> 01:07:39,140
about the states up and down

1172
01:07:39,247 --> 01:07:40,071
Those are states

1173
01:07:40,072 --> 01:07:43,427
in which sigma z had definite values

1174
01:07:43,496 --> 01:07:46,335
When you're up, sigma z is definitely one

1175
01:07:46,426 --> 01:07:49,441
when you're down sigma z definitely minus one

1176
01:07:49,553 --> 01:07:51,557
Any other vector

1177
01:07:51,662 --> 01:07:59,783
sigma z is statistically is plus or minus one, but with randomness

1178
01:07:59,858 --> 01:08:05,517
So up and down must be the eigenvectors of sigma z

1179
01:08:05,626 --> 01:08:08,288
Left and right or the states

1180
01:08:08,357 --> 01:08:10,884
in which sigma x was unambiguously

1181
01:08:10,956 --> 01:08:12,778
either plus or minus one

1182
01:08:13,038 --> 01:08:17,124
So left and right must be the eigenvectors of sigma x

1183
01:08:17,198 --> 01:08:18,701
What the sigma x is?

1184
01:08:18,810 --> 01:08:20,853
We are trying to figure out what sigma x is

1185
01:08:20,957 --> 01:08:22,178
knowing it eigenvalues

1186
01:08:22,249 --> 01:08:25,071
or its eigenvalues and its eigenvectors

1187
01:08:25,216 --> 01:08:27,857
And what are the other three

1188
01:08:27,926 --> 01:08:29,777
other two called in and out I think

1189
01:08:29,879 --> 01:08:38,831
In and out were the eigenvectors of sigma y

1190
01:08:38,972 --> 01:08:42,981
So we have three connections

1191
01:08:43,171 --> 01:08:45,628
and now a fourth connection

1192
01:08:45,734 --> 01:08:50,399
Given a general state, not an eigenvector

1193
01:08:50,505 --> 01:08:55,658
not an eigenvector

1194
01:08:55,761 --> 01:08:59,186
Given a general state A

1195
01:08:59,293 --> 01:09:04,566
somebody made an electron or a system of some sort

1196
01:09:04,671 --> 01:09:07,291
and left it in the state A

1197
01:09:07,396 --> 01:09:11,357
then we can talk about what is the probability

1198
01:09:11,468 --> 01:09:15,928
that we measure the eigenvalues of M

1199
01:09:16,069 --> 01:09:17,713
at different values

1200
01:09:17,785 --> 01:09:19,832
There may be several different eigenvalues

1201
01:09:19,943 --> 01:09:22,480
in this case they would be plus one or minus one

1202
01:09:22,589 --> 01:09:24,009
but more generally

1203
01:09:24,113 --> 01:09:28,804
thousand eigenvalues million eigenvalues whatever whatever the number is

1204
01:09:28,915 --> 01:09:35,131
Labeled by the, the label I

1205
01:09:35,242 --> 01:09:40,156
What is the probability that when you measure M

1206
01:09:40,262 --> 01:09:44,170
you get lambda I?

1207
01:09:44,242 --> 01:09:46,536
That's the last piece of thing here

1208
01:09:46,605 --> 01:09:50,325
so left-hand side we put probability

1209
01:09:50,403 --> 01:09:53,122
physical meaning probability

1210
01:09:53,227 --> 01:09:55,291
and I'll write it just as the probability

1211
01:09:55,366 --> 01:09:57,625
to measure I

1212
01:09:57,733 --> 01:09:59,379
No, to measure lambda I

1213
01:09:59,425 --> 01:10:02,804
I'm sorry, to measure lambda I

1214
01:10:02,877 --> 01:10:05,645
that the result of the experiment comes out

1215
01:10:05,715 --> 01:10:10,540
a particular eigenvalue of the observable M

1216
01:10:10,644 --> 01:10:12,402
Whatever M is

1217
01:10:12,509 --> 01:10:18,272
That is related

1218
01:10:18,340 --> 01:10:20,352
not related to, it's essentially equal to

1219
01:10:20,423 --> 01:10:23,246
Two pieces come to two steps

1220
01:10:23,353 --> 01:10:27,040
First, you calculate if you have the vector A

1221
01:10:27,115 --> 01:10:30,266
The vector which to start the state of the system

1222
01:10:30,404 --> 01:10:36,609
You take its inner product with the eigenvector I

1223
01:10:36,714 --> 01:10:39,111
the projection of A onto I

1224
01:10:39,215 --> 01:10:42,220
the component of A in the I direction

1225
01:10:42,291 --> 01:10:44,658
now that in general is a complex number

1226
01:10:44,760 --> 01:10:47,458
That is not the probability

1227
01:10:47,566 --> 01:10:50,451
You multiply it by its own complex conjugate

1228
01:10:50,519 --> 01:10:52,392
to get a real positive number

1229
01:10:52,504 --> 01:10:54,970
and that is the probability

1230
01:10:55,076 --> 01:10:57,824
We can multiply this by its complex conjugate

1231
01:10:57,931 --> 01:11:02,163
or we can just write it I, A this

1232
01:11:02,269 --> 01:11:06,460
I A and A I are complex conjugates of each other

1233
01:11:06,606 --> 01:11:12,195
We can also write this as the absolute value of I A squared

1234
01:11:12,339 --> 01:11:15,293
The individual components

1235
01:11:15,400 --> 01:11:18,233
before we square them are called amplitudes

1236
01:11:18,344 --> 01:11:20,982
They're called probability amplitudes

1237
01:11:21,089 --> 01:11:23,456
Probability amplitudes are things

1238
01:11:23,522 --> 01:11:25,865
which you multiply by their complex conjugates

1239
01:11:25,969 --> 01:11:27,054
to find probabilities

1240
01:11:27,117 --> 01:11:28,325
They are real

1241
01:11:28,435 --> 01:11:30,334
They are positive

1242
01:11:30,512 --> 01:11:36,827
and that's the fourth, the fourth postulate

1243
01:11:36,898 --> 01:11:37,915
of quantum mechanics

1244
01:11:38,017 --> 01:11:40,235
That I know are independent

1245
01:11:40,311 --> 01:11:42,047
I use them because uh

1246
01:11:42,119 --> 01:11:44,651
because they are reasonably intuitive

1247
01:11:44,799 --> 01:11:48,752
In fact, you can probably get rid of

1248
01:11:48,859 --> 01:11:52,740
at least two of them and prove them from the others

1249
01:11:52,848 --> 01:11:54,579
But we don't need to do that

1250
01:11:54,682 --> 01:11:56,622
This is a good starting point

1251
01:11:56,731 --> 01:12:03,088
Uh, and these are the basic postulates for quantum mechanics, ok?

