qm4.eng.srt

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Stanford University

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Okay, let's, let's go on

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Let's forget for the moment, the spin system

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We will come back to it, probably tonight, definitely tonight

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But let's remind ourselves of, 

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I think I wrote down four principles of quantum mechanics last time 

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And I wanna write them down again

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And then move on to a fifth principle

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In fact they are not all independent 

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It's easier for me to explain them one at a time 

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To deprive from the minimal, absolute minimal number of them

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I don't think there is any particular advantage in doing that

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So I just write down... all the principles that I wrote down

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The first one is the observable, the observables are the things you measure

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It could be called measurable, if I have, if it's up to me, I would call it measurable

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But they are called observables 

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Observables are represented... I will just use the equal sign to indicate that 

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Observables are represented by Hermitian operators

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I use the symbol L for the moment for the Hermitian operator

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Not all operators, not all linear operators are Hermitian

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And it will not always be the case that 

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the symbol L will stand for Hermitian operator

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But right now, L stand for Hermitian operator, any Hermitian operator

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Every observable is identified with a Hermitian operator

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Then the question is whether every Hermitian operator 

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is a thing which can be measured, is a generally hard question, 

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you know, what can or can't be measured 

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depends on what the materials you have available, all kind of things

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Generally, the abstract rule, or the full rule, let's call it,

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Is it any Hermitian operator is identified with a observable and 

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Some day somebody will figure out how to measure it 

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Okay, but the other way that any observable 

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is identified with a Hermitian operator, that's for sure

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The eigenvalue of a Hermitian operator, let's call them lambdas 

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Lambda is the eigenvalue of L 

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The eigenvalues represent the possible numerical values 

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that L can exhibit when it's measured

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Okay so eigenvalues, eigenvalues of L which equal lambda 

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Now they represent the possible observe... 

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or the possible outcome of an experiment 

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Possible outcome of an experiment to measure L of L measurement 

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Okay I think we have a third one along the line here 

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Physically distinguishable state, 

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Now let's just see what physically distinguishable state means

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Certainly up and down are physically distinguishable 

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And the meaning of saying two states are physically distinguishable 

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is that there exist a measurement of some kind that you can do 

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that can tell you the difference between a up state and a down state

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Another words, somebody hands you a spin from out of their pocket 

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And they say I created the spin, 

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I prepared it in a state which is either up or down

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Can you do an experiment to unambiguously tell me which one it is?

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Yes you tilt your apparatus until it's pointing along the z axis

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And you make the measurement 

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If the state was up you will get plus one

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If the state was down you will get minus one

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No ambiguity

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The same is true for left and right

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Different measurement, different thing that you will measure

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You will measure the x component, 

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then you will tilt your apparatus along the x axis

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but, same deal, 

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on the other hand, supposing somebody, that same person came along and said here

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I'm gonna give you a spin, here it is, and that spin either prepared up

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or I prepared it right 

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I'm not gonna tell you which way I prepared it 

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I'm not gonna tell you in which way I tilted the apparatus 

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when I did the preparation 

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I'm just tell you it's either left... I'm sorry it's either up or right

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Can you do a experiment which will uniquely tell you the difference?

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The answer is no, for example supposing you decide to measure the spin along the z axis

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Well if it's up you will get plus one, if it's right, 

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with fifty percent probability you will get plus one 

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So you can't be sure whether you are up or right

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>> forty five degrees 

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still you will have a probability for 

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Ya, ya, it would not be unambiguous, right, okay?

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So the next postulate

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>> excuse me, if in that statement you were assuming it was prepared along the z axis

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>> If it have been prepared along the x axis

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>> Then if you measure the x axis first, you get ...

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If you measure the x axis first you will find something else

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But you could have found out exactly the same answer if it was along the z axis

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Either case, there was some probability 

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of getting the same answer whether it was up or right

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There is no experiment you can do 

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which unambiguously will determine which it was

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That leads to the notion, I am not going to give you a more precise definition of it

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That leads to the notion of physically distinguishable states

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by which it's meant that 

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their exist an experiment or a set of experiments 

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that can unambiguously determine which of the two states you are talking about

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The next postulate is that physically distinct or physically distinguishable states 

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are represented by orthogonal vectors 

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Okay, so physically distinct, distinct, distinguishable, same thing

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It's easier to write distinct 

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This physically distinct states imply orthogonality 

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I said that observables are Hermitian operators 

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I could have just said linear operators, 

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let's leave with linear operator for the moment 

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But let's add a postulate that the result of every experiments

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Every simple primitive experiment it's a real number 

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If it's a real number... if it's a complex number 

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It really means that, two independent things are measured

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The actual result come out of your apparatus, come out of, you know

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The needle on your apparatus ,the real numbers 

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So I will let that as a postulate, the result of experiments are always real numbers

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And therefore the eigenvalues of observables are real numbers

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Now that does not prove that the operators are Hermitian, not enough

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The other added thing is the various eigen vector with different eigen value

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is physically distinguishable states, or orthogonal 

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Okay that's the third postulate

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And that tells you the observables are Hermitian operators 

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It's the Hermitian operator whose eigen values are always real

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And whose eigen vectors are orthogonal 

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That's necessary and sufficient 

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And this one like I said, they are not all completely independent to each other

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One, two, three, and let's write number four now

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I forgot what fourth is, oh, 

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Well, what was fourth? Ya

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Ah, the probability, the probability principle, okay

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The probability principle, it's also called Born's rule, that Max Born

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But it's a rule for probabilities 

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So far, none of these have much content, the added content 

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You know, it has content, but the real bite here is for the prediction of the probability 

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for various experiments 

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And the Born rule is the rule for how to calculate the probabilities 

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The Born rule says, if your system has been prepared in state |A>

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System in |A>

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And you measure L, the observable L 

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Now I'm using interchangeably the physical idea 

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with observables with the operator which represent it 

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I'm not going to make a special language where

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operator will always be called operator

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And the observable of L, same thing 

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They come in pairs 

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Alright, if you, if the system has been prepared in state |A>

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And you measure L, you measure L

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The outcome is going to be one of the eigenvalues 

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So the only question you can ask is 

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what is the probability that you get the answer lambda

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One of the specific eigenvalues

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The answer is, the probability 

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The probability that you get result lambda 

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That's the probability you get out of the various eigenvalue you get lambda

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Is equal to the square of inner product of the state vector 

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That the system was in

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With the eigen vector corresponding the normalized eigenvector 

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or vector normalized now

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The inner product of A with lambda squared 

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A with lambda, sometimes incidentally the inner product is called the overlap

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It's a measure, if we say that orthogonality is complete distinguishability

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Or unique in the complete distinguishability

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Then a lack of orthogonality represent to some extend the inability 

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to make the clear distinction between two states

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Ya, alright, so this is, question?

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>> I'm surprised to see lambda is represented as a vector 

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>> because it's a value

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No, no, alright the notation is every eigen value is associated with a eigen vector

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I will just write E vector 

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The notation is the vector labeled by lambda 

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is the eigen vector associated with the eigen value lambda

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That's a good point, you know, the problem we are trying to be a pure about notation

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is that the notation get very complicated 

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It get so complicated, it get difficult to read, 

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too many indices, too many different letters you have to use

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So, ya, sometimes we use slightly excuse the word bastardize notation

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Or where we conflate a symbol for a vector, with the symbol in this case for a eigen value

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Okay, so that's the probability principle or Born's rule 

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And just look at it for a minute 

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You ask why the square of the absolute value?

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Well in general, this overlap is neither positive nor even real in general

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Overlaps between two vectors are not necessarily real, in fact

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They are generally not, we are talking about a complex vector space

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And if the component of a vector is complex

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In general in a product that would not even be real, let alone positive

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On the other hand, the square of the absolute value of the square, 

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Square of the absolute value of the inner product, that is real

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It's not only real, it's also positive

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So this has a chance to be a candidate for the probability 

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Whereas the inner product itself doesn't

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The inner product itself is called the probability amplitude 

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The probability amplitude is a thing that you square 

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in the sense of absolute value to compute a probability 

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Needless to say, the justification for this principle in the end

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is experiment 

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On the other hand, you could ask how much can I bent them

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And still make physical sense out of prediction

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The answer is nobody has ever found a way

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to change the rule of quantum mechanics

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and still preserve a reasonably logical structure

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So you eventually get familiar with these principles

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>> Which of these principles took you from several possible outcomes

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>> to one actual outcome?

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Well, okay, if <A| is an eigenvector of L, if it is

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If the state, starting state of a system happens to be an eigenvector of L

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Then it would only have an inner product of one of the eigenvectors

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Right, so that's again, it's a good question so

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>> If you say that L lambda can't equal to...(inaudible)

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Ya, ya, that's right

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>> Excuse me, what if lambda is an eigenvalue for multiple eigenvectors?

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Ya, still the rule is the same, but right

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I will give you two answers 

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The first answer is don't worry about the case 

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with the eigenvalue maybe the same

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Because it's very special, but that's not good enough, let me...

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Let me say what the right answer is 

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The right answer is if you want the probability for a given value of lambda

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and there's more than one eigenvector with the same value of lambda

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Then you add them, you simply add the sum of the square 

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of the probability addition

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The question, they are really good

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>> What about the case <A| has overlap...

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Yeah, yeah, then you take the sum of the square of them

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You take the sum of the square of them

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If there're several different eigenvectors all with the same eigenvalues

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First of all you make them perpendicular to each other

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You can always make them perpendicular to each other

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No no, even if they are the with the same eigenvalues 

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The theorem, let's go back to the theorem

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The theorem says that the eigenvectors with different eigenvalues 

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are orthogonal 

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It doesn't tell you anything about the eigenvectors to the same eigenvalue

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That's supposing you have two eigenvectors with the same eigenvalue

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Then it's also true that any linear combination of those two vectors 

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is also a eigenvector with the same eigenvalue

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For example, let's suppose lambda, lambda 1 and lambda 2,

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Now these are not two different values of lambda

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These are the same value of lambda but two different eigenvector

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Maybe we should write it like this

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Lambda 1 and lambda 2, two different eigenvectors

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Both exhibiting the same eigenvalue

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Then you can take any... what that means is 

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L on lambda 1 equals lambda times lambda 1

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L on lambda 2 equals lambda lambda 2

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Multiply this by any complex number alpha 

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This one by any complex number beta

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And add the two equations

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What you find is that L on the linear combination

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The linear sum of alpha lambda 1 plus beta lambda 2

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is equal... on this side now you have lambda 

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times the same thing as in the bracket here

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Another word if you have two eigenvectors with the same eigenvalues

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You can add them with arbitrary coefficients and they remain eigenvectors

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Now given two distinct vectors, given two distinct vectors 

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and all the possible linear combination of them

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You can always find perpendicular combination 

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If you got this one, you got this one, now you can always add them 

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Or subtract them with coefficient to make them perpendicular 

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So the rest of the theorem is that if the two eigenvalues are the same 

251
00:19:40,472 --> 00:19:46,535
You have the freedom to choose two eigenvectors which are perpendicular

252
00:19:46,697 --> 00:19:48,396
Which are orthogonal 

253
00:19:48,586 --> 00:19:52,162
In the end of the day, to summarize the theorem

254
00:19:52,372 --> 00:19:59,637
It's that given the Hermitian operator, it's eigenvectors can be chosen

255
00:20:00,463 --> 00:20:05,287
So that they form a base, so they are ortho-normal 

256
00:20:05,466 --> 00:20:08,455
perpendicular to each other and normalized

257
00:20:08,663 --> 00:20:11,049
Okay, so where were we

258
00:20:12,307 --> 00:20:15,094
That was an interruption, but I don't remember where it was

259
00:20:15,274 --> 00:20:17,006
That's the problem with the interruption

260
00:20:19,727 --> 00:20:23,047
>> we just finished Born's rule

261
00:20:23,235 --> 00:20:25,769
Yeah, so we were talking about Born's rule

262
00:20:25,954 --> 00:20:30,395
Right oh, good, okay, so now if you happen to have several eigenvector

263
00:20:30,578 --> 00:20:32,187
Let's say lambda 1 and lambda 2 

264
00:20:32,417 --> 00:20:34,382
Both with the same eigenvalue, 

265
00:20:34,585 --> 00:20:39,787
then you construct the overlap of the state vector with lambda 1,

266
00:20:39,987 --> 00:20:45,523
you construct the overlap of the state vector with lambda 2 

267
00:20:45,699 --> 00:20:50,455
you square this one, you square this one, and you add them

268
00:20:50,647 --> 00:20:57,723
another words, you simply add the probabilities for the two possible ways

269
00:20:58,023 --> 00:21:00,273
that you can get the same lambda

270
00:21:00,491 --> 00:21:05,020
Just the sum of the probability, thank you Henry, way to late

271
00:21:05,261 --> 00:21:06,957
I already got Oreos 

272
00:21:09,658 --> 00:21:14,952
But my wife would say this is probably healthier than Oreos

273
00:21:22,474 --> 00:21:26,223
Right? You got Oreos?

274
00:21:45,776 --> 00:21:47,550
Loud and clear

275
00:21:47,854 --> 00:21:53,377
>> Can we say that the reason why those things are square regardless

276
00:21:53,377 --> 00:21:58,277
>> whether they are complex or not is sort of the probability adds to one right?

277
00:21:58,277 --> 00:22:06,421
Yes, uhm, we know it's the sum of the square of these things which add up to one

278
00:22:06,644 --> 00:22:10,309
For normalized states, for normalized vectors, that's right

279
00:22:11,269 --> 00:22:18,786
I'm not sure which one implies which, but clearly they fit together very nicely, good

280
00:22:19,021 --> 00:22:20,582
Okay, 

281
00:22:23,387 --> 00:22:29,679
>> question? Is it possible to get an unambiguous result?

282
00:22:29,871 --> 00:22:32,207
>> on an unprepared experiment?

