qm8.eng.final.srt

﻿1
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Stanford University 

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I was talking about what happens during or from beginning from before

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A measurement to after a measurement 

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We are talking about what happens to the evolution 

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of an isolated black hole left to itself 

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between the time that it was prepared and the time it was measured 

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Okay, nevertheless I could answer a question

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Is entanglement inevitably irreversible?

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If two things which are not entangled come together 

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interact with each other

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Do whatever they do and become entangled

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Is it irreversible?

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Not necessarily anything that can happen The opposite can happen

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Now we're not talking about an external apparatus measuring the system 

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we are simply talking for example about two spins 

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two spins interact with each other come together 

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go off an they are entangled 

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what happens is it possible to disentangle them by,

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for example, reflecting them, letting them come together and interact again

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Can they become disentangle?

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Yes, yes, there is no rule that says entanglement is irreversible

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On the other hand, if an entanglement happens 

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and you take one of the electrons off and throw it away 

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then the other one is stuck there left being entangled with the thing that 

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just not gonna come back, not gonna come back 

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unless, of course, somebody out there reflect it back and 

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so yeah entanglement is re- is reversible

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But as far as you're concerned,

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the measurements that you do are irreversible 

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that's a very peculiar situation, you have to think about it

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Understand what it means 

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we are gonna move off the question of these very simple systems 

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spins sometimes called qubits, tonight at some point,

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but I want to just go over just very briefly,

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I think I probably discussed this in class last time,

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but nevertheless it's so important that I wanna hammer on it, 

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first of all there was a notion of the density matrix

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if we have two systems Alice's system, and Bob system 

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and Alice's system or Bob system are far away from each other 

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so that Alice doesn't get to do any experiments on Bob system 

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Bob does not get to do experiments on Alice's system,

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but they are entangled either they are or they are not entangled 

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they may be entangled

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Is their description that Alice by herself uses to describe just her subsystem 

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the answer is yes, and the working description of her sub system is called the density matrix

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so limited review that very very quickly and then go back over some concepts 

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if the wave function of the Alice Bob system

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The combined system of Alice's subsystem and Bob's

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the wave function is called Psi of a and b,

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b for Bob a for Alice then Alice's, Alice's density matrix

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Alice's density matrix is defined to be psi star,

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the complex conjugate of the wave function,

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I know,  let's say it rho it's rho, called the density matrix,

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it's indicated by rho, it has subscript a and a prime because it's a matrix 

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but it doesn't have anything to do a Bob 

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so it's purely a matrix with respect to Alice's variables 

57
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let's indicate that by calling this Alice's density matrix 

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and its given by Psi star of a b a prime b, I was getting confused 

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a prime psi of ab, summed over Bob's variables 

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it doesn't depend on Bob's variables because they have been summed over

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when you sum over something it no longer depends on that set of variables,

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so this is only a function of Alice's variables 

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and it contains all possible all possible um, statistical information 

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that Alice can ever know about her half of the system 

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unless Bob comes in interact with her system of course 

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if Bob comes from wherever he is and give,

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her system a shove of some sort that will change her density matrix,

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but as long as Bob stays away from her system 

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Bob's in Palo Alto Alice's on Alpha Centauri 

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as long as Bob does not interact directly with her system 

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her density matrix doesn't change, no matter what Bob does

72
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let me give a little proof of that 

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let's take Bob's subsystem to not be so narrow, just to be the little spin 

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that is carrying around and also Alice's 

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let's imagine that Alice's subsystem consists of everything in Alice's neighborhood

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everything that she can have any contact with 

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everything in Bob system is everything in Bob's neighborhood,

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Maybe even including Bob, and now Bob is going to do something 

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he's going to do something interact of course he can only interact directly,

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Interact mean he may turn on a magnetic field,

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he may do all sorts of things ought to change the evolution of his sub system 

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he's gonna make a subsystem evolve in some way

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One of the ways which Bob can cause his sub system to evolve many many ways,

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but whatever they are,

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they are always governed by unitary matrices,

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let's come over here and forget Alice for a minute and ask what Bob can do is subsystem 

87
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he has a wave function, let's leave out Alice and the picture altogether 

88
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just suppose that Bob has a wave function and he's gonna manipulate the system somehow 

89
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he's gonna manipulate the system somehow 

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and send it to a new wave function 

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rule is the new wave function of his sub system 

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must be related to the old wave function by a unitary operator 

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unitary operators of the things which govern time evolution go back

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Remember the time evolution, unitary operators,

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and that means it psi of b becomes replaced by sum over b 

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some unitary matrix                                            

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b b prime psi of b prime in terms of wave functions in matrices

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This is the description of what happens to Bob's matrix  

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sorry Bob's column vector is basically called Bob's column vector 

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describing his half of the world

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or his system and to find out what happens to it no matter what he does 

102
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even if he doesn't do anything but let's suppose he does something, he does something,

103
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it generates a unitary evolution of his wave function 

104
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now how do we describe the same thing in the composite language 

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in the composite language of the wave function is psi of a and b 

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Alice's and Bob's variable 

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what happens to it when Bob manipulates his half of the world 

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exactly the same thing with Alice's variables be just spectators 

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just being passive spectators so psi of a and b just becomes psi of 

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oops, put in there b prime, sum of b prime,

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this is a function of b and you sum over b prime,

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okay that's again a sum over b prime of U of b and b prime 

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psi of a and b prime, a just passed through 

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Alice know nothing whatever about the about this whole process 

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her subsystem was not in any way influenced her index 

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her part of the wave function Her half of the dependence of the wave function 

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was uninfluenced by Bob's manipulation 

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so let's go back and take the manipulated wave function 

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This is the manipulated wave function 

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call it psi, the manipulated wave function of b and a, of a and b that's this,

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It's been modified by the action of Bob's unitary operator,

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Okay now let's recalculate let's recalculate Alice's density matrix 

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after all Alice's density matrix contains all information that you can ever know 

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about her half of the world, so let's see what happened to it

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Well we first of all have to take the manipulated psi star 

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We have to sum over b, but now we take the manipulated psi star

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Oh we hadn't written psi star,

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Sorry about that we have to write psi star manipulated of 

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and we want to know about it with index a prime and b

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a prime and b, and now we get just to complex conjugating it 

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this tells us how psi changes,

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let's just work out the complex conjugate that's the complex conjugate of a prime 

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>> Where is a and b prime? 

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this is a prime b, just a prime and b 

135
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but here since b prime is a summation variable 

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we do not want to write b prime here, 

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b prime was the summation variable used over here,

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Let's just invent another one called b double prime 

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we are gonna be, that's gonna be the summation variable in a moment 

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and U complex conjugate of b and b double prime

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U complex conjugate of course is a sum over b double prime 

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and the only reason I called it b double prime instead of b 

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was so as not to confuse it with b prime 

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here's one sum and here is a separate sum 

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Now here is something what do we know about unitary operators 

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unitary operators are operators that if we interchange b 

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if we interchange the two indices and complex conjugate what happens?

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That's called it's called the inverse operator or the Hermitian conjugate 

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for U, in general, it's called the Hermitian conjugate specifically for unitary,

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it becomes the inverse operator,

151
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Let's just call it the Hermitian conjugate 

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if I un complex conjugate and interchange these two,

153
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sorry b double prime b double prime, b 

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and this becomes Hermitian conjugate 

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alright, so this is the usual story if operators operate to the right on ket vectors,

156
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the corresponding action on bra vectors is given by the by the Hermitian conjugate 

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So now all I have to do is plug it 

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we plug this in for the, for the manipulated psi star of summation over b b double prime

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psi star of a prime and b double prime times U dagger of b double prime and b 

160
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it's a nuisance too many indices for me to be comfortable and blackboard,

161
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but I think this is worth doing 

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now what about psi, psi is the other half it comes into the density matrix,

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and that's U, b b prime psi of what psi of a and b prime, complicated little mess 

164
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oh what do we summing over, we are summing over all the b-s basically 

165
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we are summing over this b over here,

166
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let's put the b summation here, 

167
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we were summing over b double prime to get the manipulated wave 

168
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for this complex conjugate,

169
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but we also have to sum over b prime here 

170
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sum over all the bs this side 

171
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each b is a summation variable by the time we finished 

172
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Alright but now look at what we have, 

173
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it looks complicated but what is this thing over here? 

174
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Well not quite yet , I will wonder, it's essentially correct,

175
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but of course what you mean is the unit matrix

176
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This is  the product matrix of U dagger 

177
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this is a unitary matrix so U dagger is used inversed 

178
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b double prime b b b prime,

179
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this is just a Kronecker Delta b double prime b prime,

180
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it just instructs you to set b double prime equal to b prime 

181
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when you do this sum, when you do one of the two sum

182
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let's say the b double prime sum 

183
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so all of these then becomes sum over b prime,

184
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well, let's see what have we got, we set b double prime 

185
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We've done b sum over b that's done, oh sum over b we just did it,

186
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and now we set b double prime equal to b prime 

187
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psi star of a prime b prime, psi of a and b prime,

188
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that's it doesn't matter whether we call b prime b prime or whether we call it b 

189
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because it's a summation variable so let's do that 

190
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Let's just call it b, and what do we have we have exactly I hope, I pray 

191
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the original density matrix so Bob, no matter what he does 

192
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as long as what he does is described by a unitary matrix 

193
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Bob will have absolutely no effect on Alice's density matrix 

194
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does not matter whether entangled or not

195
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No way is that I specify that this was a product state  

196
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That is what physicists call locality 

197
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the inability of Bob to influence any statistical prediction of Alice's 

198
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by doing a manipulation on his own degrees of freedom 

199
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locality is not violated, this form of locality

200
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This form of locality is not violated in any way by the phenomenon of entanglement  

201
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that's the technical argument, notice that it does depend on unitarity 

202
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so what do unitarity do, unitarity had to do with conservation of distinctions 

203
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what I called the minus first law,

204
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at least in its quantum mechanical version 

205
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so funny and interesting interplay 

206
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funny, interesting interplay between the conservation of distinctions on the one hand 

207
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and locality

208
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If Bob somehow had access to changes which were not described by unitary operators,

209
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this would not have become the unit matrix over here

210
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and the final density matrix, Alice's final density matrix indeed would have been modified 

211
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this is interesting the way the whole thing holds together 

212
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this is why it's so hard to change quantum mechanics

213
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you wanna play a little game,

214
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let's let Bob tha's Bob's evolution, not quite be unitary,

215
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only approximately unitary, 

216
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well then immediately you start sending signals faster than speed of light 

217
00:18:11,784 --> 00:18:16,243
How? by instantly modifying Alice's density matrix 

218
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so things hold together tightly 

219
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and it's hard to change them 

220
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good so yes so 

221
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>>Is it correct to say that Bob's matrix is unitary?

222
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He's not able to change Alice's, Alice's system

223
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He's changed his own system by doing something 

224
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in the doing something was described by this unitary matrix 

225
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>> that's the part I confused, if you are doing something 

226
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>> Isn't he going to violate the unitarity?

227
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For example, he might be turning a magnet,

228
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He might be turning a magnet in such a way 

229
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that it has an influence on his spin 

230
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What is that influence? It will change the state of the spin

231
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For example,  if you have a spin in a given direction and it's coupled to a magnetic field

232
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Then you turn the magnetic field in some way  

233
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It will typically change the evolution of the spin 

234
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So Bob can do that, he can change a magnetic field,

235
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it's acting on certain spin,

236
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the result will be to change the way the spin evolves 

237
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that will change his own measurements of the spin of his spin 

238
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if he does act by rotating a magnetic field 

239
00:19:56,884 --> 00:20:02,193
which is mathematically described by some sort of evolution of matter what 

240
00:20:02,197 --> 00:20:05,086
some sort of evolution, it's described by unitary operator 

241
00:20:06,085 --> 00:20:13,575
then the that he does change the results of his own measurements 

242
00:20:16,711 --> 00:20:19,711
he rotates a magnetic field, but it doesn't rotated his apparatus 

243
00:20:20,278 --> 00:20:23,995
the result is he is gonna change the probability distributions of his own spin,

244
00:20:25,068 --> 00:20:28,483
but he will not change any of the probabilities of Alice's spin

245
00:20:29,051 --> 00:20:30,497
okay 

246
00:20:30,498 --> 00:20:34,515
>> So what you are saying is quantum mechanics always violates locality?

247
00:20:36,521 --> 00:20:44,423
The locality defined by the inability of Bob to influence Alice's measurements

248
00:20:46,054 --> 00:20:49,504
So what's the big deal? Is that what you are gonna ask?

249
00:20:51,734 --> 00:20:55,507
I was guessing what you are gonna ask

250
00:20:58,804 --> 00:21:02,308
>>single particle wave function, put a beam over 

251
00:21:02,949 --> 00:21:05,141
>>it goes off to two different direction 

252
00:21:05,755 --> 00:21:09,276
>>one of your detector is faster than the speed of light

253
00:21:09,696 --> 00:21:10,590
>> then the wave function collapsed

254
00:21:10,591 --> 00:21:12,620
What what what what, you do what?

255
00:21:12,630 --> 00:21:15,540
>>You detect the photon at one detector 

256
00:21:17,309 --> 00:21:20,671
>>the function of the other detector collapse instantaneously

257
00:21:22,616 --> 00:21:28,691
I tell you what happened, density matrix of the other system does not change, period

258
00:21:29,981 --> 00:21:41,594
>>

259
00:21:41,604 --> 00:21:47,635
I know I know I know people, look I'm doing my absolute best to tell it to you 

260
00:21:47,636 --> 00:21:51,960
as it really is and cut through the craps that's out there 

261
00:21:54,786 --> 00:21:56,211
cut through the crap that's out there 

262
00:21:56,830 --> 00:22:00,501
because there is a huge amount of there is a pile of mile high,

263
00:22:01,925 --> 00:22:03,309
and we somehow want to cut through that 

264
00:22:04,650 --> 00:22:07,564
this is the mathematics of cutting through it 

265
00:22:08,268 --> 00:22:12,411
now, nevertheless there was some pretty damn smart people 

266
00:22:13,110 --> 00:22:15,809
who are uncomfortable about something

267
00:22:15,903 --> 00:22:17,994
Something that they call non-locality 

268
00:22:19,003 --> 00:22:21,679
You know Einstein was no jerk, he was pretty smart 

269
00:22:23,389 --> 00:22:24,678
He ain't called Einstein for nothing

270
00:22:24,679 --> 00:22:26,529
That's my favorite joke

271
00:22:32,404 --> 00:22:35,788
John bell was pretty smart They didn't call him Einstein though

272
00:22:38,435 --> 00:22:40,807
So there's something going on, something going on

273
00:22:43,800 --> 00:22:49,397
I'm gonna get, I'm reviewing last week, so very quick 

274
00:22:49,398 --> 00:22:56,841
>>as you and that is the and idea of Locality means that 

275
00:22:57,491 --> 00:23:00,043
>> the state don't effect each other

276
00:23:00,044 --> 00:23:06,051
>>the density, and the confusing part is, 

277
00:23:07,578 --> 00:23:15,623
>>it seems two different things get mixed up and now, Is that accurate or?