1252
01:12:03,193 --> 01:12:05,307
The other postulate is that

1253
01:12:05,412 --> 01:12:07,631
when you're measuring a quantity

1254
01:12:07,702 --> 01:12:09,717
you prepare the system

1255
01:12:09,826 --> 01:12:12,713
in a state of definite value of that quantity

1256
01:12:12,786 --> 01:12:14,457
I won't bother write that down

1257
01:12:14,526 --> 01:12:15,931
When you measure something

1258
01:12:16,041 --> 01:12:18,386
you are in a fact

1259
01:12:18,491 --> 01:12:21,362
pushing it into a state of definite value

1260
01:12:21,467 --> 01:12:22,822
of that quantity depending on

1261
01:12:22,891 --> 01:12:25,475
what the outcome of your experiment is

1262
01:12:25,584 --> 01:12:28,754
If your outcome measures some particular

1263
01:12:28,824 --> 01:12:30,430
the outcome of an experiment

1264
01:12:30,503 --> 01:12:33,127
to some particular eigenvalue

1265
01:12:33,234 --> 01:12:36,010
That means you push the system into the state

1266
01:12:36,121 --> 01:12:40,518
into eigenstate of that quantity with that eigenvalue

1267
01:12:40,626 --> 01:12:41,504
I won't write that down

1268
01:12:41,609 --> 01:12:43,816
It takes too many words to write that down

1269
01:12:43,961 --> 01:12:47,278
Student: What's that P? the probability?

1270
01:12:47,383 --> 01:12:49,705
This? This

1271
01:12:49,810 --> 01:12:50,917
Student: yeah

1272
01:12:51,028 --> 01:12:53,357
Student: That's the P with?

1273
01:12:53,464 --> 01:12:55,060
Student: I can't recognize your handwriting

1274
01:12:55,167 --> 01:12:57,884
Probability

1275
01:12:58,123 --> 01:13:00,130
Probe, it's probability probe

1276
01:13:00,201 --> 01:13:04,012
It's Probably probability

1277
01:13:04,148 --> 01:13:09,441
Yeah, (laugh)

1278
01:13:09,507 --> 01:13:12,407
This is the probability of measuring lambda

1279
01:13:12,515 --> 01:13:14,070
Oh, what is lambda I?

1280
01:13:14,178 --> 01:13:17,579
Lambda I, the probability of measuring lambda I

1281
01:13:17,687 --> 01:13:26,281
is the square of the inner product of A with I

1282
01:13:26,421 --> 01:13:29,776
It's if you like it's the projection

1283
01:13:29,853 --> 01:13:31,849
of A onto the I th direction

1284
01:13:31,922 --> 01:13:33,957
of the component of A in the I direction

1285
01:13:34,031 --> 01:13:36,483
Now, there's an implication here

1286
01:13:36,589 --> 01:13:38,085
there's an implication here

1287
01:13:38,158 --> 01:13:42,675
about the nature of state vectors

1288
01:13:42,744 --> 01:13:44,177
coming from the assumption

1289
01:13:44,281 --> 01:13:47,207
that total probabilities are equal to one

1290
01:13:47,315 --> 01:13:48,864
Student: (mumble)

1291
01:13:48,865 --> 01:13:49,189
What's that?

1292
01:13:49,262 --> 01:13:51,119
Student: It has to be normalized?

1293
01:13:51,233 --> 01:13:52,800
Yeah

1294
01:13:52,905 --> 01:14:00,743
Let's work that out

1295
01:14:00,849 --> 01:14:03,920
If you want the total probability to add up to one

1296
01:14:04,029 --> 01:14:08,556
then you want the sums of all these objects

1297
01:14:08,664 --> 01:14:14,305
when you sum over I over various possible values of,

1298
01:14:14,413 --> 01:14:16,168
that you can measure

1299
01:14:16,312 --> 01:14:17,971
You want that the equal one

1300
01:14:18,079 --> 01:14:19,796
so you want the sum over I

1301
01:14:19,867 --> 01:14:26,934
for any allowable state A

1302
01:14:27,041 --> 01:14:41,747
We want the sum of A, I, I, A to equal one

1303
01:14:41,854 --> 01:14:44,096
Now I is a basis

1304
01:14:44,213 --> 01:14:47,185
and if you go up to the top equation there

1305
01:14:47,258 --> 01:14:49,283
You have equations

1306
01:14:49,394 --> 01:14:51,637
which tell you how to deal with

1307
01:14:51,710 --> 01:14:54,340
exactly this combination

1308
01:14:54,450 --> 01:14:59,887
What is the sum over I of I I A?

1309
01:14:59,960 --> 01:15:01,495
It's just A

1310
01:15:01,604 --> 01:15:03,431
So what this says

1311
01:15:03,534 --> 01:15:07,537
is that the inner product of A with itself should equal one

1312
01:15:07,647 --> 01:15:09,856
The total length of A should be equal to one

1313
01:15:09,963 --> 01:15:10,761
That's not too surprising that

1314
01:15:10,864 --> 01:15:12,311
That's what this is

1315
01:15:12,418 --> 01:15:17,671
This is the sums of the squares of the components of A add up to one

1316
01:15:17,747 --> 01:15:18,763
For ordinary vectors

1317
01:15:18,835 --> 01:15:21,970
that would say the length of the vector was one and that's what this is

1318
01:15:22,077 --> 01:15:25,550
The length of A is equal to one

1319
01:15:25,621 --> 01:15:27,851
It's equivalent in terms of the alphas

1320
01:15:27,959 --> 01:15:29,422
We got really on the alphas up there

1321
01:15:29,531 --> 01:15:38,156
and also that's equivalent to the sum of alpha star i alpha i equals one