283
00:22:33,367 --> 00:22:36,895
I'm not sure what an unprepared experiment means

284
00:22:37,067 --> 00:22:42,896
But I think what you are asking is, supposing somebody gives you a spin

285
00:22:43,102 --> 00:22:46,624
And they don't tell you anything about how they prepare it

286
00:22:46,843 --> 00:22:50,225
They just say here it is, I give it to you, I tell you nothing

287
00:22:52,206 --> 00:22:57,534
Then it is not possible to find an experiment which has an unambiguous answer

288
00:23:06,166 --> 00:23:07,959
Another good point

289
00:23:08,733 --> 00:23:17,695
Okay, now, now we need to come to a new idea and a new question

290
00:23:18,807 --> 00:23:24,421
In classical mechanics, all of these, the whole stories here 

291
00:23:24,605 --> 00:23:29,152
were summarized when I show you a heads and tails 

292
00:23:29,369 --> 00:23:32,120
Those are two possible state of a simple system

293
00:23:32,392 --> 00:23:35,628
I didn't talk about measuring it, you know

294
00:23:35,821 --> 00:23:38,116
It's also obvious that we didn't have to 

295
00:23:39,274 --> 00:23:44,417
Because we know that measuring is doesn't do very much to a system

296
00:23:44,604 --> 00:23:48,005
You measure, but it doesn't change much to a system in any way

297
00:23:48,005 --> 00:23:51,076
Here we come to much more complicated story

298
00:23:51,306 --> 00:23:56,485
But the story up till now it's really the quantum mechanic analog just specifying 

299
00:23:56,694 --> 00:23:58,105
That you have a collection of state

300
00:23:58,105 --> 00:24:03,698
The next question we ask was how to state change with time

301
00:24:03,875 --> 00:24:09,912
That we call in the classical case dynamical laws of motion

302
00:24:10,091 --> 00:24:13,024
Or dynamical laws of nature, whatever

303
00:24:13,261 --> 00:24:15,685
The rules for how a state changes 

304
00:24:15,685 --> 00:24:19,890
If you remember there was an example the coin 

305
00:24:20,098 --> 00:24:24,829
The simple example would be put the coin down and rule is nothing happen

306
00:24:25,014 --> 00:24:27,838
It just stay the same way, whatever it's attribute

307
00:24:28,023 --> 00:24:31,312
Another possible law is, in each interval of time 

308
00:24:31,537 --> 00:24:37,056
We have a stroboscopic time we can imagine

309
00:24:37,308 --> 00:24:39,495
And each individual time if flips

310
00:24:39,665 --> 00:24:42,287
Heads, Up, heads tails heads tails

311
00:24:42,549 --> 00:24:44,088
That's perfectly good law

312
00:24:45,712 --> 00:24:48,359
We talked about the concept, 

313
00:24:48,541 --> 00:24:51,487
the same concept in the context of a die with six phases

314
00:24:51,685 --> 00:24:55,640
Or any number of phases including continuous infinity

315
00:24:55,796 --> 00:25:01,346
Same basic idea, the laws of physics are updating from instance to instance

316
00:25:01,515 --> 00:25:05,120
Of a configuration, I will tell you what a configuration changes

317
00:25:05,311 --> 00:25:09,498
And if you have a good law, in classical physics it's deterministic

318
00:25:10,079 --> 00:25:13,854
It will tell you how the state changes, but also another rule

319
00:25:14,055 --> 00:25:16,151
Another rule we call reversibility

320
00:25:16,341 --> 00:25:23,136
Reversibility was basically just the idea that the state don't run into each other

321
00:25:23,337 --> 00:25:26,543
The states which are distinguishable namely, different

322
00:25:26,747 --> 00:25:29,006
Heads and tails are distinguishable, 

323
00:25:31,079 --> 00:25:37,985
two states which are different will evolve mathematically and stay different

324
00:25:38,187 --> 00:25:42,113
That's the idea of reversibility

325
00:25:42,359 --> 00:25:48,585
And when it comes down to, it's exactly that if you know the state at any instance

326
00:25:48,808 --> 00:25:50,405
You not only will know what comes next

327
00:25:50,405 --> 00:25:52,760
But you also will know what change before

328
00:25:53,013 --> 00:25:56,231
Because there is no chance that the states run together 

329
00:25:57,939 --> 00:26:01,095
And two different states giving you the same outcome

330
00:26:01,095 --> 00:26:04,395
Then you wouldn't be able to tell where you came from

331
00:26:04,395 --> 00:26:09,379
So reversibility is the idea we also call the information conservation

332
00:26:09,651 --> 00:26:13,875
And I suspect that I refer to it as minus first law

333
00:26:13,875 --> 00:26:17,205
Minus first because it comes before everything else

334
00:26:18,235 --> 00:26:24,420
Quantum mechanics also has a rule of reversibility and information conservation

335
00:26:24,603 --> 00:26:26,967
You can call it the minus first law

336
00:26:29,470 --> 00:26:33,818
But it comes down to the same thing, the state which are distinguishable

337
00:26:34,024 --> 00:26:38,098
In the sense that I said, there exist an experiment that uniquely distinguish them

338
00:26:38,275 --> 00:26:43,453
Stay, distinguishable

339
00:26:44,661 --> 00:26:48,901
So now I wanna talk about how states evolve with time

340
00:26:49,097 --> 00:26:56,562
We gonna come back and discuss this in the context of a spin

341
00:26:56,819 --> 00:26:58,530
The simplest system of quantum mechanics 

342
00:26:58,753 --> 00:27:02,501
It's a good idea to get it all down for that, but let's talk about it more generally

343
00:27:03,706 --> 00:27:11,328
Okay, here is the postulate, the postulate is that if you have a system 

344
00:27:11,576 --> 00:27:16,026
Add time, add some specific time let's call it a time zero

345
00:27:16,264 --> 00:27:20,231
And I'm going to start changing my notation for states

346
00:27:20,478 --> 00:27:25,240
Instead of calling them A and B, I'm going to start using a notation

347
00:27:25,598 --> 00:27:28,606
Which is more or less the standard notation in quantum mechanics

348
00:27:28,895 --> 00:27:35,674
Call them Psi, Psi is a Greek letter 

349
00:27:35,915 --> 00:27:40,305
And Psi is the commonly used letter to represent the state of a system

350
00:27:40,305 --> 00:27:44,586
Now the state of systems do change with time

351
00:27:44,790 --> 00:27:50,426
So you have to think of these, these vectors as being function of time that evolve

352
00:27:50,655 --> 00:27:53,905
So let's put that in here in the following way,

353
00:27:54,089 --> 00:28:00,438
The Psi of t simply represent a state which changes with time

354
00:28:01,710 --> 00:28:03,166
Which state? It could be any state, 

355
00:28:04,206 --> 00:28:07,549
it's not the same after a certain time as it were to begin with

356
00:28:07,850 --> 00:28:11,317
But we are following one specific state of the system which have been prepared

357
00:28:11,542 --> 00:28:16,233
And then allowed to evolve, by whatever means things evolve

358
00:28:16,393 --> 00:28:28,578
Alright, the postulate is, the Psi of t can be obtain from Psi of zero

359
00:28:30,188 --> 00:28:35,635
This means time zero, by an operation on Psi of zero

360
00:28:35,892 --> 00:28:41,132
A unique definite operation which is governed by the law

361
00:28:41,329 --> 00:28:44,323
By the quantum mechanical laws for that system

362
00:28:46,451 --> 00:28:50,636
So we have to do something to the vector so it doesn't stay the same

363
00:28:50,869 --> 00:28:59,563
So let's represent what we do by the operation of some kind of operator

364
00:29:01,049 --> 00:29:04,762
Now incidentally we didn't have to start with time of zero

365
00:29:05,023 --> 00:29:11,294
We could have start with time of something else and evolve by an amount t

366
00:29:11,560 --> 00:29:15,875
So this t here really represents that time of evolution from the initial state

367
00:29:16,074 --> 00:29:20,789
To the final state, not necessarily final, but the state of some time

368
00:29:22,813 --> 00:29:28,482
The basic postulate, two basic postulate, the first is that U is a linear operator

369
00:29:28,731 --> 00:29:36,514
U is a linear operator and the same U for any state, whatever the initial state is

370
00:29:36,712 --> 00:29:41,257
You put that here, and you hit it with the same U, 

371
00:29:42,385 --> 00:29:46,107
this is called a time development operator 

372
00:29:46,308 --> 00:29:54,320
It's a linear operator, and it evolve with the state from one instance to another

373
00:29:55,503 --> 00:30:01,372
You could try to ask what would happen if you make a more general hypothesis

374
00:30:01,614 --> 00:30:06,018
For example you might have a hypothesis that U depends on the initial state

375
00:30:06,321 --> 00:30:08,449
Or you might have the hypothesis that 

376
00:30:08,723 --> 00:30:11,435
U was an operation but not a linear operator

377
00:30:11,729 --> 00:30:14,186
Some other kind of operation, you can try it, 

378
00:30:14,424 --> 00:30:17,730
you will very shortly run into big troubles with all kind of things

379
00:30:17,930 --> 00:30:21,953
It's been tried, I guarantee you it has been tried and again 

380
00:30:22,181 --> 00:30:26,401
repeatedly ever so often, some physicist will try to invent something

381
00:30:26,602 --> 00:30:28,680
And then some other person would come along and say

382
00:30:28,902 --> 00:30:34,057
Look , that has a trouble of locality, and the trouble with causality

383
00:30:34,260 --> 00:30:36,529
It has trouble with probability interpretation

384
00:30:37,699 --> 00:30:43,801
You are free to try, I've given up trying, I actually never tried

385
00:30:47,316 --> 00:30:51,033
Right ,okay, so U is the linear operator 

386
00:30:51,204 --> 00:30:54,001
With all the rules for linear operators 

387
00:30:54,178 --> 00:31:01,316
But one other thing and that's this idea of conservation of information

388
00:31:01,512 --> 00:31:04,197
Or conservation of distinguishability 

389
00:31:05,415 --> 00:31:09,773
That if you have two states, two different states of the same system

390
00:31:09,973 --> 00:31:16,455
And you evolve them, let's call phi of t, alright capital Phi

391
00:31:17,687 --> 00:31:23,464
That's given by the same U of t, the same operator times Phi at time zero

392
00:31:26,063 --> 00:31:33,349
Then, if Phi to begin with, is orthogonal to Psi

393
00:31:34,856 --> 00:31:39,215
Another words if they are physically distinguishable to begin with

394
00:31:39,414 --> 00:31:42,905
Then they will remain orthogonal for all time

395
00:31:43,873 --> 00:31:49,478
>> on the left and ...(inaudible)

396
00:31:49,642 --> 00:31:50,645
say that again?

397
00:31:50,842 --> 00:31:52,134
>> The Greek letters are upper case on the left and lower case on the right 

398
00:31:52,328 --> 00:31:54,286
>> is that just?

399
00:31:54,536 --> 00:31:56,357
Because I'm sloppy 

400
00:31:56,562 --> 00:32:02,136
Alright so orthogonality breeds orthogonality

401
00:32:03,808 --> 00:32:07,746
Physically distinct states evolve into physically distinct states

402
00:32:09,475 --> 00:32:12,722
Distinguishability remains intact, 

403
00:32:12,901 --> 00:32:18,156
it's the immediate analog of the minus first law of classical mechanics

404
00:32:19,412 --> 00:32:21,529
So let's see if we can find out what it says

405
00:32:21,694 --> 00:32:26,185
Oh, incidentally it's very easy to prove from that

406
00:32:26,369 --> 00:32:31,593
What does it say? It say if Psi and Phi were initially orthogonal 

407
00:32:43,201 --> 00:32:44,938
If they are initially orthogonal 

408
00:32:45,122 --> 00:32:57,515
If, then, don't ask me to use capital letter, it's much quicker to use lower case one

409
00:32:59,883 --> 00:33:02,732
I mean capital here, I mean upper case

410
00:33:04,068 --> 00:33:06,332
Then, this is true and it remains true for all time

411
00:33:06,529 --> 00:33:13,412
It's an easy exercise I will leave it to you to prove that if orthogonality is preserved 

412
00:33:13,646 --> 00:33:15,856
That you can say a stronger thing 

413
00:33:16,091 --> 00:33:19,849
It's implied by this, you can say if you take any two vectors 

414
00:33:20,044 --> 00:33:22,035
which may or may not be orthogonal 

415
00:33:22,235 --> 00:33:26,280
The inner product between them stay constant with time

416
00:33:26,465 --> 00:33:29,826
That' s a nice little trick, if you take any two vectors

417
00:33:30,067 --> 00:33:31,837
You expand them in base vectors, 

418
00:33:32,034 --> 00:33:36,029
then you assume that base vector remains orthogonal with time

419
00:33:36,234 --> 00:33:38,630
Then you can prove that 

420
00:33:38,828 --> 00:33:42,763
the inner product of any two vectors is time independent

421
00:33:44,401 --> 00:33:51,486
So it follow from the orthogonality, the evolution of orthogonality 

422
00:33:51,685 --> 00:33:55,835
That the inner product between two states, remain

423
00:33:56,110 --> 00:33:58,228
the overlap between them remains the same

424
00:33:58,439 --> 00:34:01,034
It's essential that the overlap between two states 

425
00:34:01,235 --> 00:34:03,354
is of course a measure of their similarity

426
00:34:03,554 --> 00:34:07,673
And what it says is the degree of similarity remains the same

427
00:34:08,578 --> 00:34:13,477
Okay let's see if we can figure out what that means for you

428
00:34:15,342 --> 00:34:22,313
What it says, let's rewrite this by plugging in Phi of t and Psi of t

429
00:34:22,500 --> 00:34:26,085
The product U times Phi and U times Psi

430
00:34:26,281 --> 00:34:30,912
But before we do, we have to remember that we have to flip this equation 

431
00:34:31,094 --> 00:34:39,103
oops, it's chocolate? Yeah this is chocolate

432
00:34:39,290 --> 00:34:45,981
We have to flip it with a ket vector to a bra vector equation

433
00:34:46,183 --> 00:35:03,044
So let's do so, this is Psi of t equals Psi of zero, not times U

434
00:35:03,254 --> 00:35:09,424
But times what? U dagger, the Hermitian conjugate

435
00:35:11,164 --> 00:35:20,774
Not the complex, it's related to the complex conjugate of the Hermitian conjugate

436
00:35:21,004 --> 00:35:25,015
So now we plug this into this equations here, and what it says?