278
00:23:16,187 --> 00:23:18,765
No it's positive, but it's true

279
00:23:23,710 --> 00:23:28,656
I will try to tell you as I understand it,

280
00:23:30,257 --> 00:23:36,856
what the concrete operational facts are

281
00:23:36,857 --> 00:23:42,518
I was a little bit tired last time, and I'm not sure I was completely clear 

282
00:23:42,648 --> 00:23:45,812
So I want to go over it again because it's very very interesting

283
00:23:47,269 --> 00:23:50,592
It really is at the heart of all of the confusions about confusions

284
00:23:50,593 --> 00:23:52,262
Confusions I mean, I must say 

285
00:23:52,263 --> 00:23:55,490
including my own internal confusions about quantum mechanics 

286
00:23:56,935 --> 00:24:00,794
when I first learned about the Einstein Rosen Podolsky 

287
00:24:03,098 --> 00:24:05,858
I don't know who discovery, I think maybe Schrodinger 

288
00:24:06,709 --> 00:24:08,089
says something about entanglement,

289
00:24:09,166 --> 00:24:11,090
but it was Einstein Rosen Podolsky,

290
00:24:11,582 --> 00:24:13,611
who really called attention to peculiar was 

291
00:24:14,971 --> 00:24:18,576
basically what they said is we have a very funny situation here where 

292
00:24:18,610 --> 00:24:22,152
although we can know everything that can possibly be known about a composite system 

293
00:24:22,943 --> 00:24:24,902
we know nothing at all about its constituent

294
00:24:25,636 --> 00:24:29,256
That's crazy violates my sense of reality that violates my sense 

295
00:24:30,128 --> 00:24:32,477
of what it means to know what a system is doing 

296
00:24:33,972 --> 00:24:36,069
violates all sorts of things, when I first heard about it

297
00:24:36,070 --> 00:24:41,026
I was actually before John Bell some time in the 60s when a student 

298
00:24:41,027 --> 00:24:45,798
when I wasn't a student, or when I was a young assistant professor then

299
00:24:45,813 --> 00:24:47,412
Think about it after about a lot 

300
00:24:48,689 --> 00:24:49,440
try to figure out what it means 

301
00:24:50,512 --> 00:24:58,002
I was sort of inclined to try to think about it in terms of simulating quantum mechanics,

302
00:24:58,003 --> 00:24:58,832
and I still am

303
00:24:59,193 --> 00:25:03,465
What are the limits if if if quantum mechanics is so very strange 

304
00:25:03,999 --> 00:25:06,419
and so very different than classical physics 

305
00:25:06,486 --> 00:25:12,911
does it mean that you can't fool somebody into thinking the system is quantum mechanical 

306
00:25:12,912 --> 00:25:16,507
if all you have your resources are entirely classical 

307
00:25:17,814 --> 00:25:19,533
and the answer is yes and no 

308
00:25:20,223 --> 00:25:22,160
so I know I talked about this last time,

309
00:25:22,245 --> 00:25:23,859
but it really is at the center of thing

310
00:25:23,860 --> 00:25:26,108
Let's go back over it quick, quick as I can

311
00:25:26,641 --> 00:25:29,054
we get a build ourselves a quantum simulator

312
00:25:29,277 --> 00:25:34,732
I think somebody built one on the site I came across about an hour ago, who's that?

313
00:25:36,255 --> 00:25:36,866
Does it work?

314
00:25:37,988 --> 00:25:40,035
>>It doesn't really simulate yet

315
00:25:41,582 --> 00:25:43,531
Is that picture your picture?

316
00:25:44,175 --> 00:25:47,705
There's a picture of Einstein, Heisenberg and Einstein I don't like this

317
00:25:48,803 --> 00:25:49,334
Heisenberg's of 

318
00:25:51,257 --> 00:25:52,351
>>Einstein said,

319
00:25:52,352 --> 00:26:02,355
I will like this you are you put it,

320
00:26:02,356 --> 00:26:03,228
I think that captures it

321
00:26:04,866 --> 00:26:08,261
let's talk about the nature of quantum simulators are 

322
00:26:09,274 --> 00:26:11,506
first of all I just want to simulate a single spin

323
00:26:11,828 --> 00:26:13,165
I think I told you how to do it,

324
00:26:13,240 --> 00:26:14,494
but let's write down the steps 

325
00:26:14,938 --> 00:26:20,686
we have a computer screen, here's a computer screen on the screen

326
00:26:21,361 --> 00:26:24,858
We have a picture of the detector I will write down a detector apparatus 

327
00:26:25,292 --> 00:26:29,399
and the apparatus can be oriented in any direction in three-dimensional space

328
00:26:29,400 --> 00:26:32,109
We need to do some of three-dimensional graphics here 

329
00:26:33,843 --> 00:26:39,644
and it has a window and the window shows one of three possible states

330
00:26:40,298 --> 00:26:44,497
One is called blank, the that's the that's the detector of the apparatus 

331
00:26:44,884 --> 00:26:47,218
before its interacted with anything 

332
00:26:47,733 --> 00:26:53,258
so it can be b or plus or minus one, b is not important,

333
00:26:53,985 --> 00:26:57,436
it's just there to say before the detector interact with anything 

334
00:26:58,013 --> 00:26:59,425
There is a button over here called m 

335
00:27:01,247 --> 00:27:05,917
and the operator of the system of the computer game, 

336
00:27:06,534 --> 00:27:10,523
basically all you get to do was press M and then look at the result of 

337
00:27:11,029 --> 00:27:13,251
M stands for measure performs a measurement

338
00:27:14,031 --> 00:27:20,007
Now all I hear connected to the computer by wires of our cables,

339
00:27:20,514 --> 00:27:24,907
Connected to the screen by cables is a computer 

340
00:27:25,566 --> 00:27:27,873
and the computer stores information in its memory,

341
00:27:27,874 --> 00:27:32,397
and the basic information that it stores is two complex numbers 

342
00:27:33,029 --> 00:27:34,472
alpha up and alpha down 

343
00:27:35,201 --> 00:27:38,131
those are supposed to be the components of the wave function of the spin 

344
00:27:39,395 --> 00:27:41,034
I didn't say there is the spin here,

345
00:27:41,271 --> 00:27:42,551
but that's sort of what we are talking about 

346
00:27:43,461 --> 00:27:44,482
alpha up and alpha down, 

347
00:27:46,447 --> 00:27:52,001
the computer only can generate alphas which satisfy alpha star alpha up

348
00:27:52,304 --> 00:27:54,101
Plus Alpha star alpha down is equal to one 

349
00:27:55,545 --> 00:27:58,495
Whenever it generate them, and it can do something else 

350
00:27:58,496 --> 00:28:01,863
It can do two other things 

351
00:28:02,585 --> 00:28:08,003
it can update the alphas by solving the Schrodinger equation for them

352
00:28:08,581 --> 00:28:10,560
There's some Hamiltonian For example,

353
00:28:10,630 --> 00:28:15,704
It could be that the spins are in a, the spin is in an imaginary magnetic field

354
00:28:16,144 --> 00:28:19,181
In which case the alphas would change with time,

355
00:28:20,185 --> 00:28:24,215
the computer knows how to do that, it knows how to solve Schrodinger's equation 

356
00:28:24,328 --> 00:28:27,038
computers can solve Schrodinger's equation beautifully 

357
00:28:28,657 --> 00:28:32,408
and it updates these as a function of time 

358
00:28:33,870 --> 00:28:36,921
updates them according to the rules of quantum mechanics

359
00:28:37,223 --> 00:28:40,958
One of the thing the cables are instantaneous

360
00:28:41,228 --> 00:28:47,753
They convey information instantly and so will have one more element

361
00:28:47,754 --> 00:28:51,193
The other element will be a random number generator

362
00:28:51,488 --> 00:28:53,742
Let me just say some words about random number generators 

363
00:28:53,908 --> 00:28:56,558
random number generators sounds we are introducing quantum mechanics

364
00:28:56,843 --> 00:28:58,178
No, we don't need quantum mechanics 

365
00:28:59,129 --> 00:29:03,605
you can simulate to a very very high degree of confidence 

366
00:29:04,706 --> 00:29:07,887
random sequence of random numbers of random number generator

367
00:29:08,343 --> 00:29:12,853
Purely out of classical classical considerations you how to do it,

368
00:29:13,457 --> 00:29:15,830
you take the digits of pi whatever and 

369
00:29:16,715 --> 00:29:19,823
you make something that looks would fool anybody,

370
00:29:20,592 --> 00:29:23,207
including a sophisticated mathematician into believing 

371
00:29:24,213 --> 00:29:26,300
that you are generating a set of random numbers 

372
00:29:26,349 --> 00:29:28,695
There is a random number generator inside the computer 

373
00:29:29,769 --> 00:29:33,382
the operator starts by initializing of the system 

374
00:29:33,383 --> 00:29:42,550
and initializing the system simply means starting out some orientation for the detector 

375
00:29:42,572 --> 00:29:44,332
for the apparatus doesn't matter what,

376
00:29:45,059 --> 00:29:47,853
and setting the alphas to some initial values 

377
00:29:49,568 --> 00:29:52,833
you can plug in what he wants he wants the initial values to be,

378
00:29:53,457 --> 00:29:56,567
or the computer sets them according to whatever rule, it likes 

379
00:29:57,740 --> 00:30:03,600
then the system runs for a period of time an indetermined period of time 

380
00:30:04,801 --> 00:30:12,245
updating the alphas until the operator decides to press M 

381
00:30:13,538 --> 00:30:16,938
a certain amount of time is elapsed the operator decides how much time 

382
00:30:17,555 --> 00:30:20,339
and presses M at that point,

383
00:30:21,694 --> 00:30:26,431
the random number generator generates a plus one or minus one,

384
00:30:27,361 --> 00:30:31,220
but with probability alpha star alpha up that it's plus one,

385
00:30:32,826 --> 00:30:36,063
sorry, alpha up alpha up star plus one,

386
00:30:36,810 --> 00:30:41,283
or else alpha star alpha down that's minus one 

387
00:30:41,997 --> 00:30:45,390
remember the sum of these two alpha star alpha add up to one 

388
00:30:46,311 --> 00:30:49,059
so we went on to be a random number generator,

389
00:30:50,189 --> 00:30:56,149
the computer sends a signal with a probability of the appropriate probability 

390
00:30:57,293 --> 00:31:04,235
to the detector and the detector detects or registers what the computer sent it 

391
00:31:05,436 --> 00:31:07,689
That's one measurement

392
00:31:08,353 --> 00:31:15,104
Now as soon as that happens the computer reinitializes the alphas 

393
00:31:15,730 --> 00:31:21,087
how does it reinitialize the alphas, if the detector detected plus one,

394
00:31:22,778 --> 00:31:26,492
then it sends back a message that it doesn't really need to send back a message after all,

395
00:31:26,688 --> 00:31:28,023
this message got review here 

396
00:31:28,223 --> 00:31:32,718
so the computer over here knows whether it got plus one or minus one 

397
00:31:33,219 --> 00:31:35,991
if the detector saw a plus one

398
00:31:36,812 --> 00:31:43,087
Then it sets alpha u back to one and sets alpha d to zero

399
00:31:44,779 --> 00:31:48,165
Another words, it does what we call collapsing the wave function 

400
00:31:48,960 --> 00:31:54,316
according to the result, according to the outcome of the measurement

401
00:31:54,597 --> 00:31:59,788
We reinitializes the alphas so what was measured 

402
00:32:01,136 --> 00:32:06,330
you start with probability one for the outcome a probability zero

403
00:32:06,503 --> 00:32:09,556
what shall we call it the anti-outcome,

404
00:32:11,118 --> 00:32:13,690
and then it runs again solve the Schrodinger equation 

405
00:32:13,813 --> 00:32:18,433
does it again the operator make  whatever decision he likes 

406
00:32:19,143 --> 00:32:23,594
what makes him decide to press the M ?

407
00:32:24,199 --> 00:32:28,149
He's an experimental physicist what makes experimental physicists do what they do

408
00:32:28,150 --> 00:32:29,585
I don't really know,

409
00:32:29,822 --> 00:32:33,287
but but he does what experimental physicist 

410
00:32:33,412 --> 00:32:37,329
now and then he presses M and there's a whole operation over again 

411
00:32:37,893 --> 00:32:41,459
it oh sorry before he does,

412
00:32:43,114 --> 00:32:44,955
he may decide to rotate the apparatus 

413
00:32:46,922 --> 00:32:53,008
now the computer in here knows what the orientation of the apparatuses 

414
00:32:53,580 --> 00:32:59,774
so when it calculates it not only calculates the updated alphas,

415
00:32:59,775 --> 00:33:05,605
but it also calculates the probabilities for the rotated detector 

416
00:33:06,037 --> 00:33:07,639
to detect whatever it's going to detect 

417
00:33:08,687 --> 00:33:13,105
that's all algebra and a little bit of that's the purely algebra

418
00:33:13,514 --> 00:33:16,170
The computer makes a kind of calculations that you did 

419
00:33:17,032 --> 00:33:21,525
or presumably did, makes a kind of calculations to calculate 

420
00:33:22,123 --> 00:33:23,339
what the probability would be 

421
00:33:24,243 --> 00:33:27,911
that if the detector was rotated in some arbitrary direction,

422
00:33:28,713 --> 00:33:32,608
then the then what's the probability that it will get up or down

423
00:33:33,975 --> 00:33:36,380
Now who was it that decides to rotate the computer?

424
00:33:37,122 --> 00:33:39,531
Well it's the operator again the experimentalist 

425
00:33:40,472 --> 00:33:44,131
experimentalist not only can press M and but he can also 

426
00:33:45,442 --> 00:33:47,571
reorient the detector anytime he wants

427
00:33:49,650 --> 00:33:54,598
This is about as much as an experimental physicist can do with that spin 

428
00:33:55,750 --> 00:34:02,436
rotate the apparatus measure, rotate the apparatus again measure 

429
00:34:03,282 --> 00:34:05,991
and what's the result of the outcome of this? 