1322
01:15:38,263 --> 01:15:45,007
Well I, capital I

1323
01:15:45,078 --> 01:15:50,694
and so there is one combination of the alphas

1324
01:15:50,767 --> 01:15:53,278
this is one real equation, not complex equation

1325
01:15:53,386 --> 01:15:55,285
It's a real equation

1326
01:15:55,425 --> 01:15:58,240
and that means is one real combination of the alphas

1327
01:15:58,347 --> 01:16:00,241
which has to be constrained

1328
01:16:00,387 --> 01:16:03,881
Alphas are not completely free

1329
01:16:03,954 --> 01:16:06,391
Or the alphas are not completely free, this uh,

1330
01:16:06,499 --> 01:16:08,930
Roughly speaking, the sums of the squares are sums of them

1331
01:16:09,037 --> 01:16:12,201
times a complex conjugate has to add up to one

1332
01:16:12,308 --> 01:16:17,380
So that means is less, fewer, independent quantities

1333
01:16:17,490 --> 01:16:20,945
you thought

1334
01:16:21,052 --> 01:16:25,136
There is another fact which we have not gotten to yet

1335
01:16:25,279 --> 01:16:28,711
We will get to it

1336
01:16:28,819 --> 01:16:32,759
You could sort of see it from here

1337
01:16:32,828 --> 01:16:35,210
All probabilities any probability

1338
01:16:35,318 --> 01:16:37,024
that you ever may want to measure

1339
01:16:37,094 --> 01:16:39,808
or that you ever may want to

1340
01:16:39,852 --> 01:16:42,968
uh, to calculate or compare with experiment

1341
01:16:43,109 --> 01:16:44,927
is always connected

1342
01:16:45,035 --> 01:16:48,759
with a real quantity like this

1343
01:16:48,866 --> 01:16:51,667
The collection with such real quantities

1344
01:16:51,772 --> 01:16:56,249
is invariant under a certain operation

1345
01:16:56,356 --> 01:17:00,231
The operation is multiplying the state vector by a phase

1346
01:17:00,338 --> 01:17:04,804
Everybody know what means to multiply a number of thing by a phase?

1347
01:17:04,912 --> 01:17:08,604
You multiply by the e to the i theta

1348
01:17:08,710 --> 01:17:12,170
If you multiply all the components of A

1349
01:17:12,239 --> 01:17:14,957
by the same overall phase

1350
01:17:15,066 --> 01:17:18,738
then every probability is unchanged

1351
01:17:18,891 --> 01:17:21,374
Every probability is unchanged

1352
01:17:21,482 --> 01:17:27,911
uh, even if you rotate basis, it's unchanged

1353
01:17:27,980 --> 01:17:33,595
So physical quantities don't respond to a change of the phase

1354
01:17:33,705 --> 01:17:37,789
That means there are two fewer degrees of freedom

1355
01:17:37,889 --> 01:17:44,282
two fewer parameters in the specification of the state

1356
01:17:44,393 --> 01:17:47,028
Then the collection of complex numbers

1357
01:17:47,102 --> 01:17:50,383
that would define a vector in the abstract space

1358
01:17:50,400 --> 01:17:52,991
Two, one is the phase, the overall phase

1359
01:17:53,099 --> 01:17:55,117
and the other is the magnitude of it

1360
01:17:55,268 --> 01:17:57,570
We'll come back to that

1361
01:17:57,675 --> 01:18:00,273
Okay that uh, that's taken us a long way

1362
01:18:00,342 --> 01:18:03,140
Uh, you're probably saturated but I am not

1363
01:18:03,211 --> 01:18:04,218
so we are going to continue

1364
01:18:04,326 --> 01:18:08,547
I am in high

1365
01:18:08,688 --> 01:18:09,434
Student: Question?

1366
01:18:09,615 --> 01:18:14,376
Student: It has to be positive, can't be negative, right?

1367
01:18:14,447 --> 01:18:17,080
Yeah, this is always positive

1368
01:18:17,195 --> 01:18:22,095
Thing times its complex is always positive

1369
01:18:22,281 --> 01:18:24,815
Student: Is the evolution postulate?

1370
01:18:24,924 --> 01:18:27,131
Is what? is what?

1371
01:18:27,237 --> 01:18:28,606
Student: evolution postulate

1372
01:18:28,760 --> 01:18:31,547
We haven't, yeah, we haven't gotten to that yet

1373
01:18:31,622 --> 01:18:33,746
How things change with time, yeah

1374
01:18:33,853 --> 01:18:36,082
No, we are into the level now

1375
01:18:36,191 --> 01:18:39,443
of where we are in classical physics

1376
01:18:39,553 --> 01:18:41,366
when we just said the state of the system

1377
01:18:41,469 --> 01:18:45,573
is just a point in the, uh, among a set of things

1378
01:18:45,641 --> 01:18:46,991
That's about where we are

1379
01:18:47,097 --> 01:18:48,375
That's how far we've gotten

1380
01:18:48,483 --> 01:18:50,072
We are not talking about how things

1381
01:18:50,111 --> 01:18:51,585
how things change

1382
01:18:51,705 --> 01:18:54,161
the particular how the states of systems change

1383
01:18:54,278 --> 01:18:55,564
We will come to that

1384
01:18:55,639 --> 01:18:58,891
Student: Back to the beginning

1385
01:18:58,994 --> 01:19:03,981
Student: we had this example of measuring its q bits

1386
01:19:04,085 --> 01:19:07,846
Student: and you had some angle

1387
01:19:08,001 --> 01:19:12,704
Student: and you got to cosine theta, right? So

1388
01:19:12,778 --> 01:19:14,358
We'll do it, well

1389
01:19:14,479 --> 01:19:16,429
Student: Ok, I get my question was

1390
01:19:16,536 --> 01:19:18,374
Student: we somehow get the idea that

1391
01:19:18,447 --> 01:19:22,615
Student: the cosine theta is the average of the measurements

1392
01:19:22,685 --> 01:19:25,166
Student: shouldn't it be cosine squared?

1393
01:19:25,270 --> 01:19:25,760
Um, um

1394
01:19:25,870 --> 01:19:26,284
Student: No?