437
00:35:25,185 --> 00:35:29,389
Let's write it out, a little bit of a eraser, 

438
00:35:35,877 --> 00:35:39,157
for arbitrary state don't have to be orthogonal

439
00:35:39,359 --> 00:35:45,011
Whatever they are , they have some inner product that's equal to Psi of zero

440
00:35:47,965 --> 00:35:55,517
U dagger of t, U of t, Phi of zero

441
00:35:57,361 --> 00:36:05,094
All I've done is plug in, for Phi, U of t, Psi, U dagger of t

442
00:36:05,374 --> 00:36:11,214
Now the product U dagger times U is another operator

443
00:36:11,394 --> 00:36:12,724
What is the product of two operators 

444
00:36:12,904 --> 00:36:14,944
Because we didn't talk about this,

445
00:36:15,204 --> 00:36:17,274
But the product of two operators are two very simple concept 

446
00:36:18,689 --> 00:36:21,687
You take a linearly operator, and you apply it to a vector 

447
00:36:21,911 --> 00:36:25,477
And then you take the result , and you apply another operator to it

448
00:36:25,663 --> 00:36:31,992
The process of repeatedly applying two operators gives you an operation

449
00:36:32,177 --> 00:36:35,662
Which is called the  product of the two operates, so that's what this is 

450
00:36:35,873 --> 00:36:40,681
First you hit Phi with U and then you hit the result with U dagger

451
00:36:40,902 --> 00:36:45,481
And what this said that for any Phi and Psi

452
00:36:45,686 --> 00:36:51,828
Any Phi and Psi at all, that when you plug in here 

453
00:36:52,042 --> 00:36:54,940
U dagger and U, the product of U dagger and U

454
00:36:55,220 --> 00:37:04,732
You get something which is exactly the same as have you not evolve the state

455
00:37:11,231 --> 00:37:14,940
The inner product stays the same, oops, I think I've make an mistake

456
00:37:15,152 --> 00:37:17,388
One of these should be Phi right?

457
00:37:17,600 --> 00:37:23,466
Alright so that's a principle of conservation of overlap if you like

458
00:37:24,890 --> 00:37:29,174
Alright so here's the theorem we don't need this 

459
00:37:29,418 --> 00:37:32,438
This is again very very easily prove by going to a base 

460
00:37:32,636 --> 00:37:38,726
Very easily prove, if the matrix element, it's called the matrix element

461
00:37:38,910 --> 00:37:49,659
So the product like this, if this equation is true for any pair of vectors 

462
00:37:49,879 --> 00:37:52,770
Here we go, I will write the theorem in general

463
00:37:52,930 --> 00:37:57,404
If you have an operator, let's call it K, for U dagger U

464
00:37:57,588 --> 00:38:09,534
For any pair of vectors Psi, <Psi|K|Phi> is equal to <Psi|Phi>

465
00:38:09,798 --> 00:38:11,968
For any pair of vectors, whatever

466
00:38:12,174 --> 00:38:15,599
It follow that K is a unit operator

467
00:38:15,786 --> 00:38:19,160
The unit operator is a operator which does nothing, 

468
00:38:19,393 --> 00:38:21,250
just gives you back the same vector

469
00:38:23,967 --> 00:38:29,326
So what follow then, U dagger times U must be the unit operator

470
00:38:29,518 --> 00:38:34,655
That's the content of conservation of overlap

471
00:38:36,144 --> 00:38:45,359
U dagger of t for any t, times U of t, is equal to the unit operator

472
00:38:46,521 --> 00:38:48,817
Reading, a unit operator that does nothing

473
00:38:49,019 --> 00:38:51,183
Just gives you back the same vector 

474
00:38:52,295 --> 00:38:56,859
>> Questions? Can you say that one of the infer from the other 

475
00:38:57,039 --> 00:38:58,333
>> Is the Matrix (inaudible)

476
00:38:58,538 --> 00:39:01,910
Yes you can, yes you can, indeed

477
00:39:07,223 --> 00:39:09,071
I will examine these for a little bit now

478
00:39:09,272 --> 00:39:13,010
In particular let's examine for a very very small time

479
00:39:13,210 --> 00:39:17,471
A very small time, another words 

480
00:39:17,652 --> 00:39:21,719
if you only evolve the system by a tiny little incremental time

481
00:39:22,806 --> 00:39:26,455
Let's call the incremental time epsilon right?

482
00:39:27,303 --> 00:39:32,763
Let's write that U of epsilon, now first of all what is the U of zero?

483
00:39:34,027 --> 00:39:39,933
No time evolution at all, just one ,just the unit operator

484
00:39:40,764 --> 00:39:46,228
So for very small epsilon, U must be itself close to a unit operator

485
00:39:48,556 --> 00:39:53,303
Plus a small deviation presumably have order epsilon 

486
00:39:53,487 --> 00:39:55,151
That would be right, it would be a order epsilon 

487
00:39:55,372 --> 00:40:00,758
so it will be something of order epsilon times another operator here

488
00:40:02,823 --> 00:40:08,195
I'm going to make use of freedom with all that I have

489
00:40:08,402 --> 00:40:14,392
I'm going to put a minus sign here, we are gonna put an operator over here

490
00:40:14,584 --> 00:40:16,160
This minus sign has no content yet

491
00:40:16,363 --> 00:40:19,057
I'm gonna say the operator over here, the minus sign

492
00:40:19,247 --> 00:40:21,865
You know, just absorb in the definition 

493
00:40:22,048 --> 00:40:24,321
I'm also gonna put an i here,

494
00:40:26,562 --> 00:40:32,138
Again there's no content yet, because I haven't told you anything about H

495
00:40:32,326 --> 00:40:39,408
H is some operator, epsilon times H is just some operator

496
00:40:39,408 --> 00:40:41,239
And i times epsilon and H is just an operator

497
00:40:41,446 --> 00:40:45,878
So I have not really say anything other than a small epsilon

498
00:40:46,158 --> 00:40:51,388
The operator U is close to the identity, how close? Of the order epsilon

499
00:40:51,568 --> 00:40:56,675
And then there maybe additional things of order epsilon square epsilon cube

500
00:40:56,858 --> 00:41:00,557
But we are gonna drop that, and study this equation

501
00:41:00,802 --> 00:41:04,515
Or we should give a name to this equation here

502
00:41:04,680 --> 00:41:08,584
Or more specifically we should give a name to the operator 

503
00:41:08,811 --> 00:41:10,200
which satisfy this rule

504
00:41:11,472 --> 00:41:13,904
U dagger times U is equal to one

505
00:41:14,109 --> 00:41:20,219
Or as we said that U dagger the Hermitian conjugate of U is the inverse of U

506
00:41:22,002 --> 00:41:24,218
Such operators are called unitary 

507
00:41:30,090 --> 00:41:33,469
U is a unitary operator, so this is again

508
00:41:33,702 --> 00:41:38,499
This is a restatement of the principle of conservation of overlap

509
00:41:40,046 --> 00:41:46,408
Let's now apply this equation, this condition, to U of epsilon

510
00:41:46,625 --> 00:41:48,349
See what we find out about H 

511
00:41:49,604 --> 00:41:51,532
So let's bring it over to this blackboard

512
00:41:52,564 --> 00:41:56,880
U epsilon is equal to one minus i epsilon H

513
00:41:57,096 --> 00:41:58,431
I will do it over here

514
00:41:58,635 --> 00:42:01,969
What about U dagger? U dagger is the Hermitian conjugate

515
00:42:02,919 --> 00:42:06,688
The Hermitian conjugate of 1 is just 1

516
00:42:06,688 --> 00:42:10,847
But then we have complex conjugate plus i epsilon

517
00:42:11,029 --> 00:42:14,734
And then we have Hermitian conjugate H

518
00:42:19,134 --> 00:42:23,334
So here is U dagger, and here is U

519
00:42:23,550 --> 00:42:26,239
And now let's multiply them together, and set the result equal to one

520
00:42:26,472 --> 00:42:29,436
To order epsilon, to order epsilon

521
00:42:30,749 --> 00:42:39,908
Okay, so what do we have, we have U dagger is one plus i epsilon times H dagger

522
00:42:41,172 --> 00:42:48,219
Times one minus i epsilon H that has to equal 1

523
00:42:48,433 --> 00:42:52,918
And now just expand it out to all the epsilon one

524
00:42:53,099 --> 00:43:04,076
Plus i epsilon H dagger minus H equals one

525
00:43:04,268 --> 00:43:10,417
As I said, the approximation the epsilon is so small 

526
00:43:10,615 --> 00:43:14,002
that the square can be completely ignore compare to order epsilon

527
00:43:15,697 --> 00:43:18,819
Okay, so the ones canceled 

528
00:43:19,046 --> 00:43:28,538
And basically what we find is H dagger minus H has to be zero

529
00:43:30,109 --> 00:43:33,885
It cancel out the ones, we get zero on the right hand side

530
00:43:34,169 --> 00:43:39,080
And so to order epsilon, H has to equal H dagger

531
00:43:41,617 --> 00:43:46,467
This is the condition that H be Hermitian 

532
00:43:48,514 --> 00:43:50,869
Now I didn't call H to be Hermitian 

533
00:43:51,235 --> 00:43:54,250
I call the H for something else, I call the H for Hamiltonian 

534
00:43:56,003 --> 00:43:59,059
What it has to do with the Hamiltonian of classical mechanics

535
00:43:59,296 --> 00:44:01,432
will eventually become clear

536
00:44:01,665 --> 00:44:06,261
The Hamiltonian for classical mechanics also introduced

537
00:44:06,514 --> 00:44:11,981
to the equations of how system evolves from one instance to the next

538
00:44:12,179 --> 00:44:16,517
But of course that's not a good enough reason to call this Hamiltonian

539
00:44:16,742 --> 00:44:19,600
We will have to see the introduce equation of a way 

540
00:44:19,817 --> 00:44:24,279
very very similar to the way of Hamiltonian in classical mechanics

541
00:44:24,279 --> 00:44:30,761
It is also to think, this is Hermitian right?

542
00:44:32,675 --> 00:44:35,464
That means that if it's an observable

543
00:44:35,686 --> 00:44:36,979
It could be an observable

544
00:44:37,931 --> 00:44:43,636
Is it the Hamiltonian operator? It could be an observerble, an Hermitian observable

545
00:44:43,636 --> 00:44:46,390
It's the Hamiltonian, it's the energy

546
00:44:46,691 --> 00:44:51,036
Hamiltonian and energy are the same, this is what the energy of a system is

547
00:44:51,036 --> 00:44:56,935
It's the Hamiltonian which generate the evolution of the state vector 

548
00:44:57,119 --> 00:45:00,353
According to this rule here, ya

549
00:45:03,074 --> 00:45:04,959
It could be any Hermitian operator at the moment

550
00:45:05,162 --> 00:45:09,787
Now what determine what Hermitian operator you put there?

551
00:45:10,921 --> 00:45:13,927
Well it can be, it can be any Hermitian operator 

552
00:45:14,308 --> 00:45:15,794
associate with a system you study

553
00:45:16,028 --> 00:45:21,964
For studying a system with one spin, we want to make it up out of

554
00:45:22,202 --> 00:45:24,005
a operator associate with a spin

555
00:45:25,449 --> 00:45:27,018
So it can be any operator

556
00:45:27,206 --> 00:45:32,375
But it's either experiment, it's the same thing which

557
00:45:32,702 --> 00:45:37,365
govern how you choose Hamiltonian or Lagrangians in classical mechanics 

558
00:45:37,681 --> 00:45:41,757
Either experiment, some prejudice about the way things works

559
00:45:42,047 --> 00:45:48,077
Some symmetry principle, whatever you have that might give you a clue

560
00:45:48,293 --> 00:45:49,590
to what H is 

561
00:45:49,842 --> 00:45:52,586
At the end of the day you may have to resort to experiment 

562
00:45:52,779 --> 00:45:55,686
to find out whatever it is

563
00:45:56,781 --> 00:46:02,751
Okay, so that's right

564
00:46:02,956 --> 00:46:07,038
So now, let's rewrite our equation, let's rewrite, question?

565
00:46:07,251 --> 00:46:14,156
>> So when you do that expandsion of the, normally the H is a constant 

566
00:46:14,156 --> 00:46:18,570
>> You don't do that on first quarter, so this H would be...