430
00:34:06,237 --> 00:34:11,858
the outcome of this is indistinguishable from what the experimental physicist 

431
00:34:12,342 --> 00:34:12,934
examining the spin would see 

432
00:34:14,147 --> 00:34:15,262
so we simulated quantum mechanics 

433
00:34:17,064 --> 00:34:21,479
alright, now we want to know whether we can simulate quantum mechanics of two spins 

434
00:34:22,631 --> 00:34:27,237
one on Alice's side of the world over here and one on Bob side of the world's 

435
00:34:29,204 --> 00:34:31,305
>> I don't know what the purposes of M

436
00:34:32,701 --> 00:34:38,669
>> When he decided to push it

437
00:34:44,982 --> 00:34:47,102
Ya the computer has been updating all this time 

438
00:34:47,872 --> 00:34:52,846
>> Alright when we talk about our basic model of spin world, the Hamiltonian of it

439
00:34:54,130 --> 00:34:58,776
we did talk about Hamiltonian we talk about Hamiltonian's 

440
00:34:58,785 --> 00:35:02,752
and how Hamiltonian's update the state of the system, you're right,

441
00:35:03,081 --> 00:35:08,390
we didn't discuss how a, we didn't discuss much about Hamiltonians with spin 

442
00:35:09,279 --> 00:35:16,242
yes we did I believe I described the spin and the magnetic field 

443
00:35:16,508 --> 00:35:18,858
and how the average spin process 

444
00:35:20,234 --> 00:35:21,935
>> But we didn't have a magnetic field

445
00:35:23,039 --> 00:35:23,787
>> Last time we talk about

446
00:35:32,939 --> 00:35:35,778
right there is no real magnetic field,

447
00:35:35,779 --> 00:35:39,767
It's just the imaginary magnetic field that the computer knows about 

448
00:35:39,815 --> 00:35:41,401
and where does it come into, 

449
00:35:41,739 --> 00:35:45,481
it comes into the solution of the nature of the exact Schrodinger equation,

450
00:35:45,972 --> 00:35:53,124
okay now we do this over here and precisely the same thing over here,

451
00:35:54,214 --> 00:35:58,628
except instead of alphas we have betas, beta stand for Bob 

452
00:36:01,352 --> 00:36:04,999
exactly the same set up replicated twice 

453
00:36:12,604 --> 00:36:16,214
Bob does his measurement hours of measurements 

454
00:36:16,215 --> 00:36:19,033
And he's perfectly happy with each one says I have a spin 

455
00:36:19,374 --> 00:36:22,340
quantum mechanical, they don't talk to each other,

456
00:36:23,232 --> 00:36:25,694
and each one says my spin is in a state,

457
00:36:27,749 --> 00:36:30,824
the language that we would describe this by quantum mechanically 

458
00:36:31,533 --> 00:36:33,650
is they have simulated a product state,

459
00:36:35,206 --> 00:36:40,918
a product state in which Bob's sorry Alice's wave function is this 

460
00:36:41,659 --> 00:36:46,035
and Bob's wave function is this, and the combined system is in a product state 

461
00:36:47,321 --> 00:36:51,252
supposing now we want to simulate a any state,

462
00:36:52,193 --> 00:36:55,488
a more complete description that includes entangled states 

463
00:36:56,011 --> 00:36:57,313
well we can do it this way,

464
00:36:57,699 --> 00:36:59,577
let's start with a way sort guaranteed to work 

465
00:37:01,092 --> 00:37:04,532
we take the computer we put it in the middle of a single computer 

466
00:37:05,702 --> 00:37:10,676
we can think of a single computer as we like as a pair of computers connected by wires 

467
00:37:10,724 --> 00:37:14,132
but just the fact that they are communicating with each other interacting with each other

468
00:37:14,714 --> 00:37:20,306
Really makes in a single computer and in that single computer we now store 

469
00:37:20,779 --> 00:37:29,120
four complex numbers, the four complex numbers that we store in there are alpha up up,

470
00:37:29,821 --> 00:37:34,792
alpha up down, alpha down up alpha down down,

471
00:37:36,774 --> 00:37:40,904
these complex numbers here are not necessarily products in fact,

472
00:37:41,997 --> 00:37:42,699
let's assume the not products 

473
00:37:43,213 --> 00:37:45,124
another words at this is not a product state,

474
00:37:45,222 --> 00:37:46,884
let's assume it's a highly entangled state,

475
00:37:48,656 --> 00:37:52,084
even a highly entangled state is described by for complex numbers like this 

476
00:37:53,458 --> 00:37:57,139
that's what's in the computer, and what does the computer do with it?

477
00:37:58,519 --> 00:38:02,258
It evolves the Schrodinger equation first of all,

478
00:38:02,587 --> 00:38:06,538
we initialize same deal, we initialize we start with some alphas 

479
00:38:08,304 --> 00:38:16,547
the alphas are allowed to run according to whatever the Schrodinger equation describes them

480
00:38:16,778 --> 00:38:23,210
and at the end, either Bob or Alice or both without knowing what the other one is doing 

481
00:38:23,983 --> 00:38:27,837
press, or as the case may be, don't press their measurement buttons 

482
00:38:28,862 --> 00:38:31,611
once the measurement button has been pressed and 

483
00:38:32,704 --> 00:38:35,540
instantaneous signal goes back to the computer 

484
00:38:36,098 --> 00:38:39,000
the computer grinds out random number generator 

485
00:38:39,710 --> 00:38:44,358
and whichever side was measured sends a signal to the detector 

486
00:38:44,613 --> 00:38:48,004
with appropriate probability either plus one or minus one 

487
00:38:49,044 --> 00:38:56,119
and then reinitializes itself, reinitialize itself in a way that doesn't touch for example 

488
00:38:56,727 --> 00:39:04,081
if Alice does the experiment then it doesn't touch Bob's end of things,

489
00:39:04,972 --> 00:39:10,956
if Bob does the experiment the computer reinitializes his half of the wave function,

490
00:39:11,377 --> 00:39:15,107
which is another way of saying throws away the piece that you didn't measure 

491
00:39:15,755 --> 00:39:19,899
throws away the pieces you didn't measure and re-normalizes the wave function 

492
00:39:20,144 --> 00:39:21,748
so that the total probability is one 

493
00:39:23,641 --> 00:39:28,285
exactly the same thing is, no no real difference with a single system,

494
00:39:28,668 --> 00:39:31,278
you can simulated you can simulate a way 

495
00:39:31,620 --> 00:39:35,865
which would describe all the possible things that Alice and Bob can measure 

496
00:39:36,744 --> 00:39:40,671
into whether or not the system was, was entangled 

497
00:39:42,558 --> 00:39:45,038
where is where we get in trouble 

498
00:39:46,119 --> 00:39:51,790
we get in trouble if after initializing we tried to separate the computers and break the wires

499
00:39:53,895 --> 00:39:55,861
For example, here's what we might do

500
00:39:57,020 --> 00:40:00,865
Given these alphas we might put them into both computers 

501
00:40:02,081 --> 00:40:05,416
take the set of alphas that we generated, put this one into this computer

502
00:40:05,417 --> 00:40:10,096
This one into this computer, computers are not gonna talk to each other

503
00:40:10,948 --> 00:40:12,872
They have to have separate window number generators 

504
00:40:14,936 --> 00:40:16,894
and the random number generators will not talk to each other

505
00:40:18,094 --> 00:40:21,328
Each one contains the full entangled state, 

506
00:40:23,402 --> 00:40:25,512
and then we go through the same operation

507
00:40:26,409 --> 00:40:27,130
what's going to happen?

508
00:40:27,988 --> 00:40:30,057
The computer is simply going to forget

509
00:40:31,807 --> 00:40:34,003
For example, for a highly entangled state,

510
00:40:34,357 --> 00:40:36,134
let's say the entangled state is the singlet state 

511
00:40:38,149 --> 00:40:42,777
then the right answer would be that if Bob and Alice both measure 

512
00:40:44,318 --> 00:40:47,378
they should get opposite values for any spin

513
00:40:48,748 --> 00:40:51,714
if they measure the X component of the spin they should get opposite values,

514
00:40:52,624 --> 00:40:55,778
if they measure Y components of spin, they should get opposite values,

515
00:40:56,554 --> 00:41:01,058
but if when we separate them They have separate random number generators 

516
00:41:02,312 --> 00:41:04,661
the random number generators don't talk to each other 

517
00:41:05,373 --> 00:41:09,121
than guess what's gonna happen is gonna be no correlation of any kind 

518
00:41:09,830 --> 00:41:14,126
between what Bob measure than what Alice measures no,

519
00:41:14,147 --> 00:41:19,352
Bob makes that measure plus and Alice may also measure plus 

520
00:41:19,417 --> 00:41:22,595
because the random number generators are not connected to each other 

521
00:41:23,716 --> 00:41:27,386
So even though both computers may know about the full state of the system

522
00:41:28,656 --> 00:41:30,726
If the random number generators don't talk to each other 

523
00:41:31,564 --> 00:41:37,672
then the results of the experiments will not contain the correlation 

524
00:41:39,758 --> 00:41:43,721
the correlation that Bob and Alice could later come together and 

525
00:41:43,802 --> 00:41:48,121
report Bob would say my first experiments are plus 

526
00:41:49,477 --> 00:41:50,626
if things were working right,

527
00:41:50,820 --> 00:41:55,758
then Alice's should save my first experiments are minus and so forth and so on 

528
00:41:56,508 --> 00:42:00,196
that correlation will not be there once you try to separate the computers 

529
00:42:00,764 --> 00:42:04,327
>> will the random number be correlated if you don't separate the computers?

530
00:42:05,434 --> 00:42:10,274
you can have, you can have a single random number generator 

531
00:42:10,518 --> 00:42:12,160
that sends out signals to both computers 

532
00:42:13,141 --> 00:42:17,278
it sends out a signal plus one to this computer 

533
00:42:17,861 --> 00:42:20,240
and sends out a signal minus one for the computer 

534
00:42:21,798 --> 00:42:24,481
so all you need is a single random number generator,

535
00:42:24,738 --> 00:42:27,128
but it has to be able to talk to both computers 

536
00:42:30,298 --> 00:42:34,513
so this got to be wires in the world, this is gotta be wires in the world,

537
00:42:34,963 --> 00:42:36,576
connecting this side and this side 

538
00:42:37,349 --> 00:42:40,435
and if for no other reason than they can both 

539
00:42:40,456 --> 00:42:42,290
connect to the same random number generator 

540
00:42:42,819 --> 00:42:49,187
>>you will and will just and

541
00:42:49,188 --> 00:42:49,623
Think about it,

542
00:42:53,902 --> 00:42:54,943
that's, that's sort of right

543
00:42:55,002 --> 00:43:02,268
Once you separated them, then what happens next after some measurements are made

544
00:43:02,614 --> 00:43:05,249
doesn't preserve the relationship between them 

545
00:43:06,188 --> 00:43:08,120
they become disconnected from each other

546
00:43:09,673 --> 00:43:10,877
So think about it a little bit

547
00:43:11,708 --> 00:43:16,012
I mean even better try to simulated it, this is, I have never tried,

548
00:43:16,384 --> 00:43:19,475
but I don't think it should be that hard if you tdo a little bit of computer programming 

549
00:43:21,972 --> 00:43:27,149
I've simulated much harder things, than that when my son was a little kid

550
00:43:27,779 --> 00:43:29,905
We had this thing called BASIC anybody remember BASIC 

551
00:43:31,086 --> 00:43:34,370
and he and I designed a basketball game and have friction 

552
00:43:36,529 --> 00:43:37,832
well much harder than this,

553
00:43:38,214 --> 00:43:41,771
so that was my last experience of programming anything 

554
00:43:42,193 --> 00:43:45,145
>> I think in that scenario 

555
00:43:45,703 --> 00:43:47,750
I think you will learn a lot by trying to simulate this 

556
00:43:48,207 --> 00:43:50,805
>> you have detector to pull itself off

557
00:43:52,069 --> 00:43:54,885
>> and get a signal from Alice's

558
00:43:56,616 --> 00:43:58,569
>> some thing could be a spin

559
00:43:58,874 --> 00:44:00,286
>>let's suppose were  

560
00:44:00,287 --> 00:44:02,419
Would you speak concretely 

561
00:44:02,420 --> 00:44:05,322
we get a signal from Bob some random event what was that mean?

562
00:44:06,268 --> 00:44:09,177
Could you please be exact, very exact

563
00:44:10,380 --> 00:44:14,639
>> I'm getting two text messages, one from a one from b okay

564
00:44:14,640 --> 00:44:17,457
>> and they can have a value one or zero,

565
00:44:18,316 --> 00:44:22,261
>> okay and I get like a million of these things 

566
00:44:22,533 --> 00:44:28,949
>> I noticed in the distribution that the a signal is by 50% chance of being one or zero

567
00:44:28,987 --> 00:44:33,350
>> the b signal, it could be from a spin

568
00:44:33,898 --> 00:44:39,970
>> If the correlation turns out that whether it's a or b

569
00:44:40,608 --> 00:44:43,342
>> the other one is the same, another one was correlated 

570
00:44:43,367 --> 00:44:44,300
>> correlation of one 

571
00:44:44,683 --> 00:44:46,644
That's in a real world, a tangled state

572
00:44:48,134 --> 00:44:50,955
>> This could be, I don't know what it's coming from 

573
00:44:50,956 --> 00:44:56,026
>> just getting information so I get information a comes through

574
00:44:56,501 --> 00:44:59,909
>> I don't look at it, b comes through, It's a 50-50 chance,

575
00:45:00,148 --> 00:45:01,792
>> I don't look at it, but as soon as I get a

576
00:45:01,922 --> 00:45:04,732
>> so the wave function of b completely changes 

577
00:45:05,654 --> 00:45:08,405
>> now the example I'm using is not a state, 

578
00:45:08,625 --> 00:45:12,921
>> just Bob and Alice are twins, one's a boy one's a girl

579
00:45:13,794 --> 00:45:16,649
You got to hurry up because I have a get on particles 

580
00:45:16,761 --> 00:45:19,420
>> so there was nothing magical about that

581
00:45:19,457 --> 00:45:21,689
>> there is nothing weird about that, they are correlated like that,

582
00:45:21,787 --> 00:45:31,055
>> they are identical, so I'm addressing what is weird about that?

583
00:45:32,029 --> 00:45:33,074
If you don't find it weird good,

584
00:45:36,867 --> 00:45:41,556
What's weirder than you can't simulated classically without having a 

585
00:45:46,338 --> 00:45:50,638
central processor connected by wires, that's weird about it

586
00:45:50,733 --> 00:45:56,581
>> So doesn't wires connecting those computers represent non-locality?