1395
01:19:26,423 --> 01:19:36,842
I think you own (laugh)

1396
01:19:36,915 --> 01:19:40,021
Student: Can you say that last part one more time that we can

1397
01:19:40,092 --> 01:19:40,956
Which part?

1398
01:19:41,064 --> 01:19:41,830
Student: When you measure something

1399
01:19:41,940 --> 01:19:44,460
Student: you push it into the state you are measuring

1400
01:19:44,534 --> 01:19:50,090
No, when you measure something you get a result

1401
01:19:50,159 --> 01:19:53,620
The result is statistically determined

1402
01:19:53,729 --> 01:19:55,059
Meaning to say that you may get one result

1403
01:19:55,166 --> 01:19:59,236
you may get another result but once you get that result

1404
01:19:59,346 --> 01:20:02,958
then you know what state the system is in

1405
01:20:03,027 --> 01:20:06,069
it's in the eigenstate

1406
01:20:06,178 --> 01:20:07,939
of the quantity that you measured

1407
01:20:08,045 --> 01:20:13,411
with the eigenvalue that came up on your detector

1408
01:20:13,482 --> 01:20:18,878
Remember, when we had our apparatus

1409
01:20:18,949 --> 01:20:21,243
and we measured sigma z

1410
01:20:21,316 --> 01:20:24,529
we said that not only measure sigma z

1411
01:20:24,630 --> 01:20:28,867
but it prepare the system and state of definite sigma z

1412
01:20:28,943 --> 01:20:31,127
It leaves over in the state

1413
01:20:31,234 --> 01:20:34,377
in which sigma z is definite for the next measurement

1414
01:20:34,485 --> 01:20:39,193
That means leaves it in an eigenstate of sigma z

1415
01:20:39,299 --> 01:20:58,500
Ok, let's apply this (cough) to the spin system

1416
01:20:58,610 --> 01:21:01,290
You own very general formulation

1417
01:21:01,397 --> 01:21:04,472
and now we want to try out to see how it works

1418
01:21:04,578 --> 01:21:07,449
Alright, so I want to start with

1419
01:21:07,520 --> 01:21:10,731
the three observables sigma x sigma y and sigma z

1420
01:21:10,835 --> 01:21:13,249
and see where I can learn about them

1421
01:21:13,391 --> 01:21:18,401
First of all, sigma z

1422
01:21:18,512 --> 01:21:25,475
Now let's start in the basis associated with

1423
01:21:25,582 --> 01:21:28,260
the measurement of the spin along the z axis

1424
01:21:28,415 --> 01:21:31,301
Let's take that as appropriate basis

1425
01:21:31,491 --> 01:21:35,927
Sigma z two eigenvectors by

1426
01:21:36,035 --> 01:21:37,485
from the experiments

1427
01:21:37,593 --> 01:21:38,856
so we talked about we would have learned

1428
01:21:38,964 --> 01:21:40,649
that we've done the experiment

1429
01:21:40,759 --> 01:21:45,850
That sigma z has two eigenvalues and two eigenvectors

1430
01:21:45,958 --> 01:21:48,713
The two eigenvectors we've called up and down

1431
01:21:48,818 --> 01:21:50,352
and they are associated with sigma z

1432
01:21:50,462 --> 01:21:52,994
equals plus one and minus one

1433
01:21:53,136 --> 01:21:57,081
There must be eigenvectors of sigma z and eigenvalues

1434
01:21:57,185 --> 01:22:02,296
and so I can immediately say sigma z times up

1435
01:22:02,403 --> 01:22:08,020
is equal to up plus one time's up

1436
01:22:08,125 --> 01:22:11,755
Why? Because the measured value

1437
01:22:11,861 --> 01:22:13,875
when we measure sigma z

1438
01:22:14,025 --> 01:22:16,504
is going to be plus one

1439
01:22:16,608 --> 01:22:19,696
That's eigenvalue and up is the eigenvector

1440
01:22:19,768 --> 01:22:23,000
because in the upstate sigma z is definite

1441
01:22:23,107 --> 01:22:27,219
What about sigma z on the down state

1442
01:22:27,297 --> 01:22:29,574
What should I write there?

1443
01:22:29,679 --> 01:22:33,144
Is sigma z definite in the down state?

1444
01:22:33,249 --> 01:22:35,635
Yes, it's always minus one

1445
01:22:35,744 --> 01:22:37,243
It's going to be minus one

1446
01:22:37,311 --> 01:22:43,749
and that means the eigenvalues minus one times down

1447
01:22:43,819 --> 01:22:45,610
It's not sigma z times down equals up

1448
01:22:45,715 --> 01:22:47,267
That wouldn't be an eigenvector

1449
01:22:47,372 --> 01:22:48,875
An eigenvector has a property

1450
01:22:48,985 --> 01:22:51,388
that when the operator acts on it

1451
01:22:51,491 --> 01:22:55,209
it gives the same direction back times the eigenvalue

1452
01:22:55,322 --> 01:23:02,538
So here is what we know about sigma z

1453
01:23:02,645 --> 01:23:05,138
Let's see if we can compute its matrix elements

1454
01:23:05,249 --> 01:23:07,399
and exhibited as a matrix

1455
01:23:07,543 --> 01:23:11,003
But when I say exhibited as a matrix I have picked

1456
01:23:11,147 --> 01:23:14,042
I must pick a basis

1457
01:23:14,184 --> 01:23:18,238
I'm going to pick the basis up and down

1458
01:23:18,382 --> 01:23:21,026
First of all, one thing

1459
01:23:21,128 --> 01:23:23,709
up and down are orthogonal to each other

1460
01:23:23,817 --> 01:23:26,544
They must be according to these principles

1461
01:23:26,652 --> 01:23:29,008
Because of sigma z is an observable

1462
01:23:29,115 --> 01:23:31,571
it corresponds to Hermitian matrix

1463
01:23:31,681 --> 01:23:36,038
If there are two different eigenvalues

1464
01:23:36,151 --> 01:23:39,517
The eigenvectors must be orthogonal for Hermitian matrix

1465
01:23:39,628 --> 01:23:40,766
and so up and down

1466
01:23:40,872 --> 01:23:44,787
we first of all learned that up and down are orthogonal