567
00:46:18,769 --> 00:46:20,460
H is not a constant

568
00:46:20,690 --> 00:46:23,439
>> I mean it's not a function of time in that case

569
00:46:27,775 --> 00:46:33,555
Okay, the situation with possible explicit time dependence with H

570
00:46:33,879 --> 00:46:35,356
is the same as in classical physics

571
00:46:36,662 --> 00:46:39,582
In classical physics, H can depend on time

572
00:46:39,754 --> 00:46:43,641
For example, if you have a particle moving in a magnetic field 

573
00:46:43,866 --> 00:46:47,306
And the strength of the magnet has been vary with time 

574
00:46:47,487 --> 00:46:49,930
by changing the current to the electric magnet

575
00:46:50,189 --> 00:46:55,344
Then the Hamiltonian for the particle moving in the magnetic field is time dependent

576
00:46:55,671 --> 00:47:01,107
Otherwise if you have the parameter of the problem are not vary with time

577
00:47:01,341 --> 00:47:03,973
Then you say the Hamiltonian is not time dependent

578
00:47:04,240 --> 00:47:06,036
This is exactly the same, exact

579
00:47:06,294 --> 00:47:10,874
>>(inaudible)

580
00:47:11,084 --> 00:47:12,946
Ya, ya, that's right, that's right

581
00:47:13,197 --> 00:47:16,454
In principle, H could be a function of time 

582
00:47:16,682 --> 00:47:21,056
Let's take the case where it's not, what that correspond to 

583
00:47:21,056 --> 00:47:27,508
is the situation with parameters of the problem are not time dependent

584
00:47:27,735 --> 00:47:33,169
Whereas in classical physics we call time translation invariance 

585
00:47:33,370 --> 00:47:37,698
Remember classical physics, so time translation invariance

586
00:47:37,926 --> 00:47:40,218
Which means like every time is the same as every other time

587
00:47:40,456 --> 00:47:43,938
Okay, you do the experiment, the same experiment at a later time 

588
00:47:44,130 --> 00:47:52,571
will have the same output as the identical experiment in a earlier time

589
00:47:52,857 --> 00:48:00,747
The statement of that principle, time translation invariance 

590
00:48:00,939 --> 00:48:04,219
It's that H does not have explicit dependence on time 

591
00:48:04,431 --> 00:48:06,283
Just an operator which doesn't depend on time 

592
00:48:06,504 --> 00:48:10,933
As I said, you can imagine a situation where you ramping up current 

593
00:48:11,190 --> 00:48:16,157
in a electro magnet, or the universe is expanding, or god knows what

594
00:48:16,356 --> 00:48:20,587
There are some explicit time dependent, then H can be time dependent

595
00:48:20,804 --> 00:48:23,571
Now do you know what happen with time energy conservation

596
00:48:23,799 --> 00:48:25,813
if the Hamiltonian is time dependent?

597
00:48:27,570 --> 00:48:31,234
Down the drink, same thing here

598
00:48:31,432 --> 00:48:33,491
If the Hamiltonian is time dependent, 

599
00:48:33,657 --> 00:48:37,258
we haven't yet seen why there might be a conservation of energy, 

600
00:48:37,475 --> 00:48:39,179
we will come to that, but

601
00:48:39,399 --> 00:48:45,321
Alright, so let's come back then, to this equation over here

602
00:48:45,520 --> 00:48:49,090
>> What happens to the lon thing?

603
00:48:49,319 --> 00:48:51,665
>> This is the lon for certain 

604
00:48:53,385 --> 00:48:54,170
Say that again?

605
00:48:54,408 --> 00:48:56,057
>> This is for the infinitely small thing?

606
00:48:56,457 --> 00:49:00,006
Yeah, we will come to that, basically we just multiply them together

607
00:49:00,403 --> 00:49:06,272
We will come to that, okay, or you can think of it another way

608
00:49:06,462 --> 00:49:10,114
If you updated by a small amount of time, 

609
00:49:10,114 --> 00:49:12,000
you just do it again and again and again 

610
00:49:12,174 --> 00:49:19,022
You just do it repeatedly, but we can use this to derive 

611
00:49:19,208 --> 00:49:23,558
a differential equation for the way state change with time, let's do that

612
00:49:23,718 --> 00:49:37,108
Let's do that, okay, let's look at the state of a system at a time epsilon

613
00:49:37,385 --> 00:49:41,230
And this time epsilon could be thought of 

614
00:49:41,464 --> 00:49:45,047
as time epsilon right after some arbitrary time 

615
00:49:45,292 --> 00:49:47,837
which I'm gonna call zero, but it doesn't matter

616
00:49:48,042 --> 00:49:54,416
Psi epsilon is equal to one minus i H epsilon 

617
00:49:59,363 --> 00:50:06,114
minus i H epsilon times Psi, let's say it times zero

618
00:50:06,114 --> 00:50:14,551
Another way to say it, it's that Psi of epsilon minus Psi of zero

619
00:50:14,820 --> 00:50:17,404
This is a small incremental change in the state vector 

620
00:50:17,588 --> 00:50:27,260
That's equal to minus i epsilon operation by H on Psi

621
00:50:29,156 --> 00:50:33,740
Now as I said, this zero here doesn't have to be zero

622
00:50:33,940 --> 00:50:39,676
It could be any time as long as epsilon is an epsilon unit after the starting time

623
00:50:39,861 --> 00:50:42,348
Okay, let's divide this equation by epsilon 

624
00:50:44,388 --> 00:50:48,777
You are allowed to do this because you can multiply vectors by numbers 

625
00:50:48,970 --> 00:50:51,873
So we just multiply both side by one over epsilon

626
00:50:52,059 --> 00:50:56,561
One over epsilon times this whole thing

627
00:50:56,741 --> 00:50:59,594
Okay, so what's this thing on the left hand side?

628
00:50:59,788 --> 00:51:03,386
One over epsilon times the difference of two things

629
00:51:04,602 --> 00:51:09,434
which, by which, at two slightly different times

630
00:51:09,434 --> 00:51:13,906
That's just the time derivative of this vector 

631
00:51:15,236 --> 00:51:18,918
We haven't talked very much about taking derivatives of vectors

632
00:51:19,096 --> 00:51:21,534
But it's clear you could do that 

633
00:51:21,534 --> 00:51:24,169
You can take the difference between two vectors

634
00:51:24,361 --> 00:51:25,592
that's perfectly well defined

635
00:51:27,088 --> 00:51:29,344
They would be close to each other if epsilon is small 

636
00:51:29,529 --> 00:51:31,421
If you divide by the small epsilon, 

637
00:51:31,598 --> 00:51:33,803
you would get something which is on the right hand side

638
00:51:33,998 --> 00:51:40,658
It's just the time derivative of Psi

639
00:51:42,444 --> 00:51:50,632
The time derivative of Psi equals minus i H Psi

640
00:51:50,819 --> 00:51:54,455
At any given time, it doesn't, as I said, it doesn't have to be at time zero

641
00:51:54,659 --> 00:52:01,107
At any given time, if you want to find out how the state of the system 

642
00:52:01,286 --> 00:52:08,179
changes over a incremental time interval, this is your equation

643
00:52:09,116 --> 00:52:12,971
Completely analogous to the way you update classical systems

644
00:52:13,144 --> 00:52:20,396
But because knowing the state vector is not the same as

645
00:52:20,564 --> 00:52:26,709
knowing the values of the experimental outputs of the experiment 

646
00:52:26,877 --> 00:52:31,244
You can know the states but still there could be ambiguity or uncertainty

647
00:52:31,244 --> 00:52:32,684
on the value of the experiment 

648
00:52:32,684 --> 00:52:36,244
This tells you how states change but it doesn't tell you 

649
00:52:36,429 --> 00:52:38,165
how the result of experiment change

650
00:52:38,349 --> 00:52:41,734
The experiment were also experiment as still statistical 

651
00:52:41,905 --> 00:52:44,486
but with a continuously updating state vector 

652
00:52:44,674 --> 00:52:48,346
Okay this is, this equation have a name

653
00:52:50,163 --> 00:52:56,048
Name of this equation is the generalized time dependent Schrodinger equation

654
00:52:56,234 --> 00:53:00,197
This is the general form, this is the Schrodinger equation 

655
00:53:00,474 --> 00:53:03,499
Specific version which Schrodinger wrote down 

656
00:53:03,499 --> 00:53:05,486
was a very specific version of this

657
00:53:05,671 --> 00:53:08,706
We are gonna come to that later, and of course 

658
00:53:08,881 --> 00:53:14,626
But this is the general idea, the rate of...

659
00:53:14,811 --> 00:53:17,818
I don't need to say it again, this is what it is

660
00:53:18,006 --> 00:53:20,861
But of course in order to use it, you have to know what H is

661
00:53:21,794 --> 00:53:27,633
>>Question? So all of this depends on the irreversibility of the system

662
00:53:27,836 --> 00:53:28,641
>>Is that right?

663
00:53:28,823 --> 00:53:33,697
Yeah, or the conservation of the inner product, ya

664
00:53:33,880 --> 00:53:39,628
>> How is that rationalized when you take the up vector with the down vector

665
00:53:39,810 --> 00:53:42,817
>> you measure them along the horizontal axis 

666
00:53:42,988 --> 00:53:47,820
>> this sort of end up ...(inaudible)

667
00:53:47,986 --> 00:53:49,704
Alright, that's a good question

668
00:53:53,056 --> 00:53:57,591
You have to remember when you do that, you intervening with an apparatus

669
00:54:01,415 --> 00:54:04,839
In order to follow the system, under those circumstances, 

670
00:54:05,040 --> 00:54:08,095
you have to include the apparatus as part of the system

671
00:54:09,375 --> 00:54:15,257
You can not interact to a system and not include the things 

672
00:54:15,443 --> 00:54:17,569
which are interacting with it in the system

673
00:54:17,757 --> 00:54:20,385
Classically, it's not such a bad sin

674
00:54:20,563 --> 00:54:22,537
But quantum mechanically of something 

675
00:54:22,719 --> 00:54:25,682
interacting with the system strongly enough to affect it, 

676
00:54:25,872 --> 00:54:28,284
or strong enough to measure it

677
00:54:28,469 --> 00:54:31,475
You have to include it as part of the system

678
00:54:31,660 --> 00:54:33,908
Now this is the way a system evolve 

679
00:54:34,096 --> 00:54:38,635
If it's isolated and completely not in contact with anything else 

680
00:54:39,739 --> 00:54:43,808
We will have to come back to ask how it evolves 

681
00:54:43,808 --> 00:54:48,717
What happens when you put it in contact with a measuring apparatus 

682
00:54:48,910 --> 00:54:52,468
We will have to have a model for how measurement take place

683
00:54:52,650 --> 00:54:56,471
But this is the way the system behave if nobody disturb it 

684
00:54:56,652 --> 00:55:00,545
during the course of the evolution from time zero to time t

685
00:55:01,678 --> 00:55:04,103
We will have to come back to that 

686
00:55:04,279 --> 00:55:05,664
It's an important question

687
00:55:10,656 --> 00:55:15,792
Oh boy, yeah, really just two assumptions

688
00:55:15,965 --> 00:55:17,233
Well, maybe three, 

689
00:55:17,423 --> 00:55:22,289
First is the time evolution takes place by a linear operator call U

690
00:55:22,475 --> 00:55:26,721
And the operator is independent of the state that it's acting on

691
00:55:26,901 --> 00:55:29,017
Okay? With me?

692
00:55:31,321 --> 00:55:33,163
That's the first statement, 

693
00:55:33,340 --> 00:55:36,550
and the next statement is that inner products are conserve with time

694
00:55:36,737 --> 00:55:39,312
And that tells you U is unitary

695
00:55:40,670 --> 00:55:48,538
From that ,we went over to here, and discover that

696
00:55:48,723 --> 00:55:54,408
for a small little incremental change is governed by a Hermitian operator

697
00:55:57,753 --> 00:55:59,892
So we get to the idea of the Hamiltonian 

698
00:56:02,507 --> 00:56:05,986
And then we just rewrote that, and say that, look

699
00:56:06,186 --> 00:56:09,362
The small little incremental change could be thought of as 

700
00:56:09,542 --> 00:56:12,172
one over epsilon times the time derivative of the state vector 

701
00:56:13,362 --> 00:56:17,426
Okay so, this is the Schrodinger equation

702
00:56:23,835 --> 00:56:25,562
How do you solve the problem 

703
00:56:25,743 --> 00:56:28,346
if it's a long time interval not a short time interval?

704
00:56:28,531 --> 00:56:30,627
You basically solve this equation 

705
00:56:31,683 --> 00:56:33,562
I will talk about how you solve this equation, we will do so

706
00:56:34,690 --> 00:56:44,092
There are lots of example, but okay

707
00:56:44,288 --> 00:56:50,796
Alright, to go on, and explain what this has to do with classical mechanics

708
00:56:50,982 --> 00:56:55,532
What it has to do with the corresponding concept in classical mechanics

709
00:56:55,709 --> 00:56:58,798
We need a couple of ideas of, 

710
00:56:58,798 --> 00:57:04,368
by how you relate classical ideas to quantum mechanical ideas

711
00:57:04,368 --> 00:57:09,933
We need the idea of what it's called the expectation value of an observable

712
00:57:11,239 --> 00:57:18,647
It's a bad name, it's a bad name because the expectation value was defined

713
00:57:19,873 --> 00:57:23,418
may have very little to do with what you expect the experiment to give 

714
00:57:25,299 --> 00:57:27,732
This doesn't have much to do with quantum mechanics, 

715
00:57:27,935 --> 00:57:30,253
this just have to do with the probability theory

716
00:57:30,438 --> 00:57:31,749
I think I've talked about it before 

717
00:57:31,938 --> 00:57:35,653
But I will just remind you, if you have a probability distribution which looks like this

718
00:57:38,793 --> 00:57:42,366
Then the least likely answer that you can possibly get is right over here

719
00:57:42,540 --> 00:57:48,291
It's also the expectation value, so it's a poorly chosen terminology

720
00:57:48,477 --> 00:57:55,721
The right terminology should have sometime, some fancy people

721
00:57:55,902 --> 00:58:03,058
I won't name, Mary Youman, but some fancy people 

722
00:58:03,221 --> 00:58:09,273
like to call the expected value was even less the expected value then it is the expectation value

723
00:58:09,467 --> 00:58:13,266
It's the average value, this is average value, and 

724
00:58:13,503 --> 00:58:18,227
the average value is something could be a value the thing can't even pick on

725
00:58:18,450 --> 00:58:20,290
The possible value it could not even have 

726
00:58:20,478 --> 00:58:25,784
The average value for example, if you assign head to plus one

727
00:58:25,968 --> 00:58:29,239
and tails to minus one, and you flip the coin, what's the average? 

