1 00:00:01,890 --> 00:00:11,820 Know. Okay. 2 00:00:11,820 --> 00:00:17,910 So on Friday we began looking at operators, the connection between observables and operators. 3 00:00:18,840 --> 00:00:23,580 So the observable is the primitive start is the starting point of our discussion. 4 00:00:24,660 --> 00:00:26,190 And observable has a spectrum. 5 00:00:26,200 --> 00:00:31,500 In other words, there are possible values you can get when you measure this observable to an observable is something you can measure. 6 00:00:31,950 --> 00:00:40,829 So it has possible answers. And to each answer there is at least one state in which you are certain of getting that answer. 7 00:00:40,830 --> 00:00:49,049 So a state where there is no ambiguity, there is no question there's nothing probabilistic about the result of that measurement out of those states. 8 00:00:49,050 --> 00:00:53,370 In those numbers, we construct an operator, this animal here. 9 00:00:54,630 --> 00:01:05,370 And one one good thing about this operator, one useful aspect of it is that if you squeeze it between the between the cat, 10 00:01:05,610 --> 00:01:11,900 the state of your system and the associated bra, you get out the expectation value by. 11 00:01:16,640 --> 00:01:23,510 Of the observable cue when we're in this state. So when there is uncertainty and the result of the measurement is probabilistic, 12 00:01:23,750 --> 00:01:31,010 which normally will be the case for most states will be the case then this simple algebraic formula we showed last time, 13 00:01:31,340 --> 00:01:35,629 I think that's where we finished that. That leads to the expectation value of that measurement. 14 00:01:35,630 --> 00:01:41,690 So that's one way in which this operator Q is useful. You'll find as we go along that there are many other ways in which this operator. 15 00:01:41,690 --> 00:01:47,120 Q which for the moment is going to have a hat to distinguish it from the observable Q which is a physical, 16 00:01:47,120 --> 00:01:52,940 conceptual thing, and the operator, which is just some mathematical fiction which we're going to get used to gradually. 17 00:01:53,240 --> 00:01:58,370 The distinction will blur. But I hope when you need to, you can distinguish between the physical thing. 18 00:01:59,210 --> 00:02:06,140 So energy is the physical thing, and energy comes with an operator, which at the moment would be called hat. 19 00:02:08,450 --> 00:02:10,070 Well, actually we did introduce that. 20 00:02:10,340 --> 00:02:23,090 So the operator E hat is historical reasons called H and of course it is the operator, some over all possible energies of energy. 21 00:02:25,370 --> 00:02:31,309 Energy. So these are the states of well-defined energy and these are the corresponding energies. 22 00:02:31,310 --> 00:02:32,510 And this is the Hamiltonian. 23 00:02:37,970 --> 00:02:46,460 In honour of the Irish mathematician who introduced this into classical physics, I called the corresponding operator into classical physics. 24 00:02:47,590 --> 00:02:56,020 Okay. So any I guess you will have I hope you will recognise from endless lectures that if we have given a basis. 25 00:02:59,200 --> 00:03:03,849 Any old basis, then every operator can be turned into a matrix. 26 00:03:03,850 --> 00:03:14,110 Because given the basis, we can say, given any state find and this will be the sum a I, I can be written as this linear combination of basis vectors. 27 00:03:14,800 --> 00:03:26,860 If we use any operator queue on on up side, we're going to get some other animal fae and we can expand Fae. 28 00:03:26,920 --> 00:03:31,850 We can say that this is equal to the sum of i, i. 29 00:03:32,650 --> 00:03:38,530 And then this becomes. Q operating on the sum of a j. 30 00:03:40,120 --> 00:03:43,780 J. This being some David J. This being some Dave I. 31 00:03:43,950 --> 00:03:47,590 Right. That's just substituting in here. 32 00:03:48,430 --> 00:03:57,459 And then if I want to find out what b I is or actually what is, change this to k to make a slightly cleaner job. 33 00:03:57,460 --> 00:04:00,670 This is just a dummy index. I can call it anything I like. Let's call it k. 34 00:04:01,000 --> 00:04:08,050 If I want to find what b I is, I pick out to pick out of this sum over all the possible all the b case i, 35 00:04:08,530 --> 00:04:12,909 i of course brought through with I so I brought through with I. 36 00:04:12,910 --> 00:04:16,770 And that leads me to the conclusion that b I because this on this summit, 37 00:04:16,780 --> 00:04:21,249 we're going to have an i k here, which is going to be nothing except when k is I. 38 00:04:21,250 --> 00:04:35,559 So I get a b, I is equal to the sum of a j, the sum of I of I operate q j times a j because this is a complex number. 39 00:04:35,560 --> 00:04:45,320 So, so when we break through by I, it doesn't get in the way because I is a linear function on the on the case. 40 00:04:46,960 --> 00:04:52,910 So we can write this as the sum over j and i. 41 00:04:53,030 --> 00:04:57,790 Q i. J a j where q. 42 00:04:57,790 --> 00:05:08,560 I. J is by definition, the complex number that you get in this way by taking the JTH basis vector operating on it with the operator. 43 00:05:08,590 --> 00:05:12,430 Q And then taking the DOT product, as it were, growing through with I. 44 00:05:13,180 --> 00:05:21,490 So every operator can be represented by a matrix of complex numbers. 