1467
01:23:44,893 --> 01:23:46,780
That's one fact

1468
01:23:46,897 --> 01:23:48,331
So it doesn't matter

1469
01:23:48,437 --> 01:23:52,314
whether we write up and down or down up we can write the opposite order

1470
01:23:52,422 --> 01:23:55,208
Let's see if we can compute the matrix elements

1471
01:23:55,315 --> 01:24:00,258
The matrix elements are just the array that we get

1472
01:24:00,365 --> 01:24:03,029
when we consider the four possibilities

1473
01:24:03,136 --> 01:24:06,302
It's going to be two by two matrices

1474
01:24:06,408 --> 01:24:09,377
and the four possibilities are going to be,

1475
01:24:09,446 --> 01:24:13,007
first of all, we take the inner product with this back with up

1476
01:24:13,111 --> 01:24:14,923
That will be the up up matrix element

1477
01:24:14,995 --> 01:24:16,262
the one one matrix element

1478
01:24:16,335 --> 01:24:19,409
What is that going to be

1479
01:24:19,478 --> 01:24:23,836
To find, let's, I'm not sure I want to

1480
01:24:23,939 --> 01:24:29,700
Let's do it over here

1481
01:24:29,771 --> 01:24:33,944
Let's calculate up, sigma z, up

1482
01:24:34,049 --> 01:24:35,035
That's equal

1483
01:24:35,145 --> 01:24:40,314
Sigma z acts to the right to give us up plus one

1484
01:24:40,426 --> 01:24:46,125
Or plus is just one, times up

1485
01:24:46,198 --> 01:24:48,368
Just inner product of up with itself

1486
01:24:48,473 --> 01:24:52,659
and since up is a basis vector, it's normalized

1487
01:24:52,798 --> 01:24:55,337
and this is just equal to one

1488
01:24:55,445 --> 01:24:56,509
Its magnitude is equal to one

1489
01:24:56,651 --> 01:24:58,737
I can choose it to be equal to one

1490
01:24:58,850 --> 01:25:03,905
Uh, it's positive, yeah, it's positive, it's real

1491
01:25:04,012 --> 01:25:06,534
and it's just plain one

1492
01:25:06,645 --> 01:25:13,257
So up here in the up up, or the one one place is just one

1493
01:25:13,358 --> 01:25:17,378
Now let's worry about the off diagonal element

1494
01:25:17,443 --> 01:25:19,998
The off diagonal element

1495
01:25:20,107 --> 01:25:23,745
That's off diagonal element there

1496
01:25:23,814 --> 01:25:25,674
That's going to be equal

1497
01:25:25,775 --> 01:25:29,409
to be inner of down with up

1498
01:25:29,521 --> 01:25:34,308
What's that? It's zero over here

1499
01:25:34,376 --> 01:25:36,522
I would get exactly the same thing

1500
01:25:36,597 --> 01:25:39,593
if I put down over here and up over here

1501
01:25:39,700 --> 01:25:41,718
That would use this equation over here

1502
01:25:41,791 --> 01:25:43,630
that sigma z act on down

1503
01:25:43,739 --> 01:25:45,282
and then take its overlap with up

1504
01:25:45,389 --> 01:25:47,234
and I will get zero again

1505
01:25:47,303 --> 01:25:53,370
And the last one is down, sigma z, down

1506
01:25:53,441 --> 01:25:53,915
What is that?

1507
01:25:53,988 --> 01:25:55,005
Well you're going to here

1508
01:25:55,111 --> 01:25:58,001
sigma z times downs is minus down

1509
01:25:58,111 --> 01:26:01,437
So this is just equal to minus down down

1510
01:26:01,594 --> 01:26:04,306
What's that?

1511
01:26:04,375 --> 01:26:07,118
Minus one

1512
01:26:07,226 --> 01:26:09,321
So this is the matrix

1513
01:26:09,392 --> 01:26:14,760
That's associated with sigma z

1514
01:26:14,870 --> 01:26:18,089
When it acts on a vector on a column vector

1515
01:26:18,197 --> 01:26:19,334
in particular when it acts,

1516
01:26:19,408 --> 01:26:23,940
What is the column vector that associated with up?

1517
01:26:24,050 --> 01:26:27,636
It's just one on the top and zero in the bottom

1518
01:26:27,739 --> 01:26:30,987
Alright, let's just try this, one zero

1519
01:26:31,097 --> 01:26:35,233
What do we get? we get one times one is one

1520
01:26:35,338 --> 01:26:36,488
zero times zero is zero

1521
01:26:36,595 --> 01:26:38,305
we just get the same vector back

1522
01:26:38,408 --> 01:26:42,934
That's good, ok, same thing

1523
01:26:43,005 --> 01:26:46,065
It's guaranteed that will get the right

1524
01:26:46,182 --> 01:26:48,897
The eigenvectors are what they're supposed to be

1525
01:26:48,964 --> 01:26:50,339
Namely just up and down

1526
01:26:50,446 --> 01:26:54,459
and the eigenvalue are plus one minus one in the right way

1527
01:26:54,605 --> 01:27:02,322
Alright, so we found one of the sigma matrices, sigma z

1528
01:27:02,396 --> 01:27:04,519
is of course, well not of course

1529
01:27:04,639 --> 01:27:06,438
they called the Pauli matrices

1530
01:27:06,547 --> 01:27:08,675
sigma x sigma y and sigma z

1531
01:27:08,779 --> 01:27:14,819
Let's see what we can find out about sigma x

1532
01:27:14,929 --> 01:27:17,895
Now I'm using if I use the x basis

1533
01:27:18,000 --> 01:27:20,557
or I use the basis of left and right

1534
01:27:20,664 --> 01:27:23,117
of course I would just find exactly the same thing

1535
01:27:23,224 --> 01:27:26,342
But I want to stick with the up down basis

1536
01:27:26,641 --> 01:27:28,642
I want to stick with the up down basis

1537
01:27:28,715 --> 01:27:33,934
But find out what the matrix elements of sigma x are

1538
01:27:34,040 --> 01:27:36,258
Now what is the property of sigma x?