587
00:45:56,589 --> 00:46:01,355
Okay now we ask a lot of wires there, for goodness sakes

588
00:46:01,980 --> 00:46:05,913
Surely Bob can send the message through the wires to Alice that takes no time at all,

589
00:46:06,135 --> 00:46:07,101
because they are instantaneous

590
00:46:10,050 --> 00:46:12,998
True enough if this were a real computer set up,

591
00:46:13,163 --> 00:46:20,035
but if Bob is restricted to only do the things which quantum mechanics allows him to do 

592
00:46:21,755 --> 00:46:24,649
if he is restricted by the rules of quantum mechanics,

593
00:46:28,161 --> 00:46:31,102
then we find out, Amazingly,

594
00:46:31,753 --> 00:46:36,868
that Bob can never send a message through the wires that violates locality 

595
00:46:37,640 --> 00:46:42,761
Why? because all Bob can do is manipulate his own system with unitary operators 

596
00:46:44,807 --> 00:46:49,199
and so that's what's curious, here 

597
00:46:50,600 --> 00:46:53,681
does that mean that the world is filled up with wires that we can't see 

598
00:46:54,550 --> 00:46:55,725
I don't think that's the message

599
00:46:56,020 --> 00:46:57,359
I think the right message,

600
00:46:57,852 --> 00:47:00,672
but of course you you can hear in which is room for debate 

601
00:47:00,981 --> 00:47:02,766
what the right message to all of this is 

602
00:47:04,130 --> 00:47:08,598
the message I take away from it is basically quantum mechanics 

603
00:47:08,903 --> 00:47:11,007
cannot be simulated by a classical system

604
00:47:11,646 --> 00:47:15,497
The logic is two different, the structures of the two theories are too different 

605
00:47:16,447 --> 00:47:19,897
and I don't take away the message of the world is full of wires 

606
00:47:19,898 --> 00:47:21,180
that if only we could access them

607
00:47:21,181 --> 00:47:24,021
We could send messages back and forth, other people do,

608
00:47:25,775 --> 00:47:29,527
but I think at least if we know how to do quantum mechanics 

609
00:47:31,675 --> 00:47:34,086
will agree on the facts we may disagree on basically 

610
00:47:34,420 --> 00:47:36,637
whether there are or are not wires there 

611
00:47:36,717 --> 00:47:40,327
are there really wires there or are there really not wires there

612
00:47:41,401 --> 00:47:44,573
Well, perhaps a octopus(?) is the word really

613
00:47:45,023 --> 00:47:50,366
but this oh oh,  but there is one other element to it,

614
00:47:50,690 --> 00:47:56,267
which is quite interesting, we made an assumption that 

615
00:47:58,157 --> 00:48:02,000
Bob and Alice get kind of randomly make decisions 

616
00:48:02,104 --> 00:48:09,599
about when to press the button about what orientations about what to decide to measure 

617
00:48:10,442 --> 00:48:18,416
Bob sits here and says okay Northwest and Southeast or whatever 

618
00:48:19,060 --> 00:48:24,791
makes these random decisions according to the principle of free will, so to speak,

619
00:48:26,192 --> 00:48:31,563
that's fine as long as Alice and Bob are outside the system and not part of it

620
00:48:32,171 --> 00:48:37,388
We don't need to ask the question of consciousness in the question of free will

621
00:48:37,777 --> 00:48:41,345
we just admit that Alice and Bob can make random decisions 

622
00:48:41,557 --> 00:48:43,524
about when to were one to measure various things

623
00:48:44,072 --> 00:48:45,289
It's obviously a pretty good approximation,

624
00:48:46,825 --> 00:48:51,265
but once we take Alice and Bob and put them into the system 

625
00:48:52,532 --> 00:48:55,204
so that they are governed by the laws of physics

626
00:48:56,355 --> 00:48:58,675
Then things get a little bit touchy

627
00:48:59,138 --> 00:49:02,904
What do we mean by saying that Alice can decide to do this,

628
00:49:03,503 --> 00:49:04,793
so Bob can decide to do that,

629
00:49:05,478 --> 00:49:11,408
and that I think is where the possible loopholes in all of this can come from 

630
00:49:11,883 --> 00:49:18,409
that I don't want to get too deep into heavy philosophical discussions 

631
00:49:18,789 --> 00:49:24,329
of all sorts of psychological or questions of consciousness,

632
00:49:25,483 --> 00:49:27,015
I don't know the answers That's it

633
00:49:27,155 --> 00:49:29,197
I don't know the answers beyond what I've told you 

634
00:49:32,268 --> 00:49:33,724
I certain of two things about it

635
00:49:34,897 --> 00:49:38,281
I am certain that there are a lot of people who think they do know the answer 

636
00:49:38,423 --> 00:49:39,586
and almost certain that they don't

637
00:49:42,474 --> 00:49:49,063
>>Question? Actually not a question, if we google for random challenge 

638
00:49:49,626 --> 00:49:51,804
>> you will find the blog entries that has a set up for these

639
00:49:54,421 --> 00:49:57,340
oh really? For the entangled case?

640
00:49:57,617 --> 00:50:01,378
>> It doesn't ⋯ you can hear a program

641
00:50:02,235 --> 00:50:05,370
>> Your challenge is fill in this right place

642
00:50:05,571 --> 00:50:05,943
I see

643
00:50:06,930 --> 00:50:08,999
>> They can give you a million dollar

644
00:50:09,605 --> 00:50:15,442
>> he won't give you a million dollar, some one in Sweden will give you a million dollar

645
00:50:19,463 --> 00:50:23,978
okay that's very interesting okay so I'm not the first one to 

646
00:50:24,346 --> 00:50:26,249
that I was the first one to think about it this way 

647
00:50:26,704 --> 00:50:28,596
but have been telling us to people 50 years 

648
00:50:33,473 --> 00:50:36,432
I don't I buy probably wasn't prior to the big thought about this 

649
00:50:36,865 --> 00:50:37,944
Who knows doesn't matter 

650
00:50:39,437 --> 00:50:45,686
Let's we could spend the rest of eternity talking about spin ,why,

651
00:50:45,835 --> 00:50:48,554
because they really are interesting,

652
00:50:48,721 --> 00:50:51,322
but you know, we want to get to something else 

653
00:50:51,502 --> 00:50:54,628
we want to get the particles we want to get the more complex systems

654
00:50:54,940 --> 00:51:01,448
The most complex system we discussed is two spins fourr dimensional Hilbert space

655
00:51:01,502 --> 00:51:04,576
We've gotten about to the point where in classical physics,

656
00:51:05,424 --> 00:51:08,683
we got to the first lecture of the first lecture was about the simple, discrete systems,

657
00:51:09,580 --> 00:51:13,500
and then we rather immediately moved onto continuous motion 

658
00:51:13,764 --> 00:51:17,691
of things like particles moving continuously with an infinitive,

659
00:51:17,927 --> 00:51:21,926
and not just an infinite but a continuously infinite range of possibilities

660
00:51:23,101 --> 00:51:24,988
That's where we have to go now though

661
00:51:25,070 --> 00:51:28,383
Fortunately, we've set up so much of the apparatus of quantum mechanics,

662
00:51:28,938 --> 00:51:30,338
and we can do it fairly quickly 

663
00:51:33,117 --> 00:51:40,665
There is a little bit of subtlety about replacing sums by integrals that's about it

664
00:51:41,037 --> 00:51:42,423
That's about the nature of the subtlety 

665
00:51:43,515 --> 00:51:48,497
sums and integrals we've expressed our general principles of quantum mechanics,

666
00:51:49,772 --> 00:51:54,003
and then exhibited it in,  a variety of simple spin system

667
00:51:54,792 --> 00:52:01,226
My examples now we will plunge right in the quantum gravity 

668
00:52:03,599 --> 00:52:10,919
No we will plunge right in the problem of a single particle 

669
00:52:10,920 --> 00:52:12,546
moving on a one-dimensional axis 

670
00:52:15,485 --> 00:52:17,290
I will take the axis to be infinite

671
00:52:18,275 --> 00:52:22,585
Although we if we that's mostly because I don't have time to be more careful

672
00:52:24,620 --> 00:52:29,980
If you want to get anywhere as we gonna have to be of 

673
00:52:33,544 --> 00:52:38,019
slightly less mathematically rigorous so I have,

674
00:52:38,113 --> 00:52:39,275
and in order to be rigorous,

675
00:52:40,014 --> 00:52:44,555
you really have to contain the system in a finite volume and so forth

676
00:52:45,076 --> 00:52:47,071
But we don't need to do that, so we have a line

677
00:52:48,442 --> 00:52:51,302
a long and infinite line, one particle can exist

678
00:52:51,907 --> 00:52:52,699
We won't do two particles yet 

679
00:52:52,700 --> 00:52:55,532
one particle can exist in one dimension,

680
00:52:57,853 --> 00:53:00,607
and that line is labeled by a coordinate X

681
00:53:01,548 --> 00:53:02,650
now write down the coordinate X

682
00:53:03,515 --> 00:53:05,707
but also the state of the particle

683
00:53:07,176 --> 00:53:12,792
The states of the particle are also labeled by the position of a particle 

684
00:53:13,521 --> 00:53:17,079
now here we are departing from classical logic in classical logic

685
00:53:19,255 --> 00:53:22,930
The states of a particle are label by a position and the momentum 

686
00:53:26,152 --> 00:53:28,014
if we were talking about the states of the spin,

687
00:53:28,805 --> 00:53:33,634
we would say the state of the spin is described by all of the  

688
00:53:34,673 --> 00:53:37,378
at least two of the components of spin,

689
00:53:38,093 --> 00:53:39,497
but we can't do that in quantum mechanics,

690
00:53:40,223 --> 00:53:44,196
we can't specify two component simultaneously, same in classical mechanics,

691
00:53:44,640 --> 00:53:48,457
specifying I'm sure you know specifying the position

692
00:53:48,562 --> 00:53:49,533
The momentum is too much

693
00:53:51,840 --> 00:53:53,933
And we'll see how that works will see how that works

694
00:53:53,993 --> 00:53:55,207
but for the moment let's take that as a postulate 

695
00:53:56,557 --> 00:53:59,852
that a complete description of the state of a particle 

696
00:54:00,337 --> 00:54:05,675
you can't get anymore out is for example to know that 

697
00:54:05,676 --> 00:54:10,167
it is located at a position x, and it's located in the position x

698
00:54:10,442 --> 00:54:12,006
we will describe that by a ket vector

699
00:54:13,495 --> 00:54:18,590
The ket vector X, that's the state of a particle when it was located at position x

700
00:54:19,828 --> 00:54:22,402
now of course we can consider more complicated,

701
00:54:22,668 --> 00:54:28,062
more complicated states, linear superposition of these

702
00:54:28,063 --> 00:54:31,520
in quantum mechanics, We are always allowed to make linear superpositions

703
00:54:32,238 --> 00:54:35,041
and so the space of states is bigger than just 

704
00:54:35,042 --> 00:54:39,799
the space of particles located definite positions 

705
00:54:41,938 --> 00:54:49,442
let's call the general state psi, this is the general state of a particle 

706
00:54:49,937 --> 00:54:52,076
doesn't mean anything now just an abstract symbol 

707
00:54:52,998 --> 00:54:54,954
and it's a vector in some vector space

708
00:54:55,901 --> 00:55:00,747
The vector space now has a continuously infinite number of basis vectors,

709
00:55:02,065 --> 00:55:06,490
Now strictly speaking, that's mathematically bad thing we gonna play that game

710
00:55:07,588 --> 00:55:11,750
If you want to understand more deeply the mathematics of quantum mechanics 

711
00:55:12,479 --> 00:55:16,425
you go when you learn about Hilbert space, but we are not going to do that tonight

712
00:55:17,381 --> 00:55:19,778
So the general state of particles called psi,

713
00:55:21,657 --> 00:55:23,707
remember what the wave function of the system is 

714
00:55:23,932 --> 00:55:27,300
the wave function of the system in a particular basis 

715
00:55:27,532 --> 00:55:35,122
is the projection of the state on to the basis vectors and we call that, psi of a, 

716
00:55:38,420 --> 00:55:43,267
now we come to particle the basis of states are the position states 

717
00:55:44,209 --> 00:55:45,263
and what is the wave function? 

718
00:55:46,221 --> 00:55:53,842
the wave function is the inner product of the state vector with the eigenvectors of position,

719
00:55:54,776 --> 00:56:03,854
and that's called psi of X that's the wave function that's the classical,

720
00:56:04,273 --> 00:56:07,200
not classical, but that's the standard quantum mechanical wave function 

721
00:56:07,460 --> 00:56:12,673
of a particle, it plays the same role as in the simple systems we've discussed,

722
00:56:13,089 --> 00:56:15,118
the inner product of the state vector with a 

723
00:56:16,760 --> 00:56:20,505
Now there is a little bit more in this context,

724
00:56:20,793 --> 00:56:25,207
the probability that the system whatever it is,

725
00:56:25,805 --> 00:56:30,141
The system exhibits property a, is given by this thing squared 

726
00:56:30,453 --> 00:56:38,133
In fact it's given by psi star of a, psi of a, that's the probability that the system,

727
00:56:38,458 --> 00:56:43,218
That the outcome of an experiment to determine whether the system was at property a 

728
00:56:43,716 --> 00:56:45,942
that's a probability likewise 

729
00:56:47,022 --> 00:56:54,600
in exact parallel the probability to find a particle at x is equal to psi star of X 

730
00:56:56,029 --> 00:57:01,830
times psi of x, I really haven't done anything that we haven't done before

731
00:57:03,405 --> 00:57:08,169
Now in quantum mechanics like in any other sensible statistical theory,

732
00:57:08,621 --> 00:57:11,504
the sum of all probabilities should add up to one,

733
00:57:13,783 --> 00:57:17,094
so the sum of psi star psi should add up to one: 

734
00:57:17,986 --> 00:57:21,112
but when probabilities refer to a continuous variable 

735
00:57:22,602 --> 00:57:27,172
we don't talk about the probability that the particles was located exactly at point X

736
00:57:28,606 --> 00:57:33,276
It's too sharp, the probability the think located exactly at any given place is 0 

737
00:57:34,720 --> 00:57:39,128
we talk about the probability that's located within a range of X or better yet,

738
00:57:39,373 --> 00:57:44,555
we talk about the density of probability we talk about the probability density,

739
00:57:45,821 --> 00:57:49,981
and we integrate this over little range of X,

740
00:57:50,383 --> 00:57:53,220
to find the probability that you in that range of X 

741
00:57:54,292 --> 00:57:55,348
we do something like that here, 

742
00:57:55,654 --> 00:57:59,775
if we want to know not whether the system is definitely specify as a

743
00:58:00,115 --> 00:58:02,810
but we wanna know whether it's in range of a, we sum over them,

744
00:58:03,813 --> 00:58:07,381
but here we sort of have to do it 

745
00:58:08,659 --> 00:58:11,848
total probability equal to one that's the condition 

746
00:58:12,598 --> 00:58:20,410
that the integral of psi star of X, psi of X, d X is equal to one 

747
00:58:21,730 --> 00:58:24,498
total probability of the system has to add up to one 

748
00:58:25,866 --> 00:58:31,173
and the only as I said the only new thing here 

749
00:58:32,083 --> 00:58:33,782
is that we have to replace sums by integrals 

750
00:58:34,079 --> 00:58:36,771
and it does make mathematical subtlety that we have to worry about,

751
00:58:38,325 --> 00:58:44,509
but it, it sort of in mathematical details okay,

752
00:58:44,869 --> 00:58:50,531
what about the inner product between two state vectors one of X and one of x prime

753
00:58:52,548 --> 00:58:54,806
well here we go back for inspiration right back to 

754
00:58:59,305 --> 00:59:07,177
right back to this case

755
00:59:15,284 --> 00:59:21,107
zero, if a is not equal to a prime and one of a is equal to a prime

756
00:59:22,393 --> 00:59:24,539
We are not gonna able to quite get away with that here 