45 00:05:21,850 --> 00:05:27,010 And of course, any one of these things is called any one of those numbers is called a matrix element. 46 00:05:27,310 --> 00:05:33,220 And a lot of a lot of quantum mechanics, a lot of physics revolves around calculating matrix elements. 47 00:05:34,180 --> 00:05:37,749 So it's a word that's often used. So it's a matrix made up of matrix elements. 48 00:05:37,750 --> 00:05:41,590 These matrix elements are complex numbers. So if. 49 00:05:41,830 --> 00:05:53,620 Now another point to make is if the basis ie is the basis of the eigenvectors of Q. 50 00:05:53,800 --> 00:05:58,810 Now I forgot to last on Friday already. I think we saw I forgot to mention it just now. 51 00:05:58,810 --> 00:06:04,060 I think on Friday we saw that these things well, we defined Q this way. 52 00:06:04,310 --> 00:06:16,960 And with this definition it turned out that. Q I is an icon cat of Q and Q, I is an eigenvalue that was a consequence. 53 00:06:17,680 --> 00:06:22,569 So these physically important states are as a consequence of this definition, 54 00:06:22,570 --> 00:06:27,910 these physically important states become eigen cat's eigenvectors of the operation. 55 00:06:27,940 --> 00:06:32,290 Q And these become the eigenvalues. So now we can say something different. 56 00:06:32,290 --> 00:06:39,579 We can say Q is constructed out of its eigen kits and its eigenvalues in this manner was previously we had a physical statement that the 57 00:06:39,580 --> 00:06:46,300 operator Q was constructed out of the states in which there's no ambiguity as to the measurement and the possible results of the measurement. 58 00:06:47,830 --> 00:06:56,890 So if we use the eigen. Q I as our basis vectors, then this matrix becomes very simple. 59 00:06:57,160 --> 00:07:02,140 Then Q J is going to be, of course, I. 60 00:07:02,950 --> 00:07:08,019 Q Well, I'm going to put this in this. Q I. QJ But. 61 00:07:08,020 --> 00:07:11,760 Q on. Q Jay is necessarily Q Jay Times. 62 00:07:11,770 --> 00:07:14,800 Q Jay, this is so this becomes. 63 00:07:15,850 --> 00:07:20,470 Q Jay Times. Q I times. 64 00:07:20,500 --> 00:07:24,190 Q Jay, but this is Delta, right? Jay So this becomes. 65 00:07:24,220 --> 00:07:30,900 Q Jay Times-delta. Jay So these matrix elements vanish and less Jay is equal to I. 66 00:07:30,910 --> 00:07:35,800 When Jay is equal to I, we get the number. Q Jay, in other words, in this basis. 67 00:07:39,420 --> 00:07:43,620 Q is represented by a diagonal matrix. 68 00:07:50,310 --> 00:07:53,250 In other words, Q is going to look like The Matrix. Q. 69 00:07:53,280 --> 00:08:03,030 Q AJ is going to be Q1, Q2, Q3, all these numbers down the diagonal and nothing everywhere else and so on. 70 00:08:03,030 --> 00:08:09,720 Until we're bought, we'll run out, more to the point, run out of possible states in which Q has a well-defined value. 71 00:08:10,710 --> 00:08:23,370 Okay. As a result of that, if we do this, if we if we take the complex conjugate. 72 00:08:23,820 --> 00:08:33,060 No, no, no, but not do this. Yeah. 73 00:08:33,070 --> 00:08:35,710 All right. Note if so. So the commissioner joined, 74 00:08:35,890 --> 00:08:49,720 I think from I'm going to take it that you remember this from professor but as this lectures have you seen not joint of Q IJA of Q so the Matrix. 75 00:08:49,750 --> 00:08:53,829 Q Now we've got three things now it's a bit confusing, isn't it? 76 00:08:53,830 --> 00:08:56,980 We've got a physical quantity. Q Like the energy. 77 00:08:57,670 --> 00:09:08,380 We've got an operator. Q hat, and we've got a matrix which is in one particular set of basis vectors is representing the operator. 78 00:09:08,890 --> 00:09:15,610 So I'm a little bit short of notations. I've got a Q and a Q hat, but I will say I'm tempted to write to. 79 00:09:15,610 --> 00:09:23,080 Right. Q i. J which sometimes means the particular complex number that you will find in the I throw in the Jth column of the Matrix. 80 00:09:23,110 --> 00:09:30,160 Q But sometimes we use this notation. Q to imply the matrix that represents. 81 00:09:30,190 --> 00:09:41,889 Q Do you see that there's a there's a slight overbooking of notation here, and it's it's it's universal in in in theoretical physics. 82 00:09:41,890 --> 00:09:50,050 You can't well, nobody has a natty way of distinguishing distinguishing between the matrix and the, and the matrix elements. 83 00:09:50,440 --> 00:09:55,690 So let me just write The Matrix. Q So the emission. 84 00:09:55,750 --> 00:10:02,049 I under the Matrix. Q Is is Q Dagger and Q dagger is defined. 85 00:10:02,050 --> 00:10:09,880 So the IGF element of it is equal to is the complex conjugate of the j element of the matrix. 86 00:10:09,910 --> 00:10:11,910 Q Right. This means the complex conjugate. 87 00:10:12,880 --> 00:10:21,070 So so the commission conjugate is you you take, you know, you swap rows and columns and you take the complex conjugate. 88 00:10:21,070 --> 00:10:27,190 That's what happens with the individual elements. So let's see what happens here. 89 00:10:27,550 --> 00:10:34,210 So we we can this property doesn't depend on what basis we look at it in. 90 00:10:34,450 --> 00:10:37,540 So let's have a look at it there. So, so what is this? 91 00:10:38,440 --> 00:10:50,950 Q. J So in the basis in the particular basis of the eigenvectors of. 92 00:10:50,950 --> 00:10:54,639 Q what does this statement become? It becomes that. 93 00:10:54,640 --> 00:11:03,650 Q dagger i j is equal to we figured out what that what q g is q g turned out to be. 94 00:11:03,670 --> 00:11:09,700 Q Up there I Delta. 