728
00:58:29,432 --> 00:58:31,481
Zero, but you can't get zero, 

729
00:58:31,660 --> 00:58:34,497
so it's got nothing to do with the expected value of an experiment

730
00:58:35,697 --> 00:58:38,729
But nevertheless, it's called the expectation value 

731
00:58:38,910 --> 00:58:43,157
And we will call the expectation value, what's the average value?

732
00:58:43,346 --> 00:58:51,555
Let's talk about the average value of a observable giving a particular state

733
00:58:52,923 --> 00:58:59,684
So our state is A, I said I was gonna change the sign but 

734
00:58:59,887 --> 00:59:01,243
sometimes I will sometimes I won't

735
00:59:02,262 --> 00:59:04,219
>> expectation value 

736
00:59:04,434 --> 00:59:10,192
>> I think I just have to clear my head of statistic

737
00:59:10,410 --> 00:59:14,897
>> but with that picture you draw, I assume it's a frequency distribution

738
00:59:15,076 --> 00:59:18,217
>> So to me, the middle is not the average 

739
00:59:18,438 --> 00:59:22,547
>> but really is the expectation value is the value occur most frequently

740
00:59:22,773 --> 00:59:24,752
No, no, definitely not, this one here?

741
00:59:24,918 --> 00:59:28,728
It never occurs, never, ever ,ever

742
00:59:28,900 --> 00:59:30,960
Let's even make it worse 

743
00:59:31,143 --> 00:59:34,786
>>Okay, what do the x and y axis represent?

744
00:59:34,988 --> 00:59:36,122
>>I keep seeing

745
00:59:36,302 --> 00:59:38,832
P is the probability and x is some variable

746
00:59:38,832 --> 00:59:44,757
Something that you measure, this could be a probability of finding a particle

747
00:59:44,942 --> 00:59:46,545
with different location, okay?

748
00:59:46,730 --> 00:59:52,495
Alright, it could be, after some particular state

749
00:59:52,684 --> 00:59:57,032
The probability, let's make it symmetric about the origin

750
01:00:00,638 --> 01:00:04,072
It could be the probability distribution looks like this

751
01:00:04,072 --> 01:00:08,652
I meant to make this symmetric

752
01:00:08,652 --> 01:00:12,941
And it's really very very zero in the neighborhood of the origin here

753
01:00:16,832 --> 01:00:22,081
There's no sense in which x equal zero is 'expected'

754
01:00:22,267 --> 01:00:25,364
You can do the experiment for ever and ever and ever again 

755
01:00:25,556 --> 01:00:27,082
and you will not get x equal zero

756
01:00:27,082 --> 01:00:30,531
You will either get an x somewhere around here, or here

757
01:00:30,718 --> 01:00:35,202
So to call it the expected or the expectation value

758
01:00:35,374 --> 01:00:36,791
It's a sort or misnomer 

759
01:00:37,938 --> 01:00:42,539
It is the average value, defining average in a very particular way

760
01:00:42,721 --> 01:00:48,720
Now, for many many smooth probability distributions 

761
01:00:48,902 --> 01:00:52,830
And particular probability distribution don't have big holes in them

762
01:00:53,011 --> 01:00:54,526
Something like this I would like to say

763
01:00:54,703 --> 01:00:57,529
It can very often be the case and very often is the case 

764
01:00:57,701 --> 01:01:05,300
That the probability, that the average value is the most likely value to get

765
01:01:05,482 --> 01:01:10,868
>> Is this the (inaudible), what is the peak?

766
01:01:11,053 --> 01:01:13,652
The peak there is the most likely value 

767
01:01:13,652 --> 01:01:17,008
Yes, by definition the top of the peak is

768
01:01:17,181 --> 01:01:21,956
And in many many cases, if you have nice simple probability

769
01:01:22,140 --> 01:01:25,068
Gaussian probability, you know , bell shaped curve and so forth

770
01:01:25,248 --> 01:01:29,851
Then the expectation value or the average value 

771
01:01:30,045 --> 01:01:35,909
is usually quite close to the average and peak value

772
01:01:36,093 --> 01:01:38,302
because they are usually close to each other

773
01:01:38,302 --> 01:01:41,902
But certainly there could be violent exceptions

774
01:01:41,902 --> 01:01:46,628
The coin or the spin which can only be plus one or minus one and never zero

775
01:01:47,918 --> 01:01:51,302
The expectation value and the average value are quite different

776
01:01:51,302 --> 01:01:56,833
Okay, that's just a little sermon to give me a chance to beat my friend Mary Youman

777
01:01:57,023 --> 01:02:06,933
>> Question, if you think the spin were the apparatus perpendicular

778
01:02:07,110 --> 01:02:10,583
>> then in 70 percent if you are very unlucky, you get a million up-s in a roll

779
01:02:10,763 --> 01:02:15,027
>> the average would be up, but that's not accurate

780
01:02:15,202 --> 01:02:21,260
Ya, well, it depend on what you mean by an average

781
01:02:21,438 --> 01:02:23,573
There's a notion of average which is strictly mathematical

782
01:02:23,573 --> 01:02:24,875
I'm gonna write it down in a minute

783
01:02:26,844 --> 01:02:36,111
There is a unwritten law which I think, is safe to say nobody really understands

784
01:02:37,963 --> 01:02:39,755
It's the law of the large numbers 

785
01:02:41,323 --> 01:02:46,908
That if you do a thing enough time, how many times? I don't know

786
01:02:47,086 --> 01:02:50,116
If you do a thing enough time, you do an experiment enough time

787
01:02:50,305 --> 01:02:56,436
The average gotten by averaging the result of your experiment

788
01:02:56,635 --> 01:03:02,021
will be equal to the average defined mathematically to within what?

789
01:03:03,205 --> 01:03:04,693
To within the margin of error

790
01:03:04,869 --> 01:03:08,997
What is the margin of error? It's a specific number 

791
01:03:09,182 --> 01:03:12,240
Does it really mean every time you do it, the answer would come out 

792
01:03:12,430 --> 01:03:14,835
within the margin of error? No

793
01:03:15,009 --> 01:03:20,159
Alright, so, we are assuming standard probability theory 

794
01:03:20,334 --> 01:03:22,138
That if you do a thing enough

795
01:03:22,329 --> 01:03:24,503
Plus this extra little ingredient

796
01:03:24,721 --> 01:03:27,159
This extra physical assumption

797
01:03:27,361 --> 01:03:29,003
I don't even know what to call it 

798
01:03:29,003 --> 01:03:31,600
That if you repeat an experiment enough times

799
01:03:31,789 --> 01:03:36,504
The average of your data would be the mathematical average

800
01:03:36,709 --> 01:03:38,802
Can anybody prove that? No

801
01:03:38,978 --> 01:03:43,687
That's a law of large numbers 

802
01:03:43,871 --> 01:03:48,553
No, no, no, there's nothing you could do to prove it

803
01:03:48,743 --> 01:03:54,144
The reason you can't prove it, is because sooner or latter some where in the multiverse

804
01:03:54,324 --> 01:03:56,626
Somewhere there's gonna be an exception to it

805
01:03:56,797 --> 01:04:00,449
No matter how many time you do it, a zillion time, a zillion numbers

806
01:04:01,977 --> 01:04:07,145
It's not infinity, a zillion is one with the logarithms with a zillion of zero after

807
01:04:07,334 --> 01:04:08,507
Or something like that

808
01:04:11,161 --> 01:04:16,962
If you do an experiment, and opt for repetition of it

809
01:04:17,146 --> 01:04:20,402
Eventually somewhere some place there is gonna be a violation of it

810
01:04:20,575 --> 01:04:24,949
It is closely connected to the second law of thermo dynamics

811
01:04:25,131 --> 01:04:29,350
You know there was a principle, a second law of thermo dynamics says 

812
01:04:29,528 --> 01:04:32,411
That the entropy never decreases

813
01:04:33,596 --> 01:04:38,021
Boltzmann was never able to prove that and finally realize the right law is

814
01:04:38,203 --> 01:04:41,949
the entropy probably never decreases, except when it does

815
01:04:43,365 --> 01:04:45,839
Which is rarely, except when it does

816
01:04:46,018 --> 01:04:49,870
So I'm not going to try to give you an explanation of 

817
01:04:50,047 --> 01:04:51,806
why probability theory works

818
01:04:52,003 --> 01:04:56,318
This is to me very puzzling, but we will assume probability theory works

819
01:04:56,498 --> 01:05:00,334
The statement that it works is the same as the statement that 

820
01:05:00,526 --> 01:05:04,113
the average of your data will be, for big enough experiment, 

821
01:05:04,113 --> 01:05:06,353
would be the mathematical average

822
01:05:06,537 --> 01:05:09,603
Alright, let's come over to this side here

823
01:05:09,603 --> 01:05:11,927
First just say, supposing we have a collection of a, 

824
01:05:13,259 --> 01:05:16,163
we have a probability distribution for some variables

825
01:05:16,338 --> 01:05:18,938
Let's say for lambda, lambda, one of the eigenvalue

826
01:05:19,124 --> 01:05:20,378
By measuring the eigenvalue of L

827
01:05:20,554 --> 01:05:24,050
Supposing we have a probability for lambda, here it is

828
01:05:24,216 --> 01:05:31,410
The other thing I call prob of lambda, Probability for lambda

829
01:05:31,586 --> 01:05:37,071
Lambda can take on a whole bunch of different values

830
01:05:37,267 --> 01:05:39,326
Lambda sub ii, let's call them 

831
01:05:39,503 --> 01:05:42,375
So this becomes the probability for lambda sub i

832
01:05:42,552 --> 01:05:48,366
Then the average, the definition for the average is the sum over i

833
01:05:49,598 --> 01:05:54,799
Of lambda i times the probability for lambda i

834
01:05:55,851 --> 01:05:57,543
That's the definition of the average 

835
01:05:57,543 --> 01:06:05,623
And we will represent the average by the symbol, the bracket symbol

836
01:06:05,829 --> 01:06:09,874
Why it's labeled by a bracket symbol, we will come to it

837
01:06:10,069 --> 01:06:15,259
Okay, but that's the definition, the standard probability theory definition

838
01:06:15,448 --> 01:06:17,138
of the average of a quantity

839
01:06:17,325 --> 01:06:22,305
The quantity times the probability the quantity takes on that particular value

840
01:06:22,484 --> 01:06:26,738
Sum over i, as I said this is the standard definition of average

841
01:06:26,944 --> 01:06:35,486
Okay, what is the average in a state A of the observable lambda

842
01:06:35,661 --> 01:06:44,101
So let's say A is expanded in a bases, which bases?

843
01:06:44,288 --> 01:06:50,550
The bases of eigenvectors of L, alright?

844
01:06:50,723 --> 01:06:52,605
Remember L is the Hermitian operator

845
01:06:52,770 --> 01:06:55,621
It's eigenvectors form a bases

846
01:06:55,802 --> 01:06:59,412
It could expand vectors in ortho-normal bases

847
01:06:59,591 --> 01:07:05,228
So let's write this in the form alpha sub i summed over i

848
01:07:05,412 --> 01:07:10,788
lambda sub i, remember the lambdas are orthonormal bases

849
01:07:12,269 --> 01:07:18,287
What is the probability lambda in the state i?

850
01:07:20,724 --> 01:07:28,372
Probability for lambda i is equal to alpha star i alpha i

851
01:07:29,652 --> 01:07:37,711
Alright ? I maintain, this is the probability and the average of lambda

852
01:07:37,894 --> 01:07:41,444
Let's call the average of lambda is summed over i

853
01:07:41,624 --> 01:07:46,829
Lambda sub i alpha i star alpha i

854
01:07:49,585 --> 01:07:50,972
All I've done is plugged in

855
01:07:51,148 --> 01:07:58,863
Let me prove that this is the same thing as taking the vector A 

856
01:07:59,946 --> 01:08:05,972
bra vector A and sandwiching between the bra vector A and the ket vector A

857
01:08:06,154 --> 01:08:11,710
The operator L, the operator L is the operator whose eigenvalues are lambda

858
01:08:12,772 --> 01:08:16,721
Let's see if we can prove that, it's actually quite easy

859
01:08:16,912 --> 01:08:20,410
All we do is, let's just plug in 

860
01:08:20,592 --> 01:08:23,090
Let's come to another blackboard and plug in

861
01:08:23,270 --> 01:08:26,978
And let's see if we can prove that

862
01:08:30,898 --> 01:08:36,391
Alright, so let's calculate this creature over here

863
01:08:36,570 --> 01:08:43,598
A is the summation, and let's use also the bra vector A is 

864
01:08:43,761 --> 01:08:49,738
the summation over i of lambda sub i alpha star i

865
01:08:50,923 --> 01:08:52,355
Alright so make a sandwich

866
01:08:53,427 --> 01:08:57,475
The sandwich is the sum over i and j, why there are two sums?

867
01:08:57,732 --> 01:09:01,234
One for the bra and one for the ket

868
01:09:01,459 --> 01:09:12,066
Lambda sub j, alpha star j, this is the bra vector A

869
01:09:12,263 --> 01:09:19,809
And then we put L there, and then we put the ket vector alpha i lambda i

870
01:09:22,161 --> 01:09:23,994
It's pure plug in-ology

871
01:09:24,267 --> 01:09:29,729
Now, what is L when it act on lambda sub i?

872
01:09:33,409 --> 01:09:36,966
Lambda sub i are the eigenvectors of L

873
01:09:37,147 --> 01:09:40,221
So when L act on lambda sub i

874
01:09:40,406 --> 01:09:47,951
This left sum ij, lambda j, alpha star j alpha i

875
01:09:48,149 --> 01:09:55,673
And now L has lambda i and it gives us lambda sub i times lambda i

876
01:09:55,848 --> 01:10:00,835
Now all this junk in here is just number, for each i and j

877
01:10:01,034 --> 01:10:03,044
For each i and j this is just the number 

878
01:10:03,247 --> 01:10:07,979
So that means we approach the point to take the inner product 

879
01:10:07,979 --> 01:10:10,756
of lambda i with lambda j, what is that?