1539
01:27:36,329 --> 01:27:38,722
The property of sigma x is

1540
01:27:38,827 --> 01:27:42,126
that its eigenvectors are left and right

1541
01:27:42,233 --> 01:27:48,852
So for example, sigma x on right that equals right

1542
01:27:48,924 --> 01:27:54,583
and what about sigma x on left?

1543
01:27:54,691 --> 01:27:56,959
Equals minus left

1544
01:27:57,069 --> 01:27:59,232
Those are the two possible values

1545
01:27:59,336 --> 01:28:01,060
that sigma x can take on

1546
01:28:01,166 --> 01:28:03,748
Left and right are the eigenvectors

1547
01:28:03,859 --> 01:28:07,017
and another words the states in which the sigma x has definite values

1548
01:28:07,081 --> 01:28:08,718
plus one and minus one

1549
01:28:08,825 --> 01:28:10,922
and this is what I know about sigma x

1550
01:28:11,032 --> 01:28:15,052
But I want to find its matrix elements in the up down basis

1551
01:28:15,161 --> 01:28:18,090
So what I really want then let's uh,

1552
01:28:18,158 --> 01:28:21,587
Let's come over to here

1553
01:28:21,728 --> 01:28:25,070
and see if we can compute in the same basis

1554
01:28:25,176 --> 01:28:29,603
What do we do with Pauli matrix? Did I erase it?

1555
01:28:29,678 --> 01:28:35,018
I work so hard to get it

1556
01:28:35,164 --> 01:28:39,686
Sigma z is equal to one minus one zero zero

1557
01:28:39,756 --> 01:28:42,538
Now we want to try to get sigma x

1558
01:28:42,649 --> 01:28:54,774
So the first step is, oh here it is, to erase the blackboard

1559
01:28:54,842 --> 01:28:55,871
Let's start one by one

1560
01:28:55,980 --> 01:28:57,377
There are four matrix elements

1561
01:28:57,442 --> 01:29:00,820
Let's do it one by one

1562
01:29:00,888 --> 01:29:06,723
Up, sigma x, up

1563
01:29:06,796 --> 01:29:08,892
What we have to know is

1564
01:29:08,965 --> 01:29:16,107
that up is equal to left plus right over the square root of two

1565
01:29:16,178 --> 01:29:19,727
Remember that?

1566
01:29:19,837 --> 01:29:22,187
We originally defined

1567
01:29:22,258 --> 01:29:25,454
left and right to be up plus or minus down

1568
01:29:25,526 --> 01:29:26,558
over the square root of two

1569
01:29:26,657 --> 01:29:28,828
But then we can solve for up and down

1570
01:29:28,905 --> 01:29:30,753
in terms of left and right

1571
01:29:30,827 --> 01:29:41,283
Down, down is, left, sorry, it's right minus left

1572
01:29:41,358 --> 01:29:44,183
with the notation we had last time was down

1573
01:29:44,254 --> 01:29:51,580
sorry, is right minus left over Root Two

1574
01:29:51,650 --> 01:29:57,457
So all we have to do is to plug them in

1575
01:29:57,564 --> 01:30:01,683
On the left here we have, left

1576
01:30:01,795 --> 01:30:04,114
You know I'm gonna abuse notation

1577
01:30:04,224 --> 01:30:09,690
and just write this as left plus right over squared root of two

1578
01:30:09,739 --> 01:30:13,057
This is a meaningless symbol

1579
01:30:13,164 --> 01:30:15,586
but I think you know what I mean

1580
01:30:15,691 --> 01:30:17,759
I mean the bra vector left

1581
01:30:17,797 --> 01:30:21,211
plus the bra vector right over the square root of two

1582
01:30:21,315 --> 01:30:23,827
But too many vertical strokes and I get confused

1583
01:30:23,935 --> 01:30:26,701
So let's write, let's put right over the square root of two

1584
01:30:26,770 --> 01:30:33,254
Sigma x, left plus right over squared root of two

1585
01:30:33,403 --> 01:30:35,862
First of all, I can take out

1586
01:30:35,996 --> 01:30:37,686
This is the, what is this thing?

1587
01:30:37,783 --> 01:30:40,629
This is sigma x up and up, there it is

1588
01:30:40,699 --> 01:30:47,288
This is up and up and one goes up in here

1589
01:30:47,362 --> 01:30:49,518
Alright first of all, there's a factor of true

1590
01:30:49,622 --> 01:30:51,919
that I can take out square root of two from here

1591
01:30:51,989 --> 01:30:53,148
and the square root two from here

1592
01:30:53,217 --> 01:30:55,648
That's altogether one-half

1593
01:30:55,690 --> 01:30:57,175
Now let's look at this

1594
01:30:57,248 --> 01:31:02,812
What happens when sigma x hits left?

1595
01:31:02,884 --> 01:31:06,113
It gives back left except with the minus sign

1596
01:31:06,187 --> 01:31:11,272
What happens when sigma x hits right?

1597
01:31:11,343 --> 01:31:14,817
It gives back right with a plus sign

1598
01:31:14,886 --> 01:31:18,343
So when sigma x acts on this

1599
01:31:18,409 --> 01:31:21,943
What ones giving you can take the sigma x away

1600
01:31:22,011 --> 01:31:24,070
and just remember what it does

1601
01:31:24,178 --> 01:31:31,410
It gives right minus left instead of right plus left

1602
01:31:31,479 --> 01:31:35,834
It hits the vector right and give plus one

1603
01:31:35,905 --> 01:31:38,443
It hits the vector left and gives minus one again

1604
01:31:38,597 --> 01:31:42,139
These are not things to be numerically added

1605
01:31:42,212 --> 01:31:43,652
These are vectors

1606
01:31:43,759 --> 01:31:46,353
The right vector minus the left vector

1607
01:31:46,426 --> 01:31:49,681
and now what we have to do is take the inner products

1608
01:31:49,783 --> 01:31:54,363
Well, the inner product of right with right is what?

1609
01:31:54,433 --> 01:31:55,382
One

1610
01:31:55,490 --> 01:31:58,755
The inner product of right with the left is?