95 00:11:09,880 --> 00:11:13,390 J or Delta. It doesn't matter. Right? 96 00:11:13,390 --> 00:11:17,000 That's what we found. So that's. Q I. 97 00:11:17,020 --> 00:11:22,180 J in this particular basis, no, i sorry. J I I've swapped I hope I've swapped it over. 98 00:11:22,570 --> 00:11:28,030 And now I take the complex conjugate if. 99 00:11:32,630 --> 00:11:42,170 If Q is real, then this becomes Q II times delta AJ is equal to Q. 100 00:11:42,860 --> 00:11:55,740 AJ. So the permission now joint of Q will be Q itself, if it's possible, if all the elements in its spectrum are real. 101 00:11:57,940 --> 00:12:04,590 And traditionally people have said it's obvious that an observable is a real number. 102 00:12:04,600 --> 00:12:07,720 And I remember it was an undergraduate thinking, hang on a moment, that's ridiculous. 103 00:12:08,050 --> 00:12:11,590 The impedance of a circuit, right, is something that I have to measure. 104 00:12:12,140 --> 00:12:13,480 Yeah. Might be something you're doing. 105 00:12:13,480 --> 00:12:18,040 One of the you might have done last year in some of the electronics practical measure, the importance of this circuit at this frequency. 106 00:12:18,460 --> 00:12:24,130 It's clearly a complex number. So it's nonsense to say that observables have to be real cause they don't have to be real. 107 00:12:24,550 --> 00:12:30,070 But if they are real, then the observable will be represented by an omission matrix. 108 00:12:30,070 --> 00:12:37,850 So. So. If the spectrum. The spectrum is all real. 109 00:12:41,720 --> 00:12:44,930 Then Q Hat is mission. 110 00:12:50,520 --> 00:12:54,450 This is in the great majority of treatments. This is all back to front. 111 00:12:54,480 --> 00:13:01,950 People say it's people say that every observable is going to be represented by or associated with a machine operator. 112 00:13:02,520 --> 00:13:05,220 They then use some well-known theorem, which I'm sure you've met, 113 00:13:05,490 --> 00:13:13,110 which says that every emission operator has real eigenvalues and orthogonal eigen caps. 114 00:13:13,980 --> 00:13:17,730 And then therefore they say the eigen caps of these things are orthogonal. 115 00:13:17,760 --> 00:13:25,440 That's not the way actually the flow of the logic of the the of the flow from the real physical world into the mathematical world works. 116 00:13:25,770 --> 00:13:33,599 It's the other way. It's the real argument is that the eigen states in which the states sorry, 117 00:13:33,600 --> 00:13:38,190 the states in which Q has a well-defined value, have to be mutually orthogonal. 118 00:13:38,200 --> 00:13:41,850 Because why? Because. Q i. 119 00:13:42,560 --> 00:13:50,970 Q j. This complex number is the amplitude to get QJ given. 120 00:13:51,510 --> 00:13:55,330 Q. I. And if you know that the result of the measurement is going to be. 121 00:13:55,350 --> 00:13:59,550 Q Why this this amplitude has to vanish for any QJ not equal to. 122 00:13:59,580 --> 00:14:05,490 Q Why? So this whole functionality comes in is a physical requirement of the way we want to use the theory. 123 00:14:06,690 --> 00:14:11,069 Then if the eigenvalues, if this are all real, it's a spectrum. 124 00:14:11,070 --> 00:14:15,540 The possible results are all real. Then you end up with emission matrices, right? 125 00:14:15,840 --> 00:14:23,990 But there's no need to be working with emission matrices. If if you want to work with the complex impedance as you're observable, that's not required. 126 00:14:24,000 --> 00:14:29,579 But what you do need is this whole functionality result that is that is a consequence 127 00:14:29,580 --> 00:14:34,350 of that's a logical necessity of the way we want to interpret the mathematics. 128 00:14:38,700 --> 00:14:42,400 Okay. Now we can, of course, multiply operations together. 129 00:14:43,590 --> 00:14:50,159 So something else we can do with operators is we've got two operators, O and Q, 130 00:14:50,160 --> 00:14:55,889 we can define this animal by the rule that this multiplied object operating on any 131 00:14:55,890 --> 00:15:01,800 state of SY is simply the result of using the operations in the sequence given. 132 00:15:02,340 --> 00:15:09,900 That is to say, you use you use. Q one up sy first which makes you some cat, which you then use R on, etc. 133 00:15:10,500 --> 00:15:17,430 And when we, if we choose to look at this, if we ask, well, so what is the matrix of our. 134 00:15:17,460 --> 00:15:25,400 Q So what's the matrix of this in sum basis in any basis now it's going to be i. 135 00:15:26,790 --> 00:15:30,690 R What does this mean? 136 00:15:30,700 --> 00:15:46,080 It means our q j and into here we can stick one of our identity operators the sum of M of M. 137 00:15:47,820 --> 00:15:52,950 M. Right. We saw on Friday that this sum is the identity operator. 138 00:15:52,950 --> 00:16:06,030 You can stick an identity operator anywhere into a product. And then this becomes I o hat and m Q Hat. 139 00:16:07,050 --> 00:16:12,380 Oops. J And this now needs a sum of em. 140 00:16:13,310 --> 00:16:21,090 And what is that? This is all I m this is Q MJ So this is just the usual for a matrix product. 