880
01:10:12,050 --> 01:10:17,608
Right ,it's one or zero, depending on whether i equals j or not

881
01:10:17,786 --> 01:10:24,473
Given that, this sum, this double sum collapses to a single sum

882
01:10:24,644 --> 01:10:27,659
In which lambda i and lambda j are identified

883
01:10:27,659 --> 01:10:34,553
And it just become sum of i, lambda i, well, okay

884
01:10:35,848 --> 01:10:40,235
Let's cut through some of the steps

885
01:10:41,394 --> 01:10:45,622
If j is not equal to i, you got nothing

886
01:10:45,800 --> 01:10:49,556
If j is equal to i, the inner product is just one

887
01:10:49,733 --> 01:10:57,592
So this just becomes the sum over i, alpha star i alpha i lambda i

888
01:10:57,769 --> 01:11:03,225
Which is exactly what we have over here

889
01:11:04,897 --> 01:11:07,353
So we've prove the relationship here

890
01:11:07,529 --> 01:11:14,153
That the average of any observable is just the sandwich bra-ket

891
01:11:14,332 --> 01:11:25,470
A bra-ket, a bra-ket may sandwiching L, the observable between the bracket 

892
01:11:28,409 --> 01:11:33,369
The same bra and ket, the bra and ket associated with the state of the system

893
01:11:33,369 --> 01:11:39,949
That's where this notation of a bracket comes from

894
01:11:39,949 --> 01:11:45,036
Okay that's a good thing to know, if you want the average of a quantity

895
01:11:45,227 --> 01:11:50,943
Just sandwich it between the bracket, the bra-ket representing the state

896
01:11:52,263 --> 01:11:58,765
Any questions about that?

897
01:12:01,454 --> 01:12:03,126
Okay, I'm gonna tell you what I'm gonna do

898
01:12:03,310 --> 01:12:07,141
What we are going to do is we are going to try to find the equations 

899
01:12:08,434 --> 01:12:12,785
governing the time evolution of this average values

900
01:12:13,947 --> 01:12:18,868
The idea is under suitable reasonable circumstances 

901
01:12:19,040 --> 01:12:22,404
If the probability distribution are nicely shape 

902
01:12:22,588 --> 01:12:25,050
That they have a bell shape curve 

903
01:12:25,263 --> 01:12:32,260
that by calculating the time evolution of averages 

904
01:12:32,260 --> 01:12:36,804
You are doing something pretty close to what classical physics

905
01:12:36,981 --> 01:12:41,514
would instruct you to do, about calculating the equations of motion

906
01:12:41,684 --> 01:12:47,340
of classical systems, the way average tend to evolve with time 

907
01:12:47,340 --> 01:12:49,658
as I said, under suitable circumstances, 

908
01:12:49,848 --> 01:12:53,397
follows equations which will 

909
01:12:53,592 --> 01:12:56,646
very very much like the corresponding classical equations

910
01:12:57,998 --> 01:13:04,811
So our next goal is to try to find the rules 

911
01:13:06,036 --> 01:13:11,426
for the time evolution of these average values, expectation values

912
01:13:11,609 --> 01:13:14,971
That's not so hard, we have all the equipment we need to do that

913
01:13:16,094 --> 01:13:17,751
And let's see if we can do it

914
01:13:23,545 --> 01:13:28,489
Fine, so at any given time, at any given time, the average of lambda

915
01:13:28,670 --> 01:13:33,745
is given by Psi of t, the state vector at the time t

916
01:13:35,397 --> 01:13:40,714
The operator L representing the observables times Psi of t

917
01:13:41,801 --> 01:13:48,666
Another word, I just plug in the A, the actual time dependent state of the system

918
01:13:48,847 --> 01:13:52,866
This is the average of L as function of time 

919
01:13:54,279 --> 01:13:55,290
I don't know, what should we call it 

920
01:13:55,465 --> 01:14:03,114
Let's call it, the average of L as a function of time

921
01:14:03,302 --> 01:14:07,933
Again this is kind of a wrong notation, it's not the average of L of t

922
01:14:08,098 --> 01:14:11,725
Is the average of L as function of time 

923
01:14:14,535 --> 01:14:19,262
We could, we could put the t outside the bracket 

924
01:14:22,508 --> 01:14:26,711
That's none standard, sub t

925
01:14:26,890 --> 01:14:29,926
But why don't we just go with standard notations, standard notations

926
01:14:30,102 --> 01:14:36,504
And remember, okay, I'm not gonna do it

927
01:14:36,708 --> 01:14:41,217
I will just call it average L as a function of time, okay

928
01:14:41,399 --> 01:14:42,377
As a function of time

929
01:14:42,552 --> 01:14:45,970
It is a function of time, that's the main thing, function of time

930
01:14:46,142 --> 01:14:48,899
We do something else, we could call that L bar

931
01:14:49,079 --> 01:14:52,358
Another notation for average is just to put a bar on top of something

932
01:14:52,544 --> 01:14:55,617
That's another notation for average, standard notation 

933
01:14:55,812 --> 01:14:57,611
Or we can just call it L bar of t

934
01:14:57,791 --> 01:15:00,897
It means the average as a function of time 

935
01:15:04,586 --> 01:15:04,767
Let's do it that way, what would we like to do with this

936
00:00:00,000 --> 01:15:07,026
We would like to find the equations of motions for it

937
01:15:07,217 --> 01:15:09,858
Another words we would like to find how it changes with time 

938
01:15:10,029 --> 01:15:12,910
by differentiating with respect to time 

939
01:15:12,910 --> 01:15:16,673
I want to take this quantity and differentiate it with respect to time

940
01:15:16,867 --> 01:15:23,449
So let's call it L dot, now we have bars with dots on top of them 

941
01:15:23,634 --> 01:15:27,690
L dar of t is the time derivative of this 

942
01:15:27,690 --> 01:15:31,073
Now L of t, that's just some fix operator 

943
01:15:31,241 --> 01:15:36,714
It's a fix definite operator, All the time dependence is in the state vector 

944
01:15:36,890 --> 01:15:42,403
The state vectors change with time, the operators, those are fix

945
01:15:42,576 --> 01:15:44,090
Those are definite operators, 

946
01:15:44,090 --> 01:15:49,293
so let's see if we can take the time derivative of this here

947
01:15:50,341 --> 01:15:54,281
The time derivative act on the side of t, 

948
01:15:54,457 --> 01:15:57,348
when you take the time derivative of a product

949
01:15:57,533 --> 01:16:00,568
and this can be a product of vectors, it can be a product of numbers

950
01:16:00,746 --> 01:16:02,698
It can be a product of functions, well functions

951
01:16:04,195 --> 01:16:09,207
The rules are always the same, it's the time derivative of one times the other

952
01:16:09,387 --> 01:16:11,967
Plus the time derivative of the other time the one

953
01:16:13,375 --> 01:16:20,977
That's very general, so this is going to equal d Psi by d t, bra vector 

954
01:16:21,190 --> 01:16:34,409
L Psi plus Psi L d Psi d t 

955
01:16:34,593 --> 01:16:45,346
Oops, I've raise the very important equation, or else I bury it under the blackboard

956
01:16:45,538 --> 01:16:55,849
And I write it over here, d Psi d t, is equal to minus i H Psi

957
01:16:57,103 --> 01:16:59,713
That's what I was gonna stick in right over here

958
01:16:59,891 --> 01:17:04,302
So we begin to see that the Hamiltonian is gonna tell us something 

959
01:17:04,494 --> 01:17:06,814
about how average values change with time

960
01:17:08,125 --> 01:17:10,714
The Hamiltonians of classical physics tells us 

961
01:17:10,919 --> 01:17:13,034
how the correspondent quantities change with time

962
01:17:13,226 --> 01:17:16,008
The Hamiltonian of quantum mechanics will tell us 

963
01:17:16,182 --> 01:17:17,848
how the average change with time 

964
01:17:18,030 --> 01:17:21,977
We are going to also need the bra version of this 

965
01:17:22,170 --> 01:17:24,368
So let's write the bra version of this 

966
01:17:24,558 --> 01:17:31,637
The bra version of it is d by d t of the bra vector Psi of t

967
01:17:32,948 --> 01:17:42,709
is equal to plus i, plus i Psi H

968
01:17:42,896 --> 01:17:48,583
H is Hermitian, so we don't have to Hermitian conjugate it

969
01:17:48,774 --> 01:17:54,754
But we do have to change the sign of i, when we goes from bras to kets

970
01:17:56,050 --> 01:17:58,243
And so now we are just gonna plug in

971
01:17:58,406 --> 01:18:01,266
We are just gonna plug in and we will get L dot 

972
01:18:07,697 --> 01:18:11,050
L dot, okay, let's first do this one over here

973
01:18:11,227 --> 01:18:15,754
This one is gonna be plus Psi, there's a minus sign 

974
01:18:15,942 --> 01:18:29,699
minus sign here, minus i Psi, L H, Psi

975
01:18:32,091 --> 01:18:34,603
That's what we get from here, what about the other one

976
01:18:34,779 --> 01:18:45,995
The other one is plus i Psi H L Psi

977
01:18:51,908 --> 01:19:01,332
Or i times the expectation value, we have the Psi here 

978
01:19:01,494 --> 01:19:13,277
we have Psi here Psi here i times HL minus LH Psi

979
01:19:21,335 --> 01:19:25,206
How about HL minus LH? Is that zero?

980
01:19:26,422 --> 01:19:30,381
Why not? HL? Who cares about which way we order them?

981
01:19:30,381 --> 01:19:35,518
Operators don't necessarily commute 

982
01:19:35,706 --> 01:19:41,477
Commute means that you can freely interchange the order of them 

983
01:19:41,667 --> 01:19:44,313
In general, this is true for matrices, 

984
01:19:44,476 --> 01:19:47,834
products of matrices don't necessarily commute

985
01:19:48,000 --> 01:19:53,387
Sometime they do, and in particular an operator commutes with itself 

986
01:19:54,667 --> 01:19:59,204
L times L minus L time L that's okay, you can do it there

987
01:20:00,231 --> 01:20:03,084
L times L minus L times L is zero, okay?

988
01:20:03,276 --> 01:20:07,068
So is H times H minus H times H, but in general, 

989
01:20:09,324 --> 01:20:12,196
L times H is not equal to H times L

990
01:20:12,380 --> 01:20:14,460
Maybe, but typically not

991
01:20:17,004 --> 01:20:20,837
What is this animal called? H times L minus L times H?

992
01:20:22,325 --> 01:20:29,071
It's call a commutator, so this is the definition, given two operators

993
01:20:29,399 --> 01:20:37,381
Well, they could be just H now, the combination HL minus LH 

994
01:20:37,551 --> 01:20:47,936
is written with a square bracket, H comma L, and it's call a commutator

995
01:20:52,754 --> 01:20:57,671
Commuting has something to do with passing freely between each other

996
01:20:57,671 --> 01:21:00,584
Exactly what has to do with taking the subway of New York, I don't know

997
01:21:02,115 --> 01:21:09,016
Let's call the commutator, and so what we have here 

998
01:21:09,197 --> 01:21:18,048
is the equation that the average of L at any given time

999
01:21:19,265 --> 01:21:24,866
Sorry, the average of time derivative of the average

1000
01:21:25,043 --> 01:21:36,031
is equal to Psi, commutator HL, Psi

1001
01:21:36,031 --> 01:21:42,838
But this object over here is just the expectation value of commutator of H with L

1002
01:21:43,015 --> 01:21:47,282
So we can think of this as an equation relating 

1003
01:21:47,460 --> 01:21:51,742
the time derivative of certain average to the average of another quantity

1004
01:21:53,012 --> 01:21:56,733
We could, I did, it's plus i, plus i

1005
01:22:05,749 --> 01:22:12,702
Or we can write L dot of t is equal to i, average, 

1006
01:22:12,703 --> 01:22:18,019
times the average of the commutator HL

1007
01:22:18,237 --> 01:22:20,466
and I will indicate that by a bar on top of it

1008
01:22:20,639 --> 01:22:23,968
So we have a equation of motion now 

1009
01:22:24,148 --> 01:22:26,826
The average in terms of averages, 

1010
01:22:29,170 --> 01:22:33,921
if the probability distribution for everything are nice and peaked

1011
01:22:34,101 --> 01:22:38,669
Then we really can just say approximate nice and peaked 

1012
01:22:38,859 --> 01:22:44,131
so the average is close to the highest point on the curve 

1013
01:22:44,131 --> 01:22:50,201
And the curve is reasonably narrow, so the reason of amount of certainty

1014
01:22:50,201 --> 01:22:56,871
Then this basically says the time derivative of the classical

1015
01:22:57,057 --> 01:23:00,342
Or the time derivative of the approximate L of t

1016
01:23:00,523 --> 01:23:05,616
is equal to the approximate average of the commutator of H with L

1017
01:23:05,800 --> 01:23:09,681
So one writes, this of course is not quite right

1018
01:23:09,880 --> 01:23:16,200
but I will write it anyway, L dot is equal to i times commutator H L

1019
01:23:16,381 --> 01:23:18,543
Now this has to be taken with a little rain of salt

1020
01:23:20,151 --> 01:23:23,583
What it means is a equation among the averages

1021
01:23:23,744 --> 01:23:25,962
It means a equation among the averages

1022
01:23:26,136 --> 01:23:28,047
But it's often just written in this form

1023
01:23:28,214 --> 01:23:30,471
L dot is equal to H time L

1024
01:23:34,583 --> 01:23:40,643
Okay, now before we finish I want to remind you 

1025
01:23:40,823 --> 01:23:44,931
of a classical mechanics equation of motion

1026
01:23:44,931 --> 01:23:50,212
It's the equation of motion represented in terms of Poisson brackets

1027
01:23:50,401 --> 01:23:52,851
Do you remember Poisson brackets? Good old Poisson brackets?