757
00:59:26,652 --> 00:59:30,316
We are not going to quite be able to get away with a statement that 

758
00:59:33,508 --> 00:59:38,773
XX prime is equal to zero or one

759
00:59:40,500 --> 00:59:44,062
We are going to say that if x is not equal to X prime 

760
00:59:46,086 --> 00:59:46,322
that is zero:

761
00:59:47,827 --> 00:59:50,414
But we are not gonna say that it's equal to X prime 

762
00:59:51,662 --> 00:59:55,263
That's one, that would not lead us anywhere

763
00:59:56,764 --> 01:00:00,370
what we're going to say is that it's a function which vanishes 

764
01:00:00,862 --> 01:00:02,935
unless equals X equals X prime,

765
01:00:03,679 --> 01:00:07,636
but has area 100, we will see how that works out

766
01:00:07,942 --> 01:00:09,403
we will see what that does for us in a moment

767
01:00:10,873 --> 01:00:16,113
What it really comes down to is a different normalization here,

768
01:00:16,866 --> 01:00:19,897
but again the basic idea that states are orthogonal 

769
01:00:21,132 --> 01:00:23,784
if X is not equal to X prime that we will retain 

770
01:00:24,921 --> 01:00:28,795
but the numerical value of what we get when X is equal to X prime,

771
01:00:29,622 --> 01:00:31,094
we are gonna take this to be slightly different 

772
01:00:31,395 --> 01:00:33,038
in fact were going to invent a function 

773
01:00:33,868 --> 01:00:36,911
Not we are, Dirac did, Dirac delta function, X minus X prime 

774
01:00:36,912 --> 01:00:45,247
the Dirac delta function is not a real mathematical function,

775
01:00:45,417 --> 01:00:49,558
although by now the mathematicians of the find it to death,

776
01:00:50,062 --> 01:00:51,867
so that there is a real mathematical entity,

777
01:00:54,195 --> 01:00:59,173
but for our purposes we can think of it in a very simple way 

778
01:01:00,359 --> 01:01:06,414
it's a function which is zero , let's take delta of X 

779
01:01:07,448 --> 01:01:11,962
not delta of x, delta of x prime , it's basically built of X minus zero

780
01:01:11,963 --> 01:01:17,198
the function delta of x, it's a function which is zero unless X is equal to zero

781
01:01:17,817 --> 01:01:21,338
That's the same as saying this thing is zero unless X is equal to X prime

782
01:01:22,646 --> 01:01:29,065
so this is X it's zero everywhere except X equals X prime 

783
01:01:29,493 --> 01:01:34,610
and X equals X prime, it's a very high, narrow spike 

784
01:01:35,749 --> 01:01:40,716
how high and narrow? high enough and narrow enough for the area under is one unit

785
01:01:42,070 --> 01:01:46,706
that means if it's approximated by a narrow interval of size epsilon 

786
01:01:47,812 --> 01:01:49,964
that it has height one over epsilon,

787
01:01:52,575 --> 01:01:55,722
and then you imagine a limit when you take epsilon smaller and smaller 

788
01:01:56,275 --> 01:01:57,788
one of the epsilon higher and higher,

789
01:01:58,466 --> 01:02:03,480
always keeping the area under equal to one that's the Dirac delta function

790
01:02:03,807 --> 01:02:07,875
Dirac Delta function zero everywhere except when X equals X prime 

791
01:02:08,857 --> 01:02:11,395
infinite in some sense when X is equal to X prime,

792
01:02:12,111 --> 01:02:15,027
but in such a way that the area under it is equal to one 

793
01:02:18,028 --> 01:02:26,505
alright now let's use this to further elaborate on what the meaning of psi of x is

794
01:02:31,207 --> 01:02:32,966
I'm gonna guess and then we are gonna check

795
01:02:33,545 --> 01:02:36,515
I'm gonna guess that the meaning of psi of X is the following 

796
01:02:37,212 --> 01:02:39,554
that if I want to write a general state vector 

797
01:02:44,861 --> 01:02:50,758
Psi, that I can think of it as a sum over the basis vectors

798
01:02:51,300 --> 01:02:52,406
but not a sum, a integral, of course it's a integral

799
01:02:54,312 --> 01:02:57,160
because the basis vectors are continuous 

800
01:02:58,120 --> 01:03:05,214
d X of some coefficients, some coefficients well let's no,

801
01:03:05,355 --> 01:03:05,867
let's write it this way,

802
01:03:06,150 --> 01:03:13,207
Psi is equal to an integral d X of some set of coefficients 

803
01:03:14,731 --> 01:03:17,768
I'm gonna guess what they are, I'm gonna guess that this Psi of x

804
01:03:19,682 --> 01:03:26,020
times the Eigen times the the state

805
01:03:26,264 --> 01:03:29,657
The quantum state with a particle as it located at X

806
01:03:30,220 --> 01:03:32,170
And then we are gonna add them up for different x-s

807
01:03:32,850 --> 01:03:37,072
It's exactly the same thing that we did to ordinary quantum systems,

808
01:03:38,008 --> 01:03:39,316
except this was replaced by a sum

809
01:03:43,478 --> 01:03:46,787
Alright I'm giving you two distinct definitions of psi of X

810
01:03:47,382 --> 01:03:50,575
One is that it's the inner product of the state vector with the eigenvector 

811
01:03:51,739 --> 01:03:57,084
acts upon the eigenvector of X, the other is the expansion coefficients 

812
01:03:57,700 --> 01:04:01,083
in writing psi as a sum over vectors

813
01:04:03,093 --> 01:04:08,000
Now if you're a little bit uncomfortable with the idea of a continuous basis of vectors

814
01:04:08,210 --> 01:04:09,187
that we integrate over

815
01:04:10,269 --> 01:04:10,311
you are in good company

816
01:04:11,957 --> 01:04:17,610
The mathematics of this is, um as I said

817
01:04:17,724 --> 01:04:24,264
By now it is been made rigorous, but it's it's some, 

818
01:04:24,807 --> 01:04:27,103
it took pages and pages and pages and pages 

819
01:04:28,283 --> 01:04:31,733
in books and books and books on something called distribution theory 

820
01:04:32,093 --> 01:04:33,417
to make sense out of this with precision 

821
01:04:34,104 --> 01:04:36,613
But it's pretty intuitive, it's pretty intuitive, and we will stick with that

822
01:04:38,258 --> 01:04:40,961
so let's calculate a prime there,

823
01:04:41,167 --> 01:04:43,694
this is a sum and integral doesn't matter,

824
01:04:43,695 --> 01:04:48,396
doesn't matter what you call the integration variable, let's call it X prime,

825
01:04:49,193 --> 01:04:53,409
and now let's calculate the inner product of X with psi 

826
01:04:55,293 --> 01:04:58,000
as I said there were two different definitions of psi of X 

827
01:04:58,690 --> 01:05:02,311
let's check that they are the same, so there we have it

828
01:05:02,958 --> 01:05:11,447
integrals of X prime, psi of X prime, inner product of X with X prime

829
01:05:14,582 --> 01:05:16,962
what's the inner product of x and x prime?

830
01:05:17,116 --> 01:05:18,696
I think I wrote it someplace where is it,

831
01:05:20,594 --> 01:05:43,651
I think I erased it, it's equal to the integral of x', psi of x', delta of X minus X prime,

832
01:05:47,048 --> 01:05:51,404
Okay, what is that? First of all Delta is a function 

833
01:05:51,786 --> 01:05:55,760
which is zero everywhere except X equals X prime,

834
01:05:56,611 --> 01:05:59,236
so that means that in doing this integral here 

835
01:05:59,822 --> 01:06:03,841
you only get a contribution when X prime is equal to x

836
01:06:05,427 --> 01:06:08,848
that means the answer is gonna come out to be proportional to psi of X 

837
01:06:09,755 --> 01:06:14,665
furthermore, the area under this delta function is exactly 1 

838
01:06:16,038 --> 01:06:19,329
and we imagine that this delta function is a pinched 

839
01:06:19,784 --> 01:06:24,253
narrower and narrower high spike so the area under is one 

840
01:06:25,165 --> 01:06:30,194
then this integral would tend to psi of X

841
01:06:31,364 --> 01:06:35,278
the rule is if you have an integral over a delta function like this

842
01:06:36,406 --> 01:06:42,005
It simply picks out the value of the integration variable 

843
01:06:43,083 --> 01:06:44,710
where the Delta function vanishes 

844
01:06:45,444 --> 01:06:48,114
sorry when the argument of that Delta function vanishes 

845
01:06:48,157 --> 01:06:52,346
when X minus X prime vanishes that's where the functions is not zero 

846
01:06:54,115 --> 01:06:57,842
and the integral of a function with the Delta function just gives you

847
01:06:57,843 --> 01:07:00,394
the value of the function at the point X 

848
01:07:03,216 --> 01:07:04,389
That's a rule, we might as well write that down 

849
01:07:04,759 --> 01:07:06,594
Let's write that down on blackground as a rule 

850
01:07:06,674 --> 01:07:13,724
integral delta of X minus X prime, F of X prime is always equal to F of X 

851
01:07:14,100 --> 01:07:18,004
it picks out the value X prime equal to x

852
01:07:18,161 --> 01:07:20,573
That's part of the definition of Delta function 

853
01:07:22,003 --> 01:07:24,269
okay so it's just equal to psi of X 

854
01:07:24,889 --> 01:07:29,557
and we find consistency that the thing that we call psi of X over here 

855
01:07:30,607 --> 01:07:33,556
is the same as the thing that we call psi of X over here

856
01:07:35,478 --> 01:07:39,326
All of this has its exact parallel for the ordinary systems

857
01:07:41,211 --> 01:07:50,378
>> So I'm looking in Psi of x prime times ket vectors

858
01:07:51,572 --> 01:07:51,689
Superposing

859
01:07:52,501 --> 01:07:54,679
>>I'm trying to figure out which of these is

860
01:07:54,856 --> 01:07:56,447
>>Which are the basis vectors?

861
01:07:56,988 --> 01:07:57,858
These are the basis vectors

862
01:07:57,859 --> 01:08:05,990
let's let's compare that with, psi is equal to sum over a, a prime, 

863
01:08:06,157 --> 01:08:12,855
let's call it a prime, psi of a prime |a prime >

864
01:08:16,595 --> 01:08:21,421
okay if we used the fact that a and a prime are orthogonal 

865
01:08:21,636 --> 01:08:25,025
and do exactly the same thing except replacing sums by integrals 

866
01:08:25,755 --> 01:08:32,278
we will also find that the that the Psi that appears here is the same as 

867
01:08:32,308 --> 01:08:34,833
the inner product of psi with the basis vectors 

868
01:08:35,986 --> 01:08:38,446
So we really done nothing that we haven't done before

869
01:08:38,993 --> 01:08:42,203
and I appreciate it's a little possibly a little unfamiliar 

870
01:08:44,247 --> 01:08:48,820
but all it is, is same thing we've done before 

871
01:08:50,848 --> 01:08:54,953
>>So psi of x prime sort of plays the same rule as the alphas?

872
01:08:55,662 --> 01:08:58,701
The x prime plays the same role as a prime here

873
01:08:58,849 --> 01:09:02,398
the psi plays the same rule as psi, 

874
01:09:03,499 --> 01:09:05,198
the sum is replaced by the integral, 

875
01:09:05,616 --> 01:09:08,724
sum over a prime is replaced by integral d X prime,

876
01:09:09,698 --> 01:09:11,647
and otherwise it's exactly the same

877
01:09:15,133 --> 01:09:18,210
let's calculate the inner product,

878
01:09:19,095 --> 01:09:20,291
>>One question, 

879
01:09:21,715 --> 01:09:30,452
>>on the right hand side we have direct form in the right hand corner

880
01:09:31,709 --> 01:09:34,382
>>you got a vertical line between the x-es, 

881
01:09:36,853 --> 01:09:42,022
>>over here in the middle, there is no vertical line

882
01:09:43,166 --> 01:09:49,690
Thank you, got you got you got you

883
01:09:51,671 --> 01:09:52,836
No absolutely, 

884
01:09:56,242 --> 01:09:58,218
How about the inner product of two vectors?

885
01:09:59,432 --> 01:10:01,811
Well, the inner product of some vectors is some inner product,

886
01:10:02,038 --> 01:10:04,741
but one way of calculating the inner product,

887
01:10:04,862 --> 01:10:07,705
let's go back to ordinary the ordinary quantum mechanics

888
01:10:07,999 --> 01:10:11,705
Ordinary means for the discrete systems , we don't need this right now

889
01:10:14,915 --> 01:10:15,983
how do we calculate the inner product?

890
01:10:16,304 --> 01:10:20,640
let's take you what state inner product of psi with phi

891
01:10:21,723 --> 01:10:25,719
We first write psi and phi as sums over the basis vectors,

892
01:10:26,230 --> 01:10:30,688
so this is equal to sum over two sets of basis vectors a prime and a 

893
01:10:31,732 --> 01:10:40,794
psi star of a prime times a prime,that's bra vector we complex conjugate of course

894
01:10:40,991 --> 01:10:44,387
we deal with bra vectors that's the psi half of it,

895
01:10:46,128 --> 01:10:57,827
and then the phi half of it is the sum over a,  phi of a times a,

896
01:11:00,918 --> 01:11:04,406
phi of a being the coefficients of the phi vector 

897
01:11:04,966 --> 01:11:09,228
psi star being the coefficients of the psi, bra vector

898
01:11:10,633 --> 01:11:14,733
now phi of a is just a number, psi star of a is just a number 

899
01:11:15,278 --> 01:11:18,683
and the inner product of a prime with a is just a Kronecker Delta 

900
01:11:19,191 --> 01:11:25,151
so all of this adds up to sum of psi star of a prime 

901
01:11:26,862 --> 01:11:36,540
phi of a and then a Kronecker Delta of a and a prime which is exactly the instruction 

902
01:11:37,350 --> 01:11:38,703
to set a equal to a prime,

903
01:11:39,330 --> 01:11:40,424
I think you seen that before

904
01:11:41,819 --> 01:11:44,672
This is simply the rule about row vectors, 

905
01:11:45,433 --> 01:11:49,460
if psi star of a is represented by a row vectors psi star of a one,

906
01:11:49,671 --> 01:11:53,706
Blablablablabla, and Phi is represented by a column vector 

907
01:11:54,669 --> 01:12:00,604
and this is just a rule to take this times this plus this times this plus this times this 

908
01:12:02,170 --> 01:12:06,462
so this is the common, garden-variety quantum mechanical in a product 

909
01:12:07,753 --> 01:12:13,324
I'm not gonna we don't need to go through the same exercise here 

910
01:12:13,833 --> 01:12:17,830
we can immediately write down what the result is going to be just by analogy,

911
01:12:18,651 --> 01:12:21,048
it's almost not even an analogy,

912
01:12:21,049 --> 01:12:24,909
it's just replacing wherever we see a, put x-es 

913
01:12:25,112 --> 01:12:28,663
and wherever we see sum, put integral

914
01:12:29,994 --> 01:12:38,602
the inner product of two space vectors for a particle on the line 

915
01:12:40,717 --> 01:12:54,854
is integral d X psi star of X phi of X, this satisfies incidentally,

916
01:12:55,076 --> 01:12:57,095
one of the basic rules of inner products 

917
01:12:57,096 --> 01:13:03,194
what happens to a inner product if you interchange the two vectors?