141 00:16:21,480 --> 00:16:25,860 So it's ah, I am Q. MJ. 142 00:16:28,650 --> 00:16:36,760 And we will want to know. What the mission adjoint of this thing is. 143 00:16:37,360 --> 00:16:45,130 We want to know what our CU dagger age is. 144 00:16:46,510 --> 00:16:53,110 And so what is that going to be? It's going to be eye our hat. 145 00:16:58,030 --> 00:17:01,269 Do I want to do this? I think I probably don't. 146 00:17:01,270 --> 00:17:06,940 I think probably you've seen this done. I think what I what you've seen is done in the math physics lectures this year. 147 00:17:07,210 --> 00:17:12,910 So I think we can just remind you that this is. Q Hat dagger or hat? 148 00:17:12,910 --> 00:17:18,460 Dagger, right. The when you take the commissioner joint of a product of operators, 149 00:17:18,760 --> 00:17:23,710 you reverse the order of the, of the things in the product and dagger the individual bits. 150 00:17:28,120 --> 00:17:31,390 And I hope you've seen the demonstrate you'll find the demonstration in the book. 151 00:17:31,660 --> 00:17:37,120 If you don't if you haven't seen it, you don't recall the demonstration from Professor Lewis lectures. 152 00:17:38,140 --> 00:17:41,520 And this is all a bit dry and boring, isn't it? Okay. 153 00:17:41,620 --> 00:17:46,120 One thing you may not have seen is functions of operators. 154 00:17:53,030 --> 00:17:57,020 So in particular, for a given example, x, 155 00:17:57,380 --> 00:18:05,510 the position x on the x axis is going to become an operator and we are going to want to evaluate functions 156 00:18:05,510 --> 00:18:13,249 of X like the potential energy at the position x depends upon x and therefore is a function of x. 157 00:18:13,250 --> 00:18:19,910 So in classical physics there is a potential function V of x that tells you the potential energy at the location x. 158 00:18:21,230 --> 00:18:29,360 And since X is going to become an operator, V is going to become an operator, which is obtained by taking a function of an operator. 159 00:18:29,360 --> 00:18:32,480 So we need to know what it means to take a function of an operator. 160 00:18:32,510 --> 00:18:35,780 Another example is this going to be an operator associate with momentum. 161 00:18:36,200 --> 00:18:40,579 The kinetic energy of a particle in classical physics is p squared over two M the momentum 162 00:18:40,580 --> 00:18:44,480 squared of a over twice the mass because that's a half and V squared in classical physics. 163 00:18:45,530 --> 00:18:49,550 So P squared is a function of P very simple one. 164 00:18:49,640 --> 00:18:56,299 It's a function of P. So we need to know what it means to take a function of an operator. 165 00:18:56,300 --> 00:19:03,500 When you do statistical mechanics, you will need to there is there is a there is a quantity, a density operator, 166 00:19:03,980 --> 00:19:10,730 which for which you calculate the entropy of a system which involves a logarithm of of the density. 167 00:19:10,730 --> 00:19:15,530 OPERATOR Right. So you need to take the logarithm of something. So we need to be able to take functions of operators. 168 00:19:15,530 --> 00:19:24,200 So let's, let's decide what this means, right? So what we're going to be done, we're going to imagine we're given F of X, maybe. 169 00:19:25,010 --> 00:19:29,240 So this is just a boring at the moment. This is just a boring number. 170 00:19:30,830 --> 00:19:36,469 Suppose to be given a function. This is a boring number and that's a boring number. 171 00:19:36,470 --> 00:19:43,370 Right? I'm just giving an ordinary function of a complex valued function of a complex valued number. 172 00:19:43,400 --> 00:19:53,299 Say, all right. And let's imagine that we can tailor expand this so we can write this is f nought to value that F takes that nought plus f 173 00:19:53,300 --> 00:20:02,720 one of x the first derivative right plus a half f to x squared over two factorial is the second derivative plus a third. 174 00:20:02,720 --> 00:20:09,410 So write one of three factorial of sixth f3x cubed whoops over three factorial, etc. 175 00:20:09,410 --> 00:20:17,030 Just so we get to imagine that we're going to imagine that our function can be Telesur is expanded in detail. 176 00:20:17,030 --> 00:20:21,950 It might not be possible to expand it around the origin, but then we can expand it around some other place. 177 00:20:22,100 --> 00:20:29,150 In some little neighbourhood, physicists always assume they can expand their functions, and sometimes that leads to major disasters right there. 178 00:20:29,160 --> 00:20:39,530 Important bits of physics which happen only because you can't actually us there series expand everything in life but it's a good starting point. 179 00:20:39,530 --> 00:20:47,419 Okay, so we given this function, now we want to know what F of Q is, right? 180 00:20:47,420 --> 00:20:51,740 What is so what is F of Q? 181 00:20:51,770 --> 00:20:58,030 My the answer to that is this it's the sum of f of q i. 182 00:21:00,160 --> 00:21:03,320 Q i. Q y. 183 00:21:03,800 --> 00:21:10,040 So this is the definition when we say a function of an operator, this is what we mean. 184 00:21:13,260 --> 00:21:21,180 So what is it? This here is an operator which has. So it has the same iGen kits. 185 00:21:25,400 --> 00:21:29,240 As it's argument. All right. So a function takes an argument. The argument is an operator. 186 00:21:29,540 --> 00:21:36,080 This operator has Eigen Katz. So the function of an operator has the same like in cats by construction. 