1028
01:23:52,851 --> 01:23:59,612
What was the Poisson equation of motion

1029
01:24:01,596 --> 01:24:24,762
Do you remember? And if you write d by dt of a classical function

1030
01:24:24,996 --> 01:24:27,802
And this could be a classical function of q-s and p-s

1031
01:24:28,015 --> 01:24:30,879
In classical mechanics there are q-s and p-s

1032
01:24:33,782 --> 01:24:36,883
L incidentally is the standard notation for angular momentum

1033
01:24:37,050 --> 01:24:41,249
This could be the angular momentum, but right now I'm just using it

1034
01:24:41,430 --> 01:24:42,993
to represent any observables 

1035
01:24:44,592 --> 01:24:50,465
d by dt of L is equal to certain Poisson bracket

1036
01:24:50,647 --> 01:24:59,694
and the Poisson bracket is a Poisson bracket of L with the Hamiltonian 

1037
01:25:01,822 --> 01:25:04,911
Remember that? Probably, some of you don't 

1038
01:25:05,087 --> 01:25:16,122
As far as you could go, it's in the, this could be the electrical classical mechanics 

1039
01:25:16,122 --> 01:25:24,095
the Poisson brackets, I will not introduce it here, I just remind you of it

1040
01:25:24,272 --> 01:25:28,160
And I want you to notice the similarity, where is it?

1041
01:25:28,333 --> 01:25:35,018
I think I erased it, the similarity of this with dL by dt

1042
01:25:36,449 --> 01:25:50,777
Average value i, minus i or plus i, Psi i times the commutator of H with L

1043
01:25:50,963 --> 01:25:54,087
Now incidentally, okay, let's just point out one thing 

1044
01:25:54,269 --> 01:26:02,152
Poisson brackets have the property that if you interchange the two expressions

1045
01:26:02,337 --> 01:26:04,472
What happens to the Poisson brackets?

1046
01:26:05,712 --> 01:26:09,304
It just change sign, so do commutators 

1047
01:26:09,487 --> 01:26:14,286
H L minus L H, if you interchange the order of them change sign

1048
01:26:14,467 --> 01:26:21,144
So we can also write this as minus L with H

1049
01:26:21,325 --> 01:26:26,159
Commutator of L with H, as a remarkable similarities

1050
01:26:26,340 --> 01:26:37,362
and it suggest that we identify the Poisson bracket with minus sign 

1051
01:26:37,544 --> 01:26:43,332
times a commutator, now I've left out a discussion or something

1052
01:26:43,510 --> 01:26:48,801
I just realized it, just now, I completely forgot what happen to H bar?

1053
01:26:49,841 --> 01:26:51,937
What happen to H bar?

1054
01:26:52,127 --> 01:26:56,953
We've been working in units in which H bar is one

1055
01:26:57,921 --> 01:27:01,593
I've been working in units in which H bar plus constant is one 

1056
01:27:01,787 --> 01:27:07,210
The question is if we were to reinstate Planck's constant and make it not equal to one

1057
01:27:07,399 --> 01:27:09,013
Where would it go in these equations?

1058
01:27:09,196 --> 01:27:12,277
And, what's it?

1059
01:27:12,454 --> 01:27:15,779
Yes it goes into Schrodinger's Equation, that's correct

1060
01:27:15,947 --> 01:27:19,514
But how do we trace it, let's see, how do we trace it?

1061
01:27:19,692 --> 01:27:25,434
Here's the equation, I told you H has significance as a Hamiltonian

1062
01:27:25,654 --> 01:27:27,514
Therefore it has a unit of energy

1063
01:27:28,604 --> 01:27:34,697
So let's look at this equation, sign cancels out on both sides

1064
01:27:34,870 --> 01:27:36,275
Just as far as unit goes 

1065
01:27:36,469 --> 01:27:39,039
Unit of this side is the same as unit of this side

1066
01:27:39,222 --> 01:27:44,988
So the right hand side of the equation, apart from the sign has a unit of energy

1067
01:27:45,180 --> 01:27:48,869
The left hand side of the equation has a unit of inversed time

1068
01:27:49,030 --> 01:27:53,817
Inversed time and energy don't have the same units 

1069
01:27:55,018 --> 01:27:59,356
Inversed time is inversed seconds, and energy is joule, so whatever

1070
01:27:59,573 --> 01:28:05,077
So we have an equation here which doesn't make good dimensional sense

1071
01:28:06,352 --> 01:28:12,359
We need a constant in here and the constant we have to put the constant in here

1072
01:28:12,542 --> 01:28:14,647
let's see where does the constant goes? 

1073
01:28:14,845 --> 01:28:16,257
Does it go to the right hand side or left hand side?

1074
01:28:17,889 --> 01:28:22,062
Where does the Planck's constant go? Right hand side or left hand side?

1075
01:28:23,412 --> 01:28:25,125
What? >> Either

1076
01:28:25,335 --> 01:28:27,828
Well, if it goes in one place with one thing 

1077
01:28:27,997 --> 01:28:29,287
it would go to inverse in the other place

1078
01:28:29,287 --> 01:28:34,490
So, here is all you have to know is that 

1079
01:28:34,688 --> 01:28:37,114
Planck's constant has a unit of energy times time

1080
01:28:37,312 --> 01:28:40,549
Planck's constant has a unit of energy times time 

1081
01:28:42,141 --> 01:28:46,114
So, let's see so energy times time, so we want to put Planck's constant here

1082
01:28:47,294 --> 01:28:49,697
Energy time times over time is equal to energy

1083
01:28:49,697 --> 01:28:53,181
That's where Planck's constant goes into these equations

1084
01:28:54,785 --> 01:28:58,306
I apologize for having completely forgotten about it

1085
01:28:59,616 --> 01:29:08,245
But that's where it goes, and if we follow through the equation 

1086
01:29:08,476 --> 01:29:10,317
No doubt it appears over here 

1087
01:29:13,774 --> 01:29:18,194
I think it appears in the denominator here 

1088
01:29:20,505 --> 01:29:26,066
Let's see, I got everything right except when I have H I should have H over h bar

1089
01:29:27,683 --> 01:29:30,850
Let's rewrite it this way, H over h bar 

1090
01:29:32,458 --> 01:29:36,769
So in all my equations, wherever we wrote H, the Hamiltonian

1091
01:29:36,964 --> 01:29:40,175
We really should have, if we want to keep the unit straight, 

1092
01:29:40,361 --> 01:29:43,768
H over Planck's constant, and that goes straight to here

1093
01:29:43,951 --> 01:29:48,177
So really the right equation including the constants 

1094
01:29:48,364 --> 01:29:56,069
is D L dt is one over h bar here

1095
01:29:57,068 --> 01:29:59,372
One over Planck's constant

1096
01:30:01,868 --> 01:30:04,349
Planck's constant is a small number 

1097
01:30:06,628 --> 01:30:08,189
So this looks like it was very big 

1098
01:30:09,445 --> 01:30:12,101
But in the other hand, this is L times H minus H times L

1099
01:30:12,972 --> 01:30:15,397
In classical physics, that would be zero

1100
01:30:16,493 --> 01:30:22,534
So whatever this is, classically zero, in quantum mechanics, 

1101
01:30:22,701 --> 01:30:25,544
it must be a small correction to the classical physics 

1102
01:30:25,725 --> 01:30:29,516
To extend that quantum mechanics is a small correction to the classical physics

1103
01:30:29,695 --> 01:30:35,480
And so the commutator is itself something which is typically small in the order of h bar

1104
01:30:35,654 --> 01:30:38,501
So that allows us then, to guess, 

1105
01:30:38,687 --> 01:30:45,618
to guess an identification between commutators and Poisson brackets

1106
01:30:45,793 --> 01:30:51,938
Now I'm, this is none intuitive at the moment 

1107
01:30:52,101 --> 01:30:55,314
It's just looking at the two equations, and say

1108
01:30:55,504 --> 01:30:57,546
Look is there going to be any connection 

1109
01:30:57,728 --> 01:30:59,690
between quantum mechanics and classical mechanics?

1110
01:30:59,874 --> 01:31:04,342
Then there must be a connection between the Poisson brackets

1111
01:31:04,519 --> 01:31:08,887
And between the Poisson brackets and the commutators 

1112
01:31:09,087 --> 01:31:19,482
Namely Poisson brackets must be equal to minus i over h bar times commutator

1113
01:31:20,922 --> 01:31:22,570
Or it's usually written in the other way, 

1114
01:31:22,753 --> 01:31:29,039
commutator is i h bar times Poisson brackets

1115
01:31:30,223 --> 01:31:35,439
Commutator is a small thing because there is a h bar there 

1116
01:31:36,758 --> 01:31:39,510
Poisson brackets in classical physics is not a small thing

1117
01:31:39,691 --> 01:31:42,246
It's a number, I mean it's some characteristic quantities 

1118
01:31:42,492 --> 01:31:44,006
which is neither small nor big

1119
01:31:45,157 --> 01:31:48,206
And in the same unit, the unit that you do classical physics 

1120
01:31:48,403 --> 01:31:52,305
Standard units, meters, seconds, whatever

1121
01:31:52,487 --> 01:31:57,270
Commutators are very small, that's because they are almost zero

1122
01:31:57,476 --> 01:31:59,643
A times B minus B times A is almost zero

1123
01:31:59,835 --> 01:32:06,642
Okay, no I don't think so, we have a minus i on this side

1124
01:32:06,821 --> 01:32:10,609
Minus i over h bar multiply by h bar 

1125
01:32:13,595 --> 01:32:15,444
And now multiply both side by i

1126
01:32:15,624 --> 01:32:18,877
And use that minus sign, times plus i is one 

1127
01:32:23,046 --> 01:32:26,396
It was Dirac who recognized the connections 

1128
01:32:26,585 --> 01:32:29,498
and built his quantum mechanics out of it

1129
01:32:29,673 --> 01:32:34,810
He built his quantum mechanics out of making an identification of Poisson bracket

1130
01:32:34,999 --> 01:32:40,035
with commutators, but of course the foundation of the subject

1131
01:32:41,203 --> 01:32:46,261
quantum mechanics is a consistent subject 

1132
01:32:46,461 --> 01:32:48,990
without having to make any reference to classical physics, 

1133
01:32:49,182 --> 01:32:50,899
the right way to think about things is 

1134
01:32:51,072 --> 01:32:53,815
quantum mechanics comes before classical mechanics

1135
01:32:53,989 --> 01:32:56,463
Commutators before Poisson brackets 

1136
01:32:56,641 --> 01:33:02,793
You have to derive classical physics is an approximation to quantum mechanics

1137
01:33:02,976 --> 01:33:05,801
So here we see some formal similarities, at this point here

1138
01:33:05,981 --> 01:33:11,743
We have no idea why this commutator should have features 

1139
01:33:11,945 --> 01:33:13,415
in common with a Poisson bracket

1140
01:33:13,676 --> 01:33:17,404
Before we go, let me just say, let me just write down 

1141
01:33:17,588 --> 01:33:20,392
some of the features of commutators and of Poisson brackets 

1142
01:33:20,575 --> 01:33:21,545
To remind you

1143
01:33:34,505 --> 01:33:37,853
And again all of this should be at this stage should be a little mysterious

1144
01:33:38,027 --> 01:33:40,807
Why is this funny connection between Poisson brackets?

1145
01:33:42,625 --> 01:33:44,575
But let's pursuit it anyway 

1146
01:33:44,750 --> 01:33:50,568
Okay, I remind you about Poisson brackets

1147
01:33:50,749 --> 01:33:55,848
First of all if I have two quantities, I will call them A and B now

1148
01:33:56,031 --> 01:33:58,462
Or A and B that I used, what the hell?

1149
01:33:58,646 --> 01:34:00,293
I will just call them L and H, 

1150
01:34:00,467 --> 01:34:02,821
it doesn't have to be the Hamiltonian and anything special

1151
01:34:02,995 --> 01:34:04,533
They are just L and H

1152
01:34:04,712 --> 01:34:11,455
Alright the first thing is, the Poisson brackets of L with H is equal to 

1153
01:34:11,631 --> 01:34:13,967
minus of Poisson bracket of H with L

1154
01:34:17,679 --> 01:34:25,880
The commutator of L with H is minus the commutator of H with L

1155
01:34:26,792 --> 01:34:29,628
So that's a parallelism that we could begin with

1156
01:34:29,628 --> 01:34:34,889
Now let's take some other properties of Poisson brackets

1157
01:34:36,709 --> 01:34:39,191
There's a lot of properties of Poisson brackets

1158
01:34:39,354 --> 01:34:42,095
But let me just write one or two more 

1159
01:34:43,480 --> 01:34:47,850
If I have two operators, or two of not operators 

1160
01:34:48,038 --> 01:34:52,986
But two quantities, let's call them L and M, okay,

1161
01:34:53,167 --> 01:34:58,731
L times M is itself a classical quantity, 

1162
01:34:58,914 --> 01:35:03,159
and I can compute it's Poisson bracket with H

1163
01:35:03,338 --> 01:35:05,159
Does anybody remember the answer?