918
01:13:03,721 --> 01:13:09,653
what's the relation between the inner product Psi-Phi and the inner product Phi-Psi?

919
01:13:10,491 --> 01:13:15,107
complex conjugation, now what will happen in here if you interchange psi with phi

920
01:13:16,256 --> 01:13:21,326
we wind up writing phi star here and psi here 

921
01:13:21,693 --> 01:13:24,546
that again would be the inner product of the complex conjugate 

922
01:13:26,142 --> 01:13:27,724
and for exactly the same reasons 

923
01:13:28,804 --> 01:13:38,835
So this is equal to this for both discrete systems and for these continuous systems

924
01:13:41,050 --> 01:13:44,906
in a product the inner product of a vector with itself

925
01:13:46,516 --> 01:13:51,525
is the norm of the vector just as in previous examples 

926
01:13:52,303 --> 01:13:57,835
you normalize a vector to one to express the fact that the probability adds up one,

927
01:14:00,360 --> 01:14:09,888
so not much difference, just a difference of notation 

928
01:14:10,968 --> 01:14:15,731
and the what's I think what gets to be confusing about it 

929
01:14:15,929 --> 01:14:17,462
is you know both of these things,

930
01:14:17,712 --> 01:14:21,135
you know about these simple row vectors and column vectors 

931
01:14:21,352 --> 01:14:22,561
and you know about functions,

932
01:14:23,185 --> 01:14:25,576
but you are not used to thinking of functions as vectors 

933
01:14:26,644 --> 01:14:28,177
or maybe you are, if you are good,

934
01:14:28,817 --> 01:14:31,386
but if you are not use to thinking of functions as vector, 

935
01:14:32,210 --> 01:14:34,170
as function of psi of x-es, 

936
01:14:34,974 --> 01:14:35,816
then you have to get used to it

937
01:14:36,509 --> 01:14:39,963
but functions are vectors in a vector space, you can add them,

938
01:14:40,532 --> 01:14:42,467
you can multiply them by complex constants 

939
01:14:44,419 --> 01:14:48,004
and you can add them, you can multiply by numbers,

940
01:14:48,845 --> 01:14:53,006
and so they satisfy the axioms of a vector space 

941
01:14:54,089 --> 01:14:56,673
space of functions is a vector space 

942
01:14:57,717 --> 01:14:58,990
now sometimes you may want

943
01:15:00,034 --> 01:15:05,246
>>In class of mathematics, would you put the limit of integration

944
01:15:05,302 --> 01:15:06,650
>>minus and plus infinity?

945
01:15:07,602 --> 01:15:11,621
Good, which incidentally said something

946
01:15:13,561 --> 01:15:19,650
It says, we shouldn't be thinking about functions which would grow at infinity for example

947
01:15:20,613 --> 01:15:26,707
we should be only thinking about functions which are square-integral-able

948
01:15:26,803 --> 01:15:29,161
Now we are gonna have to, we gonna have to modify that 

949
01:15:29,855 --> 01:15:32,096
square integral means you square them

950
01:15:32,563 --> 01:15:35,032
This is called squaring them that the integral exists 

951
01:15:35,730 --> 01:15:38,516
functions which grow at infinity won't do that

952
01:15:39,102 --> 01:15:41,792
Those are crazy because it would say that,

953
01:15:42,389 --> 01:15:45,656
if you have functions which grew with infinity where would the probability be?

954
01:15:46,106 --> 01:15:49,561
a probability will be all located off at infinity

955
01:15:49,958 --> 01:15:51,007
And that's not what we want

956
01:15:51,520 --> 01:15:53,883
we want to be talking about probability distributions 

957
01:15:54,073 --> 01:15:56,585
which I have a total finite probability under them 

958
01:15:58,366 --> 01:16:04,735
so that means we should take perhaps with a grain of salt

959
01:16:05,253 --> 01:16:07,073
The idea that we can write down any function,

960
01:16:07,611 --> 01:16:10,778
but for the moment we are we are gonna be a little bit loose about that

961
01:16:12,309 --> 01:16:19,775
okay, I think I will slow down here for a few minutes 

962
01:16:19,776 --> 01:16:23,479
# and digest it, and ask a few good questions 

963
01:16:26,554 --> 01:16:29,389
No, no no no no, 

964
01:16:31,460 --> 01:16:33,117
you know a you have a know I had that 

965
01:16:33,463 --> 01:16:36,277
I'm I meant about this, but okay 

966
01:16:36,470 --> 01:16:41,493
>> This is quick, it seems like everything must be entangled with something

967
01:16:41,714 --> 01:16:42,995
>> almost all the time, is that?

968
01:16:43,335 --> 01:16:49,604
Oh yeah, entangled with all sort of things in the environment,

969
01:16:50,244 --> 01:16:53,623
and we haven't talked about the importance of a complex environment 

970
01:16:55,711 --> 01:16:58,875
it's not so important in the very foundations of the subject,

971
01:16:59,307 --> 01:17:01,956
but it is important to why we see the world the way we are,

972
01:17:01,979 --> 01:17:05,738
there were constantly being bombarded and getting getting entangled with things

973
01:17:07,276 --> 01:17:11,021
>>Is there a .. For physicisits?

974
01:17:11,022 --> 01:17:14,426
Oh boy, certainly

975
01:17:14,928 --> 01:17:20,348
>> so okay, it's not just some, it didn't just the Dirac function all about it

976
01:17:21,654 --> 01:17:24,614
oh you mean is that, I see what you mean, 

977
01:17:25,258 --> 01:17:29,200
You mean to do anything else besides clean up the .. I don't know

978
01:17:30,987 --> 01:17:35,434
I think it's make rigorous as a set of concepts that physicists use over time

979
01:17:38,973 --> 01:17:44,358
Has it really seems no no

980
01:17:45,846 --> 01:17:53,410
The theory of distributions, ya I will tell you about that with physcists

981
01:17:53,575 --> 01:18:00,391
It made them argue less with mathematicians

982
01:18:02,335 --> 01:18:04,592
was one of the many examples were mathematicians 

983
01:18:04,990 --> 01:18:07,377
that drag screaming and kicking with physicists 

984
01:18:09,428 --> 01:18:10,965
of course the opposite also happens 

985
01:18:11,483 --> 01:18:16,422
but that was an example where mathematicians first looked at this thing

986
01:18:17,836 --> 01:18:20,075
so what are you crazy that doesn't make any sense

987
01:18:21,623 --> 01:18:28,155
It's not rigorous and in time they came to peace with it 

988
01:18:28,964 --> 01:18:32,889
through defining it, in a wrong way

989
01:18:33,378 --> 01:18:39,715
No, I don't think the mathematical structure of it has influenced physics 

990
01:18:39,982 --> 01:18:42,237
as much as you might've expected

991
01:18:44,755 --> 01:18:49,289
Unfortunately, because there are big problems of confusion 

992
01:18:50,541 --> 01:18:55,077
related to it in quantum field theory, and it hasn't resolve those problems okay,

993
01:18:58,787 --> 01:19:02,139
okay, next place we want to go is observables 

994
01:19:04,051 --> 01:19:07,521
We want to talk about observables and give some examples of observables 

995
01:19:10,013 --> 01:19:12,408
observables are linear operators,

996
01:19:14,348 --> 01:19:17,564
what are linear operators acting on the space of functions?

997
01:19:18,985 --> 01:19:22,792
Their operations on the state on a function that gives another function 

998
01:19:25,076 --> 01:19:33,647
linear operator, let's call it M, for some reason 

999
01:19:33,768 --> 01:19:37,873
I saved L to stand for observables 

1000
01:19:40,275 --> 01:19:43,294
and I've used previously M for general linear operator 

1001
01:19:43,995 --> 01:19:45,828
So I continue to do, so whatever M is,

1002
01:19:46,294 --> 01:19:51,845
it's a machine and thought of as an abstract operator on it's space

1003
01:19:52,711 --> 01:19:56,289
it takes the state vector into another state vector,

1004
01:19:56,985 --> 01:20:00,042
psi prime psi, or a vector and a vector space,

1005
01:20:00,941 --> 01:20:04,701
but it also can be thought of as a thing which acts on wave functions 

1006
01:20:04,763 --> 01:20:07,649
every state has a wave function that describes it

1007
01:20:08,405 --> 01:20:14,269
And the, the machine, or, that was why we call it M

1008
01:20:14,269 --> 01:20:17,075
Yes I remember now, you call it M for Machine, 

1009
01:20:17,584 --> 01:20:22,929
the machine can also be thought of as acting on a wave function 

1010
01:20:23,759 --> 01:20:24,697
to give another wave function,

1011
01:20:26,430 --> 01:20:28,197
so let's think of it that way,

1012
01:20:28,198 --> 01:20:30,061
let's think of it in concrete terms 

1013
01:20:31,874 --> 01:20:36,198
as an action that we do on a wave function to produce another wave function

1014
01:20:36,889 --> 01:20:37,899
Now it's linear,

1015
01:20:39,168 --> 01:20:42,394
linear means that when it acts on the sum of two functions

1016
01:20:42,847 --> 01:20:44,376
It gives back the sum of the results, 

1017
01:20:45,114 --> 01:20:49,036
it means that if it acts on a function which is a numerical product

1018
01:20:49,412 --> 01:20:55,533
a numerical multiple of some function it gives back the same numerical multiple

1019
01:20:55,929 --> 01:21:01,491
all have ideas of what a linear operator are, continued to be exactly the same 

1020
01:21:03,324 --> 01:21:08,220
let's talk about some linear, oh, okay that's that's the notion of a linear operator 

1021
01:21:08,766 --> 01:21:10,657
Now what about a Hermitian linear operator 

1022
01:21:10,879 --> 01:21:13,464
I will tell you the definition of Hermitian we will,

1023
01:21:13,920 --> 01:21:15,205
We've talk about it before,

1024
01:21:15,469 --> 01:21:17,170
but I'm gonna give it to you in a condensed form 

1025
01:21:18,405 --> 01:21:23,559
the condensed form is also true for all of the vector spaces we studied 

1026
01:21:24,396 --> 01:21:30,875
and you may recognize it, if not, go back and prove that 

1027
01:21:30,876 --> 01:21:33,524
it follows from the previous definitions

1028
01:21:34,388 --> 01:21:38,574
a Hermitian operator Hermitian for operators is, roughly speaking,

1029
01:21:38,728 --> 01:21:44,484
the analog of real, as opposed to complex okay,

1030
01:21:44,788 --> 01:21:47,806
here's what's true about Hermitian operators if you have a Hermitian operator,

1031
01:21:47,819 --> 01:21:49,311
which I will now call L

1032
01:21:52,037 --> 01:21:53,223
then it has a property that 

1033
01:21:53,566 --> 01:21:58,058
if you take if you act on wave function psi or or state vector psi,

1034
01:21:59,090 --> 01:22:10,312
and take it's inner product with phi, that that is equal to psi L phi,

1035
01:22:10,836 --> 01:22:12,690
but not quite, I will have to do with it,

1036
01:22:14,112 --> 01:22:16,990
complex conjugate it, now if L not,

1037
01:22:17,061 --> 01:22:19,493
let's go back a step supposing L was not Hermitian 

1038
01:22:19,949 --> 01:22:22,179
would this be right? not quite

1039
01:22:22,812 --> 01:22:26,266
What do we have to do? We have to put a dagger here

1040
01:22:27,976 --> 01:22:30,302
another words we have to roughly speaking,

1041
01:22:30,303 --> 01:22:38,664
complex conjugate L, if L is Hermitian in the rule for that 

1042
01:22:39,503 --> 01:22:42,195
is that it's made with matrix elements it's it

1043
01:22:42,582 --> 01:22:47,415
It's so when sandwiched when sandwiched in opposite order

1044
01:22:48,769 --> 01:22:54,142
You don't have to complex conjugate L because it's already the analog of real

1045
01:22:55,092 --> 01:22:57,707
this is a condition that an operator is Hermitian 

1046
01:22:58,236 --> 01:23:01,809
that's important because Hermitian operators are the class of observables,

1047
01:23:03,487 --> 01:23:05,846
so let's work out one or two Hermitian operators 

1048
01:23:06,060 --> 01:23:09,985
or while we are at it, guess we may need this 

1049
01:23:11,090 --> 01:23:13,663
so let's work out a couple,

1050
01:23:16,694 --> 01:23:19,744
First of all, one the simplest things that I can do to a wave function 

1051
01:23:23,037 --> 01:23:25,610
I don't know there are many simple things that can do it,

1052
01:23:27,004 --> 01:23:31,248
but a very simple thing is to multiply it by x 

1053
01:23:33,013 --> 01:23:36,778
and even simpler thing would be to multiply it by simple number,

1054
01:23:38,502 --> 01:23:41,541
but let's take slightly more complicated multiply it by X 

1055
01:23:42,331 --> 01:23:44,076
multiplying by X give back a function 

1056
01:23:44,566 --> 01:23:46,742
so this is it at minimum an operation on a function

1057
01:23:47,033 --> 01:23:53,358
is it linear? Sure it is, if x multiplies psi of x, let's say, or Phi of x

1058
01:23:54,956 --> 01:23:59,539
it gives x psi of x plus x phi of x

1059
01:24:00,995 --> 01:24:04,601
another words if it acts on the sum it gives a sum of the results 

1060
01:24:05,524 --> 01:24:09,250
If it acts on a numerical multiple of the Psi,

1061
01:24:09,357 --> 01:24:12,425
it gives back the numerical multiple, so linear operator 

1062
01:24:13,377 --> 01:24:18,946
question is it Hermitian? is it a Hermitian operator?