187 00:21:36,770 --> 00:21:45,389 But the eigenvalues. Ah, the given function of the old eigenvalues. 188 00:21:45,390 --> 00:21:48,510 And can you see that this is always, this is guaranteed to work because, 189 00:21:49,050 --> 00:21:59,850 because we started with a function of let's even imagine this is a real valued function of a real value function on the real, on the real variable. 190 00:22:00,180 --> 00:22:03,899 So then this is just going to be some real number for every this will be some real numbers. 191 00:22:03,900 --> 00:22:07,980 So this is a perfectly well-defined thing. But actually it would all work perfectly fine with complex numbers. 192 00:22:08,370 --> 00:22:14,340 Complex valued functions of a complex argument. So this is what we mean by a function of an operator. 193 00:22:19,200 --> 00:22:22,890 I'm going to it's a it's a it's a problem. I mean, I'm leaving it as a problem. 194 00:22:23,640 --> 00:22:34,680 You can now you can now show. So on some problem set, it's a problem to show that this definition is the same as. 195 00:22:39,480 --> 00:22:55,110 F of Q is equal to f nought times the identity plus f one times q plus f two over two Q times q plus. 196 00:22:55,260 --> 00:23:02,729 Right. So if you if you've got the tensor is expansion, then you know what this stuff means, right? 197 00:23:02,730 --> 00:23:07,889 Because we know what it is to multiply an operator on itself. We may not know what it is, take the logarithm of an operator, 198 00:23:07,890 --> 00:23:13,920 but we do know what it is to multiply an operator on itself as many times as it usually will want, because we've defined multiplication of operators. 199 00:23:14,460 --> 00:23:19,260 So this right hand side has a well defined meaning, and you should. 200 00:23:19,680 --> 00:23:24,749 And it's an exercise to prove it's not not desperately difficult to prove that this animal on 201 00:23:24,750 --> 00:23:30,840 the right that we're defining here has as eigenvectors these animals and its eigenvalues, 202 00:23:30,840 --> 00:23:34,290 these animals, and therefore these two definitions coincide. 203 00:23:35,310 --> 00:23:38,730 But this is the more general definition, because this doesn't assume that we can do any. 204 00:23:38,780 --> 00:23:45,689 Taylor is expanding. This does. But when you can do a tiny series expansion or somehow express F in terms of algebra, 205 00:23:45,690 --> 00:23:50,040 which has meaning for operators, which is just to allow which is which is to say only multiplication. 206 00:23:50,040 --> 00:23:56,390 For example, you can't divide one operator by another operator. That doesn't necessarily mean anything, but you can multiply them together. 207 00:23:56,400 --> 00:24:00,570 So when you when this definition works, then this one is the same as this one. 208 00:24:00,570 --> 00:24:04,380 And that's an exercise that I would encourage you to be able to do. 209 00:24:06,400 --> 00:24:08,560 But will not take time to do it now. 210 00:24:12,200 --> 00:24:17,870 Because we're setting up this mathematical apparatus, and I'm sure you're all dying to do a bit of physics, and I am, too. 211 00:24:20,210 --> 00:24:23,420 But we do have to cover a couple of little things here. Comitatus. 212 00:24:25,320 --> 00:24:28,520 Oh, actually, perhaps. Perhaps we should. 213 00:24:28,790 --> 00:24:38,180 We should this time. I moved over here. Okay. 214 00:24:38,290 --> 00:24:46,490 So in some sense, the big news with operators is that a B hat is not necessarily equal to be had a hat. 215 00:24:47,200 --> 00:24:53,380 You know, this already is in as much as, you know, the matrix multiplication doesn't compute generally. 216 00:24:53,980 --> 00:24:58,990 So when you're multiplying matrices together, you don't expect the product this way in the product that way to agree. 217 00:24:58,990 --> 00:25:09,330 And we've agreed that operators once once we take a particular basis, vector system of basis vectors can be represented by matrices. 218 00:25:09,340 --> 00:25:12,580 So it's not surprising that there is this non-core mutability. 219 00:25:12,850 --> 00:25:15,700 And the elementary techs claim this is the key thing about quantum mechanics. 220 00:25:15,700 --> 00:25:18,880 They claim this is not the key thing about quantum mechanics, non commuting. 221 00:25:19,060 --> 00:25:25,270 Things occur also in classical physics. And we'll see we'll we'll see that concretely as we go down the line. 222 00:25:25,780 --> 00:25:37,120 However, it is a fact that these operators do not commute. And we we spend a great deal of time calculating this animal, which is which is AB minus B. 223 00:25:38,770 --> 00:25:43,810 Okay. So the definition of A and B, A comma B in a square bracket is that it means just this. 224 00:25:45,370 --> 00:25:48,819 Now we have some obvious results. 225 00:25:48,820 --> 00:26:01,450 We have that A comma, B plus C, the comitato of A with B and C, the result of adding B to C is clearly the sum from this definition. 226 00:26:01,450 --> 00:26:14,180 It follows that it is just this sum. The little one. 227 00:26:15,770 --> 00:26:23,570 Oh, yeah. We have this obvious result that Abby is able to be a plus, a comma, B, 228 00:26:25,850 --> 00:26:33,799 one of the reasons why we need to know the value as you will see why we need to know the value of a commentator is because we often need to swap. 229 00:26:33,800 --> 00:26:38,750 We need to want to whatever. We often want to swap the order in which operators are around. 