1164
01:35:09,007 --> 01:35:12,944
I will remind you and tell you, look it up, this is important

1165
01:35:15,136 --> 01:35:20,687
L is a Poisson bracket of L with H times M

1166
01:35:20,849 --> 01:35:28,238
Plus L times a Poisson bracket of M with H

1167
01:35:29,710 --> 01:35:36,243
Now in classical physics the order multiplication doesn't matter

1168
01:35:38,427 --> 01:35:41,443
So you could bring the M on the left on the right it doesn't matter

1169
01:35:41,639 --> 01:35:43,844
You can bring the L on the left or right it doesn't matter

1170
01:35:44,028 --> 01:35:49,418
But this is the relationship for the product of two variables

1171
01:35:49,608 --> 01:35:54,179
The Poisson bracket, go back, go back to the last quarter lecture

1172
01:35:54,372 --> 01:35:57,757
And check this out, this is one of the defining 

1173
01:35:57,935 --> 01:36:01,525
Not one of the defining, one of the properties of Poisson bracket

1174
01:36:01,740 --> 01:36:04,753
You can do the same, you can ask the same question over here

1175
01:36:04,939 --> 01:36:11,947
Supposing you have LM and you take it's commutator with H

1176
01:36:13,140 --> 01:36:18,551
What do you get? This is an exercise, this is an exercise

1177
01:36:18,761 --> 01:36:26,366
All you do is you write, that this is equal to LMH minus HLM

1178
01:36:27,439 --> 01:36:29,647
And now you start just juggling and 

1179
01:36:29,821 --> 01:36:32,423
adding same term, subtracting the same term

1180
01:36:32,607 --> 01:36:37,364
And eventually you will discover that this is the same as this

1181
01:36:38,244 --> 01:36:44,260
Same relationship, this is equal to L times a commutator of M with H

1182
01:36:45,492 --> 01:36:54,695
Plus commutator of L with H times M

1183
01:36:56,711 --> 01:37:00,808
The way to do this is just write LMH minus HLM

1184
01:37:00,994 --> 01:37:03,640
And then write out this guys over here

1185
01:37:03,812 --> 01:37:08,205
This is L, MH, minus LH, M

1186
01:37:08,382 --> 01:37:13,295
Plus LH, M minus HL, M

1187
01:37:14,932 --> 01:37:20,082
Four terms here, two cancel in pairs, and you will just get this back

1188
01:37:20,256 --> 01:37:23,405
It's two one step thing , very easy thing

1189
01:37:23,563 --> 01:37:33,091
Okay, so you begin to see some pattern that Poisson bracket and commutator

1190
01:37:33,272 --> 01:37:34,764
must somehow be closely related

1191
01:37:34,950 --> 01:37:42,560
The last thing I'm gonna show you tonight is about energy conservation

1192
01:37:42,765 --> 01:37:47,601
Now we are not going to consider 

1193
01:37:47,786 --> 01:37:50,625
the most general concept of energy conservation

1194
01:37:50,826 --> 01:37:56,152
We're simply going to ask, is the average of the energy conserve with time?

1195
01:37:57,232 --> 01:37:59,248
We have not even really got into 

1196
01:37:59,447 --> 01:38:03,188
what it would mean for energy itself to be conserve in any deeper way

1197
01:38:03,367 --> 01:38:06,684
But we can ask we have all the ingredients now

1198
01:38:06,876 --> 01:38:14,485
Which would allows us to ask how is the average of

1199
01:38:16,884 --> 01:38:20,021
How is the average of any quantity changes with time

1200
01:38:21,093 --> 01:38:24,877
No, here it is over here, we can now specifically ask 

1201
01:38:25,049 --> 01:38:27,222
how the average of the energy change with time?

1202
01:38:27,222 --> 01:38:38,873
Let's do it, the energy I claimed is Hamiltonian, or it's definition if you like

1203
01:38:39,064 --> 01:38:40,578
What is energy? It's Hamiltonian 

1204
01:38:40,774 --> 01:38:43,850
So let's check and ask whether the energy is conserve 

1205
01:38:44,032 --> 01:38:45,472
if energy is the Hamiltonian

1206
01:38:45,472 --> 01:38:53,238
This equation would read, the time derivative of the average of the Hamiltonian

1207
01:38:53,423 --> 01:38:58,640
is equal to minus i over h bar, 

1208
01:38:58,824 --> 01:39:03,192
times the commutator of the Hamiltonian with itself

1209
01:39:03,370 --> 01:39:08,665
The Hamiltonian with itself is H times H minus H times H

1210
01:39:08,861 --> 01:39:11,649
H times H is the same as H times H

1211
01:39:13,104 --> 01:39:15,681
Quantum mechanics is not that weird, you know it's weird

1212
01:39:15,874 --> 01:39:20,560
But H times H is the same as H times H, so this is zero

1213
01:39:24,345 --> 01:39:26,073
Whatever the Hamiltonian is

1214
01:39:26,247 --> 01:39:32,563
It's average, it's average, it's expectation value with time is always zero

1215
01:39:34,323 --> 01:39:38,371
So it's conserve in that sense, at minimal that sense

1216
01:39:38,551 --> 01:39:42,120
There is stronger sense in which it's conserve, which we will come to it

1217
01:39:42,296 --> 01:39:48,885
But even in this weaker sense, it's kind of interesting and it tells us

1218
01:39:49,070 --> 01:39:52,356
on some kind of right track would be able to 

1219
01:39:52,536 --> 01:39:55,433
compare quantum mechanics and classical mechanics

1220
01:39:56,889 --> 01:39:59,395
# Okay, I'm gonna stop now and take a few questions

1221
01:40:05,675 --> 01:40:10,416
>> the bar notation (inaudible)

1222
01:40:10,597 --> 01:40:11,224
Up here?

1223
01:40:11,401 --> 01:40:16,203
>> Is the bar notation mean average?

1224
01:40:16,391 --> 01:40:19,551
Probably, I just mean the average, I mean it's the average that define

1225
01:40:24,228 --> 01:40:30,536
define the summation of probability of lambda times lambda, okay

1226
01:40:30,708 --> 01:40:37,038
So I don't know what's the standard symbol in statistic or average in that sense

1227
01:40:38,214 --> 01:40:44,839
>> The first moment of the distribution you knew that 

1228
01:40:49,607 --> 01:40:53,863
The first moment of the distribution, ya, okay,

1229
01:41:00,271 --> 01:41:05,222
I used to just calling it lambda

1230
01:41:05,405 --> 01:41:06,454
>> including brackets?

1231
01:41:06,631 --> 01:41:13,039
E bracket of lambda, that stands for what?

1232
01:41:13,232 --> 01:41:15,544
>> the average of expected value?

1233
01:41:15,720 --> 01:41:17,999
The expected value even though it's not the expected value 

1234
01:41:22,648 --> 01:41:27,178
You never see the expected value as a bar on top of it?

1235
01:41:28,179 --> 01:41:33,366
>> I mean (inaudible)

1236
01:41:33,547 --> 01:41:37,042
Okay, ya, when I use the term average I will mean this

1237
01:41:37,229 --> 01:41:39,718
It's also call the expectation value, 

1238
01:41:39,913 --> 01:41:45,106
and I will freely interchange the Dirac notation 

1239
01:41:45,284 --> 01:41:50,428
for the average with the bar notation

1240
01:41:53,787 --> 01:42:01,070
And that's not, if that's not standard according to statistic books, too bad

1241
01:42:01,263 --> 01:42:05,996
>> So here you wrote down the Schrodinger equation

1242
01:42:06,169 --> 01:42:12,245
>> and then you wrote down the ...

1243
01:42:12,427 --> 01:42:14,645
>> How did you go to left hand equation to right hand equation?

1244
01:42:14,817 --> 01:42:17,653
>> Is that complex conjugation or Hermitian conjugation?

1245
01:42:17,827 --> 01:42:23,852
Hermitian conjugation, but remember H is Hermitian, we proved that

1246
01:42:24,050 --> 01:42:30,877
That was the condition of unitarity, that H is Hermitian

1247
01:42:31,071 --> 01:42:35,230
>> But how can you interchange the order of Psi H 

1248
01:42:35,407 --> 01:42:36,966
>> but not d/dt with Psi?

1249
01:42:37,146 --> 01:42:38,086
Where, where, where, where, where?

1250
01:42:39,214 --> 01:42:43,152
>>On the left side of the equation you got d/dt with Psi

1251
01:42:43,337 --> 01:42:46,096
>>d/dt is the operator 

1252
01:42:46,275 --> 01:42:47,520
This one here? >> Yeah

1253
01:42:48,706 --> 01:42:51,954
The time derivative of a bra vector is a bra vector

1254
01:42:52,135 --> 01:42:57,067
Look the time derivative is just another way of talking about difference

1255
01:42:57,259 --> 01:42:59,352
The difference of two bra vectors is a bra vector

1256
01:42:59,536 --> 01:43:02,316
The difference of tow ket vectors is a ket vector

1257
01:43:04,029 --> 01:43:07,503
And the limit difference becomes the time derivative 

1258
01:43:07,686 --> 01:43:11,437
So the derivative of a bra vector is a bar vector 

1259
01:43:11,612 --> 01:43:15,079
The derivative of a ket vector is a ket vector

1260
01:43:15,262 --> 01:43:17,942
If you want to take this ket equation 

1261
01:43:18,144 --> 01:43:20,950
What is that bra that goes with this ket?

1262
01:43:21,154 --> 01:43:24,063
The bra, the dual of this ket is 

1263
01:43:24,242 --> 01:43:26,480
the time derivative of the corresponding bra vector

1264
01:43:30,823 --> 01:43:33,239
It's just the think of this as taking the difference of two vectors

1265
01:43:34,799 --> 01:43:36,223
It's just the difference of two vectors

1266
01:43:38,615 --> 01:43:41,959
Alright, exactly

1267
01:43:41,959 --> 01:43:47,724
And again, quantum mechanics is weird, 

1268
01:43:47,900 --> 01:43:50,709
but taking the difference of two vectors

1269
01:43:50,709 --> 01:43:53,029
and replace it by a derivative 

1270
01:43:53,029 --> 01:43:57,408
when the difference is small, that's perfectly legitimate

1271
01:43:57,592 --> 01:44:04,783
>> (inaudible) energy conservation

1272
01:44:04,944 --> 01:44:06,246
Well, yeah

1273
01:44:06,432 --> 01:44:10,952
>>so we ignore this so far?

1274
01:44:11,142 --> 01:44:12,000
No

1275
01:44:12,197 --> 01:44:15,616
>> So why it is that energy

1276
01:44:15,802 --> 01:44:16,960
How come we getting to what ?

1277
01:44:17,136 --> 01:44:20,256
>> Energy conservation is, how do you know the energy is conserved?

1278
01:44:20,435 --> 01:44:24,062
Well, we don't know it's correspond with energy until we do a lot more

1279
01:44:24,246 --> 01:44:27,553
We don't know, we can find it to be energy Hamiltonian,

1280
01:44:27,737 --> 01:44:31,053
but that's not satisfying, we want to see in cases 

1281
01:44:31,201 --> 01:44:33,926
where we understand the classical physics

1282
01:44:34,096 --> 01:44:36,574
And we understand what energy is

1283
01:44:36,752 --> 01:44:41,186
We are gonna want to see that the quantum mechanics of the same system

1284
01:44:41,359 --> 01:44:45,163
classical system is an approximation to the quantum mechanical system

1285
01:44:45,341 --> 01:44:46,359
Physics is really quantum mechanical 

1286
01:44:46,359 --> 01:44:50,365
Under certain condition, such as the system is heavy

1287
01:44:51,711 --> 01:44:56,724
Basically heavy is big, large masses

1288
01:44:56,903 --> 01:45:01,388
Then the system behaves approximately classicaly

1289
01:45:01,571 --> 01:45:04,789
which means that the average values move in ways 

1290
01:45:04,789 --> 01:45:07,669
which are consistent to what classical equations of motion

1291
01:45:07,669 --> 01:45:12,314
We are gonna want to see when we study such big heavy systems

1292
01:45:12,507 --> 01:45:18,071
That the thing we call quantum mechanical Hamiltonian

1293
01:45:18,269 --> 01:45:24,918
is essentially closely connected with the thing what we call classical Hamiltonian

1294
01:45:25,086 --> 01:45:27,599
But we will come to that, but we can't do all thing all in one night

1295
01:45:27,599 --> 01:45:31,420
At the moment, I'm just showing you similarities

1296
01:45:31,595 --> 01:45:33,908
And if you wanna do better than that 

1297
01:45:34,072 --> 01:45:37,381
Whether we will actually get to do better than that I don't know

1298
01:45:37,559 --> 01:45:42,611
We can't do everything, but that's the logic

1299
01:45:46,199 --> 01:45:50,731
>> way back to the electron spin, I think you left off in the last lecture

1300
01:45:50,938 --> 01:45:55,919
>> You want to show the probability (inaudible)

1301
01:45:56,085 --> 01:46:00,659
Well, yeah, right, I've save some time to do that 

1302
01:46:00,659 --> 01:46:02,055
but I think we've done enough for tonight

1303
01:46:03,236 --> 01:46:05,998
Who was it that send in this solution to the...

1304
01:46:07,226 --> 01:46:17,023
Yeah, well done, we will do it, but I just run out of steam

1305
01:46:19,559 --> 01:46:22,055
Right we will come back to that, we are gonna come back to that 

1306
01:46:23,543 --> 01:46:26,727
>> So how can your postulate work with that without a collapsed postulate?

1307
01:46:26,904 --> 01:46:30,723
>> How can your postulate work without a collapsed postulate?

1308
01:46:30,909 --> 01:46:34,278
Well we really want to derive a collapsed postulate

1309
01:46:34,459 --> 01:46:39,234
The derivation is deeply connected with something else we have not come to yet

1310
01:46:39,417 --> 01:46:41,920
I think we will come to it next time, entanglement 

1311
01:46:46,117 --> 01:46:49,461
The measurement process is really a process of 

1312
01:46:49,634 --> 01:46:53,512
establishing entanglement between system and the apparatus

1313
01:46:53,694 --> 01:46:59,312
And until we talk about that, the entanglement process

1314
01:47:01,848 --> 01:47:08,429
I think until we do, we simply have to adopt the language of collapsed

1315
01:47:08,629 --> 01:47:11,918
But next time, next time, next time, remind me about collapsed

1316
01:47:12,083 --> 01:47:17,022
And course to collapsed, Okay

1317
01:47:17,200 --> 01:47:22,422
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