1611
01:31:58,969 --> 01:31:59,881
Zero

1612
01:31:59,991 --> 01:32:02,138
The inner product of left with right is zero

1613
01:32:02,210 --> 01:32:05,061
and the inner product of left with left

1614
01:32:05,135 --> 01:32:05,952
Student: One or minus one?

1615
01:32:06,024 --> 01:32:09,069
Minus one when you account for this minus

1616
01:32:09,138 --> 01:32:13,116
The result is they cancel, right?

1617
01:32:13,190 --> 01:32:14,488
One minus one

1618
01:32:14,560 --> 01:32:21,906
Equivalently the inner product of uh, of uh, of uh

1619
01:32:21,978 --> 01:32:23,436
Well, ok, zero

1620
01:32:24,021 --> 01:32:26,883
So first of all those zero are over here

1621
01:32:26,953 --> 01:32:30,812
That's the up up matrix element

1622
01:32:30,882 --> 01:32:33,335
It's easy to get the down down matrix element

1623
01:32:33,416 --> 01:32:34,661
To get the down down matrix element

1624
01:32:34,768 --> 01:32:36,323
you would do exactly the same thing

1625
01:32:36,430 --> 01:32:39,556
you except you start with right minus left

1626
01:32:39,659 --> 01:32:45,145
right minus left, right minus left

1627
01:32:45,215 --> 01:32:46,555
sigma x

1628
01:32:46,663 --> 01:32:50,095
and then you would say when sigma x hits right

1629
01:32:50,200 --> 01:32:51,412
it's plus right

1630
01:32:51,523 --> 01:32:54,058
When it hits left, it gives minus left

1631
01:32:54,129 --> 01:32:55,479
When sigma x acts

1632
01:32:55,480 --> 01:33:00,609
it takes right minus left to right plus left

1633
01:33:00,718 --> 01:33:05,366
and what's the inner product of right plus left with my right minus left

1634
01:33:05,414 --> 01:33:08,052
Zero again

1635
01:33:08,162 --> 01:33:09,574
Right with right is one

1636
01:33:09,684 --> 01:33:10,973
Left left is minus one

1637
01:33:11,081 --> 01:33:12,959
The cross term is just zero

1638
01:33:13,063 --> 01:33:15,781
So this is zero over here

1639
01:33:15,854 --> 01:33:19,945
Now, we are going to get zeros, but that can't be right

1640
01:33:20,016 --> 01:33:29,213
So let's try, let's try up down

1641
01:33:29,321 --> 01:33:30,835
Alright, we try to up down

1642
01:33:30,937 --> 01:33:32,705
Let's put up down there

1643
01:33:32,778 --> 01:33:37,164
Up will be right plus left, will have sigma x

1644
01:33:37,214 --> 01:33:40,919
and then down was right minus left

1645
01:33:41,066 --> 01:33:44,582
What happens when sigma x is right plus left?

1646
01:33:44,655 --> 01:33:49,963
It makes right minus left

1647
01:33:50,076 --> 01:33:52,787
and now what's the inner product?

1648
01:33:52,828 --> 01:33:55,207
Inner product of right with the right is one

1649
01:33:55,314 --> 01:33:59,457
The inner product of minus left with minus left is also plus one

1650
01:33:59,559 --> 01:34:02,538
So they add, and again

1651
01:34:02,608 --> 01:34:05,619
The cross terms between right and left are zero

1652
01:34:05,690 --> 01:34:09,955
So we get two divided by two which is one

1653
01:34:10,060 --> 01:34:14,357
Is one up here and if we work it out

1654
01:34:14,467 --> 01:34:19,270
we will get another one over here

1655
01:34:19,409 --> 01:34:28,441
So now we've got sigma y

1656
01:34:28,549 --> 01:34:32,279
Let's just stop for a minute and say what is it that we have

1657
01:34:32,391 --> 01:34:36,525
We have a pair of linear operators, a pair of matrices

1658
01:34:36,667 --> 01:34:41,286
whose eigenvectors are

1659
01:34:41,396 --> 01:34:45,196
This fellow's eigenvectors here are up and down

1660
01:34:45,304 --> 01:34:48,473
This fellow's eigenvectors are right and left

1661
01:34:48,580 --> 01:34:52,596
They are exactly what we were talking about on the previous blackboard

1662
01:34:52,672 --> 01:34:57,345
The states in which the observables

1663
01:34:57,421 --> 01:35:01,969
have unambiguous results are the eigenvectors

1664
01:35:02,039 --> 01:35:04,073
That's what we are now

1665
01:35:04,149 --> 01:35:07,063
The last one is sigma y, Ok?

1666
01:35:07,135 --> 01:35:11,180
What do we know about sigma y? I am not

1667
01:35:11,291 --> 01:35:15,033
In and out are the eigenvectors

1668
01:35:15,102 --> 01:35:17,133
Student: So the one what we did is just sigma x, right?

1669
01:35:17,201 --> 01:35:19,103
The one we did is sigma x

1670
01:35:19,177 --> 01:35:21,497
Student: Yours is sigma y

1671
01:35:21,605 --> 01:35:23,206
Oh, sorry

1672
01:35:23,307 --> 01:35:26,462
Student: Yes

1673
01:35:26,598 --> 01:35:29,213
I don't think I wrote sigma y

1674
01:35:29,284 --> 01:35:31,590
I think I just failed to cross the x

1675
01:35:31,699 --> 01:35:36,227
I think, maybe I wrote sigma y

1676
01:35:36,295 --> 01:35:38,655
Alright, the next one is sigma y

1677
01:35:38,764 --> 01:35:40,273
and here I'm going to leave some of the words,

1678
01:35:40,342 --> 01:35:41,572
some of the algebras

1679
01:35:41,679 --> 01:35:44,885
It's very easy to you but let's see what the uh,

1680
01:35:44,954 --> 01:35:51,079
What the rules are

1681
01:35:51,154 --> 01:36:02,530
Oh, incidentally, no, I think we are Ok

1682
01:36:02,604 --> 01:36:07,404
Okay so there was in and out

1683
01:36:07,556 --> 01:36:10,986
I cannot remember which one was which

1684
01:36:11,088 --> 01:36:14,065
So I am going to fake it

1685
01:36:14,173 --> 01:36:16,051
and if the comes out wrong, it comes out wrong