230 00:26:39,050 --> 00:26:43,670 And the way to do it is to write that ap b.a plus this comitato, which is obviously true either way. 231 00:26:43,670 --> 00:26:49,070 I think if it is, this adds in the thing that I should have had and takes away the thing that I've put in that I'm not entitled to have. 232 00:26:49,340 --> 00:26:56,660 But it's obvious, right? And now finally, a less obvious result, which is that a B, 233 00:26:57,260 --> 00:27:09,319 the product de B commuted with C is equal to A comma C with B standing by on the outside of the COMITATO plus excuse me, 234 00:27:09,320 --> 00:27:14,960 plus A with C, comma B like this. 235 00:27:17,130 --> 00:27:21,150 It's easy to prove this. I encourage you to prove it. I'm not going to take time to do it. 236 00:27:21,480 --> 00:27:32,010 All you have to do is write down what this is from that definition and then insert two extra terms which cancel each other and you will. 237 00:27:32,010 --> 00:27:37,469 You'll find you can arrange it like this. I would say it should be become a see you. 238 00:27:37,470 --> 00:27:42,870 Absolutely right. Thank you very much for that. The other one I got, right? 239 00:27:42,870 --> 00:27:46,560 Yeah. Okay. So what is this analogous to? 240 00:27:46,770 --> 00:27:50,430 This is analogous to D by D C of a B. 241 00:27:50,910 --> 00:27:58,110 If I have to do a differential of a product with respect to C, then that is equal to the A by de c. 242 00:27:58,650 --> 00:28:01,650 B. Plus a. D. 243 00:28:01,650 --> 00:28:06,180 B. D. C. Right. 244 00:28:06,690 --> 00:28:09,600 This is the rule for differentiating a product. And can you see the mirror there? 245 00:28:09,960 --> 00:28:20,070 The idea is that taking the comitato of something with C is analogous to taking the derivative of something with C and this is no accident. 246 00:28:20,340 --> 00:28:24,270 This, for a mathematician in certain contexts is called a derivative. 247 00:28:27,150 --> 00:28:38,280 And the and the and the rules that we are familiar with here is that you, you, you, first of all, if you have a product, 248 00:28:38,280 --> 00:28:43,890 you can get the result by having this operation happen on the first thing while the second stands idly by. 249 00:28:44,310 --> 00:28:49,950 And then you have two. You let the first one ads stand idly by and then you work on the second one. 250 00:28:49,950 --> 00:28:57,120 So here we have you work on the first one second standing idly by, and then you work on the second one with the first one standing idly by. 251 00:28:57,720 --> 00:29:03,360 The only material difference between these formulae is that this formula is left invariant. 252 00:29:03,360 --> 00:29:08,340 If I move B over here, or if I move over there or whatever, I change the order here. 253 00:29:08,580 --> 00:29:12,420 It won't make any difference because these are ordinary, boring multiplications of complex numbers. 254 00:29:12,780 --> 00:29:17,729 But here it is. It does make a difference like this. A comma C is an operator. 255 00:29:17,730 --> 00:29:21,460 It's the difference of two operators. So it's an operator. And, 256 00:29:21,800 --> 00:29:25,950 and therefore it isn't clear that I can swap the order of this operator in 257 00:29:25,950 --> 00:29:28,650 this operator in the order in which you write these things down is important. 258 00:29:30,460 --> 00:29:36,460 So these these rules should be kind of should be you should make sure you understand where they come from. 259 00:29:36,640 --> 00:29:37,840 You should memorise them. 260 00:29:38,440 --> 00:29:48,910 And broadly speaking, once you once you've got these three rules on board, you never need to look inside a commentator and use this relationship here. 261 00:29:49,180 --> 00:29:55,180 It's bad practice, by and large, when you're doing computations to expand commentators to see what's inside them. 262 00:29:55,450 --> 00:30:05,890 In the same way, I would say as this rule here of course, comes from looking at A, B evaluated at C plus delta C minus A, 263 00:30:05,920 --> 00:30:12,760 B evaluated at C, all over delta C limit, all this stuff, you know, using this stuff, you can prove this. 264 00:30:13,390 --> 00:30:19,390 But we once you've got the rules of calculus, you don't do this expanding stuff anymore. 265 00:30:19,600 --> 00:30:23,230 You just, you know, that's what lies underneath it. That's the justification. 266 00:30:24,190 --> 00:30:31,750 But you don't go back to that every time you have to do a calculation, every time you have to differentiate the x of x cubed, 267 00:30:31,960 --> 00:30:38,980 you do not write that this is x plus delta x cubed minus minus x cubed all over 268 00:30:38,980 --> 00:30:43,300 delta x cubed and come to the conclusion that it's about three x squared to you. 269 00:30:45,160 --> 00:30:52,600 So please don't resist the temptation to to expand out to commentators, write the contents of a commentator out. 270 00:30:53,140 --> 00:31:00,160 There are times when ultimately you have to do that, but most of the time you don't and try and avoid doing it by using these rules here. 271 00:31:01,300 --> 00:31:04,100 Okay. Okay. 272 00:31:04,240 --> 00:31:18,970 I'm going to need one result which combines these statements and those statements we're going to need very shortly to calculate what F, B, comma A is. 273 00:31:19,090 --> 00:31:23,499 So I've got it. So I will want the commentator concretely. 