1063
01:24:19,658 --> 01:24:22,488
okay so let's check, here's what we have to take

1064
01:24:22,918 --> 01:24:36,554
we have to take psi inner product of x times phi, that's this one,

1065
01:24:36,935 --> 01:24:39,749
oh sorry, I think I want to do it the other way

1066
01:24:40,589 --> 01:24:45,388
Phi X psi, phi X psi,

1067
01:24:48,412 --> 01:24:50,869
and ask whether this is true or not

1068
01:24:51,740 --> 01:24:52,577
We will check in a minute

1069
01:24:54,742 --> 01:24:56,358
No X on them, X is a function x, 

1070
01:24:57,814 --> 01:25:02,555
we multiply one function by another function that takes functions to functions

1071
01:25:03,460 --> 01:25:10,089
It's not a number, it's a variable, any given value of it is a number,

1072
01:25:11,118 --> 01:25:18,133
but it's a machine that takes a function psi of x and replaces it by X psi of X 

1073
01:25:18,756 --> 01:25:25,700
alright, so let's calculate, the wave function of the ket vector

1074
01:25:26,299 --> 01:25:33,797
is x psi of x, the wave function of the bra vector is phi star of X

1075
01:25:37,146 --> 01:25:42,236
the inner product is, put brackets around it bracket

1076
01:25:43,817 --> 01:25:47,629
This is a ket vector inside the bracket, and this is a bra vector 

1077
01:25:47,830 --> 01:25:51,250
and the rule for taking the inner product, oops, I think I interchanged it

1078
01:25:51,768 --> 01:25:55,575
But we will take the inner product is to integrate,

1079
01:25:56,848 --> 01:26:01,060
So let's get off the bracket here and just integrate dx

1080
01:26:04,693 --> 01:26:10,236
Oops, sorry,dx, that's in this order

1081
01:26:10,921 --> 01:26:14,206
What about this order? this order over here,

1082
01:26:14,207 --> 01:26:15,865
the inner product without the star without the star

1083
01:26:17,009 --> 01:26:19,624
the thing we are gonna compare with is the same thing 

1084
01:26:21,618 --> 01:26:30,161
except with psi star x phi of x, well look at it 

1085
01:26:30,170 --> 01:26:32,738
is this integral complex conjugate of this?

1086
01:26:34,319 --> 01:26:38,258
Yes, it is, the complex conjugate of phi star phi, 

1087
01:26:38,862 --> 01:26:42,068
the complex conjugate of psi is psi star and what about X?

1088
01:26:44,375 --> 01:26:47,887
X is it's own complex conjugate so yes indeed 

1089
01:26:49,226 --> 01:26:55,188
X is an operator which satisfies the basic rule of Hermitian operator 

1090
01:26:55,737 --> 01:26:59,146
that if you interchange the psi and the phi, it complex conjugates 

1091
01:27:01,822 --> 01:27:07,832
if, what observable does X correspond to? 

1092
01:27:10,009 --> 01:27:15,020
position, it is the position, it's the Delta function 

1093
01:27:15,483 --> 01:27:21,739
a eigenvector of position well maybe we will come to that next time 

1094
01:27:22,259 --> 01:27:25,257
I think I want yesterday, but we will come back to that,

1095
01:27:25,978 --> 01:27:28,397
X is an observable, what does it correspond to?

1096
01:27:28,731 --> 01:27:30,385
it corresponds to position of the particle

1097
01:27:31,005 --> 01:27:35,802
we will prove that by showing that the eigenvectors X 

1098
01:27:36,495 --> 01:27:38,801
that the vectors X are eigenvectors of that operator,

1099
01:27:39,096 --> 01:27:41,752
but not right now, we will do that the next time

1100
01:27:42,464 --> 01:27:43,365
It's easy, trivial

1101
01:27:44,150 --> 01:27:49,524
>>Question, so when you write x up there

1102
01:27:49,587 --> 01:27:52,947
>>It's implied that there really an x?

1103
01:27:52,948 --> 01:27:55,850
The X is a function of x,

1104
01:27:56,344 --> 01:27:59,902
X? X is an n alphabet, yes,

1105
01:28:00,143 --> 01:28:02,915
It's implied that x is a function of x

1106
01:28:04,789 --> 01:28:06,247
it has a whole bunch that's right,

1107
01:28:06,248 --> 01:28:09,399
it itself is a function which is multiplying another function 

1108
01:28:12,104 --> 01:28:14,885
so it's an operation on a function which gives another function 

1109
01:28:18,438 --> 01:28:19,964
that's exactly right,

1110
01:28:21,709 --> 01:28:23,890
Let's take another operation or operator 

1111
01:28:24,770 --> 01:28:28,794
differentiation by differentiate a function they get a function 

1112
01:28:29,509 --> 01:28:31,342
that the first test of whether it's an operator 

1113
01:28:32,581 --> 01:28:35,453
If I differentiate the sum of two functions I get the sum of derivatives 

1114
01:28:36,563 --> 01:28:39,771
by differentiate a numerical multiple of the function 

1115
01:28:39,865 --> 01:28:41,739
I get numerical multiple of the derivative 

1116
01:28:42,212 --> 01:28:46,775
it's a linear function, is it a Hermitian operator?

1117
01:28:47,347 --> 01:28:51,432
So, here we have to, here we have to get ourselves a fresh blackboard

1118
01:28:53,320 --> 01:28:58,068
and check once and for all whether it's well, check whether it's Hermitian

1119
01:29:00,616 --> 01:29:07,795
Okay, what happened, what is, let's call it derivative operation 

1120
01:29:08,941 --> 01:29:16,790
let's call it capital D, D on psi gives us some psi prime,

1121
01:29:20,379 --> 01:29:21,463
what about wave function,

1122
01:29:22,998 --> 01:29:26,541
D when it acts on psi of x, by definition now,

1123
01:29:26,819 --> 01:29:29,388
this is the definition of how D acts,

1124
01:29:29,868 --> 01:29:32,376
it simply gives the d Psi by dx, 

1125
01:29:35,077 --> 01:29:37,997
I may sometimes lapse and going to call this partial derivative with respect to X 

1126
01:29:38,821 --> 01:29:40,915
it's fine doesn't matter, there is only one X

1127
01:29:42,626 --> 01:29:49,316
so it just differentiates , it creates a new function, it's a linear operator 

1128
01:29:50,627 --> 01:29:57,144
is it Hermitian? okay so let's write down the two Psi of the Hermitian condition here 

1129
01:29:57,342 --> 01:29:59,555
and see if they are the same, that's all, that's all there is to it

1130
01:30:01,324 --> 01:30:07,705
left side is integral phi star of X 

1131
01:30:08,807 --> 01:30:16,236
times the operation D on psi which is just d psi by d X 

1132
01:30:18,540 --> 01:30:21,171
what's the other way of doing it on the right hand side?

1133
01:30:21,350 --> 01:30:26,889
the other way of doing it is integral psi star, not not doing the same thing 

1134
01:30:26,926 --> 01:30:31,822
doing something different, psi star of X, d phi by dx

1135
01:30:33,315 --> 01:30:37,453
and the question is whether this is the complex conjugate of this?

1136
01:30:38,044 --> 01:30:40,598
so let's write equals with a question mark

1137
01:30:41,450 --> 01:30:53,086
No no complex conjugate, so let's, let's do this integral over here

1138
01:30:58,406 --> 01:31:04,549
First of all a thing in the bra-ket here is just equal to integral 

1139
01:31:04,623 --> 01:31:10,526
psi of X, d phi star by dX, 

1140
01:31:12,224 --> 01:31:14,899
just complex conjugate, the complex conjugate of a product 

1141
01:31:15,945 --> 01:31:17,860
is just a product of complex conjugate 

1142
01:31:18,768 --> 01:31:22,647
The thing in the bra-ket psi of x d phi star by dx,

1143
01:31:23,177 --> 01:31:27,913
When we are asking whether this is equal to this, they look similar,

1144
01:31:28,254 --> 01:31:30,151
this has a psi, this has a psi

1145
01:31:30,915 --> 01:31:33,049
This have a phi star, this has a phi star,

1146
01:31:33,718 --> 01:31:36,659
but unfortunately this one has psi differentiated

1147
01:31:38,216 --> 01:31:41,772
and this one has psi, you know what I mean, psi and phi

1148
01:31:43,169 --> 01:31:44,794
How do you go from this to this? 

1149
01:31:46,612 --> 01:31:51,474
integration by parts, and integrate like this,

1150
01:31:51,475 --> 01:31:54,675
Assuming, what do we assuming when we say we can integrate by parts 

1151
01:31:58,103 --> 01:32:02,850
we are assuming that far away the integral, the functions go to zero,

1152
01:32:04,722 --> 01:32:09,782
That the end point contributions from integration by part don't give us anything 

1153
01:32:10,913 --> 01:32:13,694
well of course the functions must go to zero far away 

1154
01:32:13,839 --> 01:32:16,612
because the integral of the total thing must be one

1155
01:32:17,235 --> 01:32:19,584
so we can integrate by parts,

1156
01:32:19,843 --> 01:32:22,744
anybody know what integrating by parts is? If not find out fast,

1157
01:32:23,540 --> 01:32:25,808
It's simply a rule, I will tell you what the rule is 

1158
01:32:26,395 --> 01:32:29,649
you can shift the derivative from one if you have a product 

1159
01:32:31,469 --> 01:32:38,012
dx, that's dX here, if you have a product psi of X time derivative of phi star of X 

1160
01:32:38,746 --> 01:32:43,138
then you can simply shift the derivative from one function to the other,

1161
01:32:43,355 --> 01:32:46,990
but it's at a cost, the cost, the minus sign right?

1162
01:32:48,024 --> 01:32:57,665
good, so this is equal to minus the integral phi star of x

1163
01:32:59,161 --> 01:33:04,732
d psi by dX, oops, we are in bad shape

1164
01:33:06,231 --> 01:33:08,807
This is integral phi star d phi dx,

1165
01:33:08,932 --> 01:33:12,030
this is minus the integral of d phi star d psi dx

1166
01:33:12,341 --> 01:33:14,435
they are not equal, in fact worse than that, 

1167
01:33:15,498 --> 01:33:17,692
they are absolutely the opposite of each other 

1168
01:33:18,767 --> 01:33:22,904
They are opposite in the sense of the negative of each other 

1169
01:33:23,741 --> 01:33:26,829
can we fix it? Well no we can't fix this,

1170
01:33:27,263 --> 01:33:30,479
but we can redefine an operator which is Hermitian,

1171
01:33:31,243 --> 01:33:35,027
to redefine operator which is Hermitian, it's not derivative,

1172
01:33:36,461 --> 01:33:39,110
it is minus i, the minus sign it doesn't matter,

1173
01:33:39,697 --> 01:33:43,296
you can put i or minus I, d by dx, 

1174
01:33:47,139 --> 01:33:52,178
so let's go through it again and check whether now it's Hermitian 

1175
01:33:52,179 --> 01:33:53,896
This is the new thing that we gonna call 

1176
01:33:54,534 --> 01:33:55,490
and we are gonna give it a different name 

1177
01:33:55,638 --> 01:33:57,815
I'm not gonna call it d, because it's not derivative

1178
01:33:58,753 --> 01:34:00,150
it's minus i times derivative, 

1179
01:34:00,159 --> 01:34:01,993
anybody have a good suggestion for what I call this?

1180
01:34:02,710 --> 01:34:16,761
p brilliant, p for, momentum Momentum 

1181
01:34:18,007 --> 01:34:21,328
and if I really wanted to be momentum with units of momentum 

1182
01:34:21,543 --> 01:34:27,408
what else would I put there? Where? right or left?

1183
01:34:28,939 --> 01:34:31,527
I will put on left but you better put it downstairs

1184
01:34:32,652 --> 01:34:34,345
Alright but we are not gonna worry about h bar 

1185
01:34:36,094 --> 01:34:39,144
We are gonna use our usual convention of ignoring H bar for the moment 

1186
01:34:40,950 --> 01:34:42,664
p is minus i, d by dx, 

1187
01:34:42,979 --> 01:34:49,363
let's see what happens, over here we have to replace this by an i over here,

1188
01:34:53,259 --> 01:34:55,346
minus i thank you, minus i,

1189
01:34:58,976 --> 01:35:04,137
what about this one? with complex conjugate it and everything 

1190
01:35:06,053 --> 01:35:10,265
we better change the signm because the complex conjugate of minus i is plus i,

1191
01:35:11,786 --> 01:35:24,421
so, let's put it this way, minus i, and we should get plus i here if it's Hermitian

1192
01:35:24,422 --> 01:35:27,621
Why? because in the process of taking the complex conjugate 

1193
01:35:28,339 --> 01:35:31,154
every i gets converted to a minus i,

1194
01:35:31,336 --> 01:35:33,833
and every minus i gets converted to a plus i,

1195
01:35:34,907 --> 01:35:36,044
Is it true now?

1196
01:35:38,905 --> 01:35:43,470
Yeah, why, because this is equal to minus this

1197
01:35:44,033 --> 01:35:45,330
That's what we proved over here 

1198
01:35:46,364 --> 01:35:52,089
okay so we stick an i here, because i changes sign on the complex conjugate

1199
01:35:52,849 --> 01:36:01,427
this little operation here tells us that minus i d by dx is a Hermitian operator

1200
01:36:02,308 --> 01:36:08,238
A little odd isn't it, that minus i, d by dx should somehow be a real thing 

1201
01:36:08,600 --> 01:36:12,603
should represent the real thing, and d by dx should not 

1202
01:36:13,263 --> 01:36:19,044
but, let's, let's check something out, let's check something out right now 

1203
01:36:22,122 --> 01:36:25,338
let's just, we are gonna finish up in a moment 

1204
01:36:28,373 --> 01:36:30,934
we haven't talked about expectation values yet tonight,

1205
01:36:31,299 --> 01:36:33,371
but I think you can guess how to think about them 

1206
01:36:33,675 --> 01:36:37,968
expectation values in the state psi of an observable L 

1207
01:36:38,983 --> 01:36:43,952
I just the sandwich composed of psi on both sides and L in the middle 

1208
01:36:44,926 --> 01:36:47,542
so let's take the expectation value of momentum,

1209
01:36:48,952 --> 01:36:56,763
it's integral psi star of X, psi of X and then according to the rule here 

1210
01:36:56,764 --> 01:37:08,009
p is minus i d by dx, is it real or imaginary?

1211
01:37:09,300 --> 01:37:15,082
you are wrong, it's real, so let's prove that it's real, I hope it's real

1212
01:37:15,320 --> 01:37:16,063
God hope it's real, okay

1213
01:37:20,056 --> 01:37:21,975
So let's integrate by parts

1214
01:37:24,761 --> 01:37:27,138
This is equal, first of all, let's take out the minus i,

1215
01:37:27,324 --> 01:37:37,137
minus i, integral psi star of X d Psi by dx, now let's integrate that by parts

1216
01:37:38,919 --> 01:37:46,069
This is equal to minus i, and then we can switch the derivative 

1217
01:37:47,933 --> 01:37:53,099
we can switch the derivative to the star variable, integral

1218
01:37:54,364 --> 01:37:59,539
d psi star by dx times psi of X,

1219
01:38:00,903 --> 01:38:07,541
but then in doing the integration by parts, we have to change the sign 

1220
01:38:09,329 --> 01:38:12,027
so, here we have a little thereom, 

1221
01:38:12,522 --> 01:38:14,555
the integral of this thing here with a minus i 

1222
01:38:15,388 --> 01:38:18,621
is the integral of this thing here with a plus i

1223
01:38:27,758 --> 01:38:30,478
What's the complex conjugate of this whole thing? 