1686
01:36:16,159 --> 01:36:30,172
Uh, I think in was up, plus, I think, i times down

1687
01:36:30,245 --> 01:36:36,121
and out was up minus i times down

1688
01:36:36,194 --> 01:36:56,950
and I left out a factor of square root of two

1689
01:36:57,095 --> 01:36:57,884
Incidentally you can check

1690
01:36:58,028 --> 01:37:09,022
that in and out are orthogonal to each other

1691
01:37:09,165 --> 01:37:11,079
Ok, these are in and out

1692
01:37:11,185 --> 01:37:14,107
Can we find a matrix

1693
01:37:14,178 --> 01:37:22,987
which in and out are eigenvectors

1694
01:37:23,058 --> 01:37:26,119
with eigenvalue plus one and minus one

1695
01:37:26,227 --> 01:37:26,819
It's getting late

1696
01:37:26,885 --> 01:37:28,707
It's very easy

1697
01:37:28,779 --> 01:37:29,962
leave it to you

1698
01:37:30,069 --> 01:37:32,401
I will simply tell you what that matrix is

1699
01:37:32,558 --> 01:37:34,570
It's unique

1700
01:37:34,675 --> 01:37:38,308
If you know the eigenvectors and if you know the eigenvalues

1701
01:37:38,454 --> 01:37:38,944
It's unique

1702
01:37:39,015 --> 01:37:43,157
One way of doing it is to solve for up and down

1703
01:37:43,262 --> 01:37:45,726
in terms of in and out

1704
01:37:45,796 --> 01:37:47,186
Once you solved for up and down

1705
01:37:47,289 --> 01:37:50,517
in terms of in and out and you know how the matrix acts on in and out

1706
01:37:50,589 --> 01:37:52,082
you've finished

1707
01:37:52,151 --> 01:37:56,804
The last one in the multiple here

1708
01:37:56,908 --> 01:38:01,447
Now I am writing sigma y

1709
01:38:01,546 --> 01:38:03,709
Similar to this one

1710
01:38:03,855 --> 01:38:10,684
But it's all minus i and i

1711
01:38:10,758 --> 01:38:12,803
Alright, so a little exercise

1712
01:38:12,851 --> 01:38:15,242
You don't really need to go and prove this

1713
01:38:15,350 --> 01:38:16,882
You can do a different thing

1714
01:38:16,989 --> 01:38:19,056
You can take this matrix

1715
01:38:19,163 --> 01:38:23,586
and show that these are the eigenvectors of it

1716
01:38:23,693 --> 01:38:25,228
Just show that these are the eigenvectors of it

1717
01:38:25,338 --> 01:38:27,084
If you take this matrix

1718
01:38:27,194 --> 01:38:29,225
and show that if its eigenvectors 

1719
01:38:29,296 --> 01:38:29,713
What is that mean?

1720
01:38:29,819 --> 01:38:34,880
It means its eigenvectors are one i and one minus i

1721
01:38:34,989 --> 01:38:37,860
The square root of two doesn't matter

1722
01:38:37,968 --> 01:38:39,224
Eigenvectors are eigenvectors

1723
01:38:39,331 --> 01:38:42,420
They don't care about the square root of two

1724
01:38:42,564 --> 01:38:45,939
These are the three Pauli matrices

1725
01:38:46,047 --> 01:38:48,067
They represent the components of the spin

1726
01:38:48,176 --> 01:38:49,848
in different directions

1727
01:38:49,956 --> 01:38:53,377
Uh, I think, uh, that's enough for tonight

1728
01:38:53,482 --> 01:38:54,626
What I was going to do

1729
01:38:54,733 --> 01:38:57,325
and I have in my notes was exactly this question

1730
01:38:57,433 --> 01:39:00,373
what happens if you have a component of spin

1731
01:39:00,480 --> 01:39:03,322
along an arbitrary direction

1732
01:39:03,426 --> 01:39:06,266
Can we show using what we know?

1733
01:39:06,337 --> 01:39:08,446
But the problem, I will do this next time

1734
01:39:08,601 --> 01:39:10,225
We will do this next time

1735
01:39:10,329 --> 01:39:14,877
If we have a component of spin along an arbitrary direction

1736
01:39:14,983 --> 01:39:18,094
and we want to calculate using the principles

1737
01:39:18,201 --> 01:39:22,680
We want to calculate what the probability is

1738
01:39:22,822 --> 01:39:27,820
for the measurement of sigma z in various, either plus or minus

1739
01:39:27,821 --> 01:39:29,000
We should be able to do it

1740
01:39:29,071 --> 01:39:30,849
We have enough information

1741
01:39:30,920 --> 01:39:33,429
I think it's probably more than

1742
01:39:33,537 --> 01:39:35,627
than what we can get handle for tonight

1743
01:39:35,732 --> 01:39:37,870
We did a lot for tonight

1744
01:39:37,973 --> 01:39:40,561
Probably more than reasonable

1745
01:39:40,707 --> 01:39:43,758
Student: Am I wrong to make the u be negative

1746
01:39:43,829 --> 01:39:47,094
Student: to solve the problem?

1747
01:39:47,169 --> 01:39:49,358
Student: Is one of the u is negative?

1748
01:39:49,467 --> 01:39:57,649
No, you know you're missing? You're missing to have to complex conjugate

1749
01:39:57,759 --> 01:40:00,977
If you want to inner product of in with out

1750
01:40:01,081 --> 01:40:03,033
you multiply one times one

1751
01:40:03,141 --> 01:40:08,824
and then add I, not times minus sign but i times plus i

1752
01:40:08,933 --> 01:40:17,511
Yeah, yeah, the only way is that somebody like me know so quickly

1753
01:40:17,613 --> 01:40:18,976
why what you're thinking

1754
01:40:19,082 --> 01:40:26,356
is having made the same mistake of course, (laugh)

1755
01:40:26,357 --> 01:40:30,838
Sometime in the past, Ok

1756
01:40:30,983 --> 01:40:32,226
Alright, we've finished with tonight I think

1757
01:40:32,333 --> 01:40:35,010
and I left these questions

1758
01:40:35,117 --> 01:40:38,000
For more, please visit us at stanford dot edu