274 00:31:23,500 --> 00:31:29,559 This is going to be V of X and I'm going to want to take the commentator with the momentum operator and these things. 275 00:31:29,560 --> 00:31:34,990 These all need hats, I suppose. Yeah. And those things up there need hats, but you're managing them all. 276 00:31:35,890 --> 00:31:39,040 So this is I'm going to want to, I'm going to want to calculate something like this. 277 00:31:39,040 --> 00:31:42,249 So let's see what this comes to. In order to see what it comes to, 278 00:31:42,250 --> 00:31:50,470 I'm going to imagine that I can expand F in this manner so I can write this as a f nought 279 00:31:52,900 --> 00:32:03,940 times the identity plus f one times B plus f two over two times B squared plus blah blah. 280 00:32:03,970 --> 00:32:13,870 Right. The title sir is expansion of F around the origin commuted with a so now I can use 281 00:32:14,260 --> 00:32:18,820 that second rule there that second rule to do the commentator of this product. 282 00:32:19,180 --> 00:32:24,280 The comment this is a boring number, right? This is a number and this is the identity operator. 283 00:32:24,280 --> 00:32:26,980 So this isn't a number. That's a number, but this is the identity operator. 284 00:32:27,190 --> 00:32:33,429 And the identity operator obviously commutes with everybody because I times a is going to 285 00:32:33,430 --> 00:32:40,209 be this is going to be a same as a times I is going to be a so the commentator so so I 286 00:32:40,210 --> 00:32:44,530 use the second rule to say that the commentator of this sum with a is the sum of the 287 00:32:44,530 --> 00:32:51,940 commentators of this thing with a vanishes and this thing with a so that's going to be F one, 288 00:32:53,560 --> 00:32:58,840 B hat, comma, a hat. This comes outside the comitatus. 289 00:32:58,840 --> 00:33:02,620 Or maybe I should have added that to the rule list there because it's a boring number. 290 00:33:05,950 --> 00:33:18,399 But I think it's it's kind of an obvious principle plus f two over two factorial a B squared comma a plus 291 00:33:18,400 --> 00:33:28,750 F three over three factorial B cubed comma a plus plus plus plus plus plus plus right until your board. 292 00:33:31,360 --> 00:33:38,340 So that's that's the middle rule used. Now we use the last rule to say that this is f one. 293 00:33:38,350 --> 00:33:58,030 Well this is, this is just to repeat, but this B squared is B times B so I can expand this into F two it into B hat, comma, a hat, b hat plus. 294 00:33:58,630 --> 00:34:06,280 Right? So it was b b commuting with a so I worked on the first B while the second B stood idly by and now 295 00:34:06,280 --> 00:34:15,099 I have to put down the first B standing idly by and have the second B worked on by a plus dot, 296 00:34:15,100 --> 00:34:18,090 dot, dot, plus F three, etc. Right. 297 00:34:18,100 --> 00:34:24,160 Which is going to involve three terms because it'll be b b b committee with a, so it'll be three things to consider. 298 00:34:27,680 --> 00:34:42,770 And this is as far as I can go in general. But in an important case, if the hat A hat commutes with B. 299 00:34:44,150 --> 00:34:53,300 So if this. Commentator So be hat a hat commentator is an operator. 300 00:34:54,170 --> 00:34:55,640 This is the difference in two operators. 301 00:34:56,390 --> 00:35:07,220 So if this operator commutes with B hat, then this B camera and this B camera and this one could all be taken outside. 302 00:35:07,640 --> 00:35:32,900 And I have that. So under this condition, that F of B hat commuting with a hat is equal to B hat, comma, a hat times F one plus, f two plus. 303 00:35:33,050 --> 00:35:40,460 And you can, you see it will be F three over two because the f three would have been over three factorial, but we would have had three terms. 304 00:35:40,940 --> 00:35:45,830 Oh, sorry. This is going to be times B this silly me, this is going to be times b hat. 305 00:35:46,640 --> 00:36:03,700 This is going to be times B squared plus. So this is what this will all reduce to, which can be more conveniently written as as d f by DP. 306 00:36:09,910 --> 00:36:14,440 So this is an operator. Oops. Sorry. Yeah. 307 00:36:14,460 --> 00:36:22,720 It doesn't matter which order I put it in. This is an operator. And that Taylor series is the Taylor series for Def by the x. 308 00:36:23,920 --> 00:36:29,740 So I can write this stuff. Here is the F by D, B, and then here is my B camera. 309 00:36:29,980 --> 00:36:34,450 And I was momentarily panicked about having written this in front of this. 310 00:36:34,690 --> 00:36:41,170 But we've agreed that this operator computes with B, that was the condition under which we were making this further development. 311 00:36:42,070 --> 00:36:47,709 And if this thing commutes with B, it commutes with every function of B in particular, 312 00:36:47,710 --> 00:36:52,970 it commutes with the F IDB, which is a function of B, so it doesn't matter which order I put this in. 313 00:36:52,990 --> 00:37:01,660 So this is a function. Which means it has the same I can cats. 314 00:37:07,930 --> 00:37:13,390 So that's a result we're going to want. And there's one other thing that now needs to be discussed. 315 00:37:16,720 --> 00:37:24,910 Which is the physical implications of a commuting would be so if they had a B hat. 316 00:37:27,440 --> 00:37:32,300 Equals nought. We say commuting observables. 