1224
01:38:31,569 --> 01:38:38,023
To complex conjugate it, you replace psi star by psi, or psi is over here,

1225
01:38:39,045 --> 01:38:45,007
you replace d psi by, d psi by d x by d psi star by dx 

1226
01:38:45,736 --> 01:38:48,193
and you replace minus I by plus i 

1227
01:38:49,441 --> 01:38:52,117
this over here is the complex conjugate of this 

1228
01:38:53,404 --> 01:38:55,962
Can everybody see that, or should I explain it further? 

1229
01:38:56,101 --> 01:38:58,737
let's put a minus i out here and write,

1230
01:38:58,738 --> 01:39:02,350
this is equal to d psi, this just gives us d psi by dx

1231
01:39:07,223 --> 01:39:08,859
minus i, d psi by dx, 

1232
01:39:14,135 --> 01:39:20,571
this is clearly the complex, the integral are complex conjugate of each other 

1233
01:39:21,688 --> 01:39:24,588
d psi by dx is this complex conjugate of this 

1234
01:39:24,989 --> 01:39:26,162
that one is the complex conjugate of this 

1235
01:39:26,985 --> 01:39:29,984
and i and minus i are complex conjugate of each other 

1236
01:39:31,401 --> 01:39:35,950
so given that this is the complex conjugate of this and they are equal to each other 

1237
01:39:36,933 --> 01:39:41,045
what you say about a thing which is equal to its own complex conjugate? It's real 

1238
01:39:42,731 --> 01:39:44,741
so we prove rather startling fact, 

1239
01:39:47,081 --> 01:40:03,387
that the integral of psi star of X minus i d psi by dx is real not yes,

1240
01:40:03,427 --> 01:40:09,935
I did, and again is because when you integrate by parts you pick up an extra minus sign

1241
01:40:11,168 --> 01:40:14,858
so instead of this thing being minus it's complex conjugate, 

1242
01:40:15,106 --> 01:40:16,767
it's plus it's complex conjugate,

1243
01:40:18,264 --> 01:40:19,225
okay so what do we learn?

1244
01:40:19,637 --> 01:40:22,159
We learned that this is a Hermitian operator 

1245
01:40:23,660 --> 01:40:27,940
it's a Hermitian operator and therefore an observable in quantum mechanics

1246
01:40:29,792 --> 01:40:31,607
The next time we will explore a little more 

1247
01:40:31,736 --> 01:40:34,279
We will discuss what this has to do with Fourier transforms 

1248
01:40:35,556 --> 01:40:37,846
We will discuss questions about what the probability 

1249
01:40:38,092 --> 01:40:40,742
for finding the particle at different locations are, 

1250
01:40:41,109 --> 01:40:42,545
but then we will also talk about 

1251
01:40:42,616 --> 01:40:45,309
what the probability for having different momentum are 

1252
01:40:46,613 --> 01:40:48,883
And then we move to the Schrodinger equation 

1253
01:40:49,153 --> 01:40:54,244
We will move to the evolution of the, of the wave function

1254
01:40:54,311 --> 01:40:59,349
This, after all, now is really in the classic sense of the word wave function 

1255
01:41:01,491 --> 01:41:04,633
and we will think about how it evolves with time

1256
01:41:06,880 --> 01:41:08,335
Oh one of the thing we, 

1257
01:41:09,341 --> 01:41:11,136
we could do it tonight or think I think would probably 

1258
01:41:13,348 --> 01:41:16,486
at the point of diminishing returns, Are you exhausted?

1259
01:41:18,183 --> 01:41:20,771
we could do one more thing let me do one more thing for you

1260
01:41:22,647 --> 01:41:25,540
Let's calculate an interesting quantity 

1261
01:41:25,718 --> 01:41:28,203
and why it's an interesting quantity will come through in a moment 

1262
01:41:29,234 --> 01:41:34,904
so let's take a quantity which seems very dull, but it's interesting and calculate it,

1263
01:41:36,445 --> 01:41:40,101
it's the commutator of x with p 

1264
01:41:48,728 --> 01:41:52,594
take any wave function, whatever it is, psi of X anyone 

1265
01:41:54,692 --> 01:41:56,880
And we are gonna do two things with it,

1266
01:41:57,656 --> 01:41:59,430
We are first gonna multiply it by X, 

1267
01:42:00,246 --> 01:42:03,478
and then we are gonna multiply this by minus i d by or 

1268
01:42:03,741 --> 01:42:05,779
operate would like minus i d by dx, 

1269
01:42:07,872 --> 01:42:12,628
this can be called p x operating on psi

1270
01:42:14,259 --> 01:42:19,292
This is P this is X in the order first X then P 

1271
01:42:21,693 --> 01:42:23,394
then we are gonna calculate it the other way

1272
01:42:23,503 --> 01:42:27,290
XP on psi, what was that?

1273
01:42:27,291 --> 01:42:29,564
the other half of the blackboard over here, 

1274
01:42:30,192 --> 01:42:37,530
we are gonna write x times minus i d by dx psi of x 

1275
01:42:38,365 --> 01:42:41,780
and see what we get, in fact what we are gonna do is

1276
01:42:42,031 --> 01:42:45,746
We are gonna do the commutator, which means we are gonna calculate this one 

1277
01:42:45,747 --> 01:42:49,885
with a plus, I'm gonna calculate this one with a plus sign, 

1278
01:42:49,994 --> 01:42:51,142
and this one with a minus sign 

1279
01:42:53,317 --> 01:42:56,254
but before we subtract them ,let's just calculate them

1280
01:42:57,868 --> 01:43:00,871
Alright, this is just minus i

1281
01:43:01,887 --> 01:43:03,716
and now the derivative of a product

1282
01:43:05,565 --> 01:43:06,575
The derivative of the product

1283
01:43:06,831 --> 01:43:13,831
This derivative here now has to act on the whole result of x acting on psi of x

1284
01:43:13,832 --> 01:43:17,770
x acting on psi of x gives a new wave function,

1285
01:43:18,563 --> 01:43:19,833
and then P act on it

1286
01:43:20,477 --> 01:43:26,111
So what you get when you hit the product? You get two terms

1287
01:43:26,738 --> 01:43:30,882
The first term is the derivative of X times psi derivative of x is just one,

1288
01:43:31,468 --> 01:43:33,466
So the first term is just Psi of x

1289
01:43:34,648 --> 01:43:41,330
and the second term is plus X d psi by dx, everybody agree with that?

1290
01:43:48,828 --> 01:43:52,926
the derivative hit X in left psi, and then the derivative hit psi and left x,

1291
01:43:53,938 --> 01:43:56,096
This is the Psi over here, now what about the psi over here?

1292
01:43:57,068 --> 01:44:01,356
here, the d by dx doesn't act on x, it just act this way 

1293
01:44:01,657 --> 01:44:10,134
it just acts on psi of X and gives us minus i d by dx psi of x

1294
01:44:14,787 --> 01:44:26,767
Sorry, X minus i X d psi by dx, now let's subtract them,

1295
01:44:27,757 --> 01:44:31,285
let's take this one with minus, and this one with plus 

1296
01:44:31,838 --> 01:44:35,715
this corresponds to taking the commutator of X with P 

1297
01:44:37,650 --> 01:44:41,992
acting on psi of X, commutator of X with P is another operator 

1298
01:44:42,256 --> 01:44:46,262
and it acts on psi of X, what we get when we subtract?

1299
01:44:46,975 --> 01:44:49,193
let's see, this one now has to come with a plus sign

1300
01:44:49,194 --> 01:44:54,657
since we are subtracting ,some sort of subtracting we have to change sign here 

1301
01:44:55,716 --> 01:45:04,578
and this term cancels,  this term cancels goes away,

1302
01:45:04,681 --> 01:45:07,955
and all were left with a psi of X, 

1303
01:45:07,957 --> 01:45:15,281
that's all that's left with this term cancels, now we just left i psi of X i psi of X

1304
01:45:15,992 --> 01:45:17,653
alright so here's what we've found 

1305
01:45:18,337 --> 01:45:22,542
for any psi of X whatever for any psi of X 

1306
01:45:22,832 --> 01:45:28,854
the commutator of X with P acting on psi of X just gives

1307
01:45:29,718 --> 01:45:34,045
If it's a plus i, I think it's a plus i, psi of x

1308
01:45:34,964 --> 01:45:42,321
now if an operator acting on any wave function gives another operator,

1309
01:45:45,858 --> 01:45:49,168
but if the operator, XP acting on any wave function 

1310
01:45:49,865 --> 01:45:52,134
is the same as i acting on that same wave function 

1311
01:45:52,457 --> 01:45:53,642
for any wave function whatever 

1312
01:45:54,793 --> 01:45:58,325
it follows that this operator is equal to this operator

1313
01:45:58,326 --> 01:46:04,973
Another way of saying it is if XP minus i

1314
01:46:05,422 --> 01:46:10,489
Put all in a big bracket, when hits any wave function at all give zero

1315
01:46:12,814 --> 01:46:14,786
then it means that this is zero, 

1316
01:46:15,595 --> 01:46:19,464
there is only one operator which give zero when it acts on every state

1317
01:46:20,181 --> 01:46:27,338
On every vector, there is no vector which escapes when this thing hits it

1318
01:46:28,993 --> 01:46:32,007
The implication is that this thing is zero, 

1319
01:46:32,877 --> 01:46:38,910
This is the only one operator which annihilates which kills every vector to zero

1320
01:46:39,286 --> 01:46:44,050
XP, minus, commutator XP is equal to i

1321
01:46:55,818 --> 01:47:01,565
that I put back the h bar, there would be an h bar here, i h bar 

1322
01:47:06,788 --> 01:47:11,240
we've seen things like this before, you may have to go back to classical mechanics 

1323
01:47:11,734 --> 01:47:13,920
we've seen things before, like this,

1324
01:47:13,921 --> 01:47:17,329
we've seen Poisson brackets which had similar relationships 

1325
01:47:19,024 --> 01:47:24,874
Poisson bracket of X with P is equal to one 

1326
01:47:25,654 --> 01:47:31,381
earlier in this course we discussed the potential relationship 

1327
01:47:31,951 --> 01:47:37,281
we had reason to discuss why commutators have to do with the Schrodinger's equation 

1328
01:47:37,433 --> 01:47:40,632
go back but we discussed the relationship, 

1329
01:47:41,121 --> 01:47:45,503
it might be a relationship between commutators of Poisson brackets in particular,

1330
01:47:45,920 --> 01:47:48,010
the relation involved this i h bar

1331
01:47:50,674 --> 01:47:53,979
this is actually the way Dirac began his discussion of quantum mechanics

1332
01:47:55,026 --> 01:47:58,835
He said there must be something analogous to a Poisson bracket in quantum mechanics 

1333
01:47:59,988 --> 01:48:03,982
and the only thing to think of which had the right algebraic structure 

1334
01:48:04,019 --> 01:48:11,235
anti-symmetric, had a certain mathematical structure was the commutator 

1335
01:48:11,905 --> 01:48:17,211
and he postulated that commutators and the Poisson brackets were related with each other

1336
01:48:17,621 --> 01:48:18,896
To an i h bar,

1337
01:48:19,179 --> 01:48:22,312
and then proceeded to go exactly backwards from the way that we went,

1338
01:48:23,435 --> 01:48:26,735
he said, if this is a if this is the Poisson bracket 

1339
01:48:27,062 --> 01:48:30,561
then this must be the commutator, then there must be an operator P 

1340
01:48:32,161 --> 01:48:37,052
was commutator x with i, and we went back through the operation 

1341
01:48:37,603 --> 01:48:41,238
and discovered that P was minus i d by dx 

1342
01:48:42,298 --> 01:48:43,883
so we went the opposite direction

1343
01:48:44,009 --> 01:48:46,617
We discovered P by just playing games, 

1344
01:48:46,618 --> 01:48:49,971
let's say how about this operator d by dx have to do?

1345
01:48:50,353 --> 01:48:53,145
What we've found is that it was, it was not Hermitian 

1346
01:48:53,381 --> 01:48:55,784
we have to multiply it by i to make it Hermitian

1347
01:48:56,895 --> 01:48:59,941
We explore it a little bit found out some interesting properties of it,

1348
01:49:00,381 --> 01:49:06,241
but the most interesting property is that it's commutator is equal to i h bar

1349
01:49:06,242 --> 01:49:13,418
which suggests that it's connected with the Poisson bracket 

1350
01:49:13,419 --> 01:49:16,166
the classical Poisson bracket being one 

1351
01:49:18,845 --> 01:49:24,295
of course the upshot of it, of course we need to do a lot more to convince ourselves

1352
01:49:25,082 --> 01:49:30,192
the operator minus i d by dx really behaves like quantum mechanical momentum 

1353
01:49:31,453 --> 01:49:33,932
You can't just say it looks like a Poisson bracket 

1354
01:49:33,933 --> 01:49:35,385
it must be the quantum mechanical momentum 

1355
01:49:36,224 --> 01:49:39,434
we have to show that wave packets move with velocities 

1356
01:49:40,819 --> 01:49:46,249
that the governed by the by the by this that thing must be conserved 

1357
01:49:47,303 --> 01:49:49,297
that it has all the properties of momentum,

1358
01:49:49,298 --> 01:49:50,650
but we will 

1359
01:49:51,605 --> 01:49:56,734
>> Is there some connection between these abstract operators

1360
01:49:56,892 --> 01:50:04,661
>> minus i derivatives and the body of mass

1361
01:50:05,817 --> 01:50:08,246
Well we would like to show that in some way that 

1362
01:50:08,247 --> 01:50:14,965
it is mass time velocity, so what we will try, we will eventually show 

1363
01:50:15,548 --> 01:50:18,618
that if you have a wave packet another words,

1364
01:50:18,618 --> 01:50:22,692
a psi of X which is a concentrated distribution like this,

1365
01:50:23,219 --> 01:50:26,033
and that wave packet moves along the x-axis 

1366
01:50:27,226 --> 01:50:33,404
with a velocity which is given by the expectation value of this momentum operator 

1367
01:50:33,405 --> 01:50:37,913
divided by the mass, that's what we would like to show 

1368
01:50:37,914 --> 01:50:47,504
the velocity with which this wave packet propagates along the axis of velocity 

1369
01:50:47,914 --> 01:50:49,331
just that we call the group velocity

1370
01:50:49,766 --> 01:50:53,739
Velocity that the group of waves move along times the mass of the particle 

1371
01:50:54,876 --> 01:50:57,372
is equal to the expectation value of the momentum 

1372
01:50:59,165 --> 01:51:04,176
once we do that, then I think we are, and I think that's conserved 

1373
01:51:04,825 --> 01:51:07,453
that'll take us a long way of calling it momentum 

1374
01:51:09,617 --> 01:51:13,852
at some point we have to say quantum mechanics is more fundamental in classical mechanics 

1375
01:51:13,852 --> 01:51:17,852
and the definition of momentum should be true quantum mechanics,

1376
01:51:17,914 --> 01:51:20,231
and then we should show how it behaves in a certain way 

1377
01:51:20,448 --> 01:51:22,951
when particles are heavy and so forth 

1378
01:51:26,000 --> 01:51:31,900
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