317 00:37:41,270 --> 00:37:44,299 Then the mathematicians assure us we have a theorem. 318 00:37:44,300 --> 00:37:48,320 And the theorem is that in this case. 319 00:37:51,530 --> 00:37:54,620 There is a complete set. 320 00:37:59,280 --> 00:38:11,690 Of mutual aid cats. We'll call these mutual Asian cats. 321 00:38:11,720 --> 00:38:15,770 Just I. That is to say, for each and every one of these, 322 00:38:15,770 --> 00:38:33,320 it is true that a hat on I is equal to a i i and simultaneously be hat on I is equal to some number by when I when to operate is commute. 323 00:38:33,410 --> 00:38:37,130 There's a theorem that states this. What does that mean for the real physical world? 324 00:38:37,820 --> 00:38:42,830 What that says in the real for the real physical world is there is a complete set of states, 325 00:38:44,090 --> 00:38:50,749 these states in which the result of making a measurement of a is definitely known 326 00:38:50,750 --> 00:38:55,700 and simultaneously the result of making it a measurement of B is certainly known. 327 00:38:55,700 --> 00:39:00,260 So there is a complete set of states in which there is no ambiguity. 328 00:39:00,290 --> 00:39:04,070 There is nothing probabilistic about the result of measuring either of these quantities. 329 00:39:05,810 --> 00:39:14,720 It's very important to bear in mind that in complete, we're not merely saying that there is a state or ten states with this property. 330 00:39:15,050 --> 00:39:22,430 There are enough states with this property that any state can be written as a linear combination. 331 00:39:23,660 --> 00:39:27,260 Whatever GI of these objects right there complete. 332 00:39:27,980 --> 00:39:42,230 That's what completeness means that any state. So there is a complete set of states in which there's absolute certainty. 333 00:39:42,440 --> 00:39:44,750 It does not mean that. 334 00:39:47,950 --> 00:39:58,220 The fact that I definitely there's no uncertainty in the value that B takes implies that there's no uncertainty in the value that takes. 335 00:39:58,460 --> 00:40:03,560 That is not that does not follow from the commuting of A and B as we will see it. 336 00:40:04,790 --> 00:40:13,250 It may well be the case is that there are estates in which B definitely has a value, for which A, the outcome of measurement of A is uncertain. 337 00:40:17,400 --> 00:40:26,740 So it's so the the result of two observables commuting their operators commuting is slightly technical because it involves this complete statement. 338 00:40:26,760 --> 00:40:34,140 It is that there is a complete set of states in which the outcomes of the measurements of both observables are certain. 339 00:40:35,310 --> 00:40:45,690 Okay. Now if a comma be not equal to zero, what does this mean? 340 00:40:45,750 --> 00:40:53,940 All it means is that there is at least one cat. 341 00:40:56,400 --> 00:41:00,810 Such that a comma be. 342 00:41:05,440 --> 00:41:11,350 There may be an infinite number of cats such that a bee operates on them and produces nothing. 343 00:41:12,160 --> 00:41:16,000 But there is one. There is at least one. If you say that these operators don't compute, 344 00:41:16,000 --> 00:41:23,440 you're saying or asserting that there is at least one cat where the commentator operating on it doesn't produce nothing. 345 00:41:26,150 --> 00:41:30,470 So it. If so, what does this imply? 346 00:41:30,500 --> 00:41:35,630 It implies that there is no complete. 347 00:42:04,210 --> 00:42:13,570 So it's a very weak and not emotionally striking result that there just isn't a complete set of states in which they're both. 348 00:42:14,380 --> 00:42:20,490 They both have definite values. There may be a very large number of states in which they do have definite values simultaneously. 349 00:42:20,500 --> 00:42:24,280 So it is not a statement that you can't know the value of this simultaneously. 350 00:42:24,280 --> 00:42:30,640 With the value of that, we'll come across a counterexample next term, I guess a very important counterexample. 351 00:42:31,450 --> 00:42:38,349 So don't run away. It's a very it's a very, very widely held misconception that if two operators don't commute, 352 00:42:38,350 --> 00:42:41,370 you can't know the value of the one and the value of the other. 353 00:42:41,380 --> 00:42:46,870 That's just not true. The statement is that there isn't a complete set of states with that nice property. 354 00:42:55,390 --> 00:43:04,900 Okay. We've just got time to start on the next really important section, which is about time evolution. 355 00:43:05,350 --> 00:43:14,630 Maybe. Maybe it's time to move over here. Okay. 356 00:43:14,690 --> 00:43:18,620 So physics is about prophecy. It's prophecy that works. It's about predicting the future. 357 00:43:18,620 --> 00:43:26,689 That's what it's about. And therefore the core of it is equations of motion, Newtonian mechanics. 358 00:43:26,690 --> 00:43:29,960 We think of usually as to do with F equals Emma. 359 00:43:30,230 --> 00:43:34,490 It's, it's making it seem to what the acceleration is when you can calculate the acceleration and you know the, 360 00:43:34,940 --> 00:43:36,769 and you know the initial position and velocity, 361 00:43:36,770 --> 00:43:42,020 you can predict where your by your missile is going to be at some future time or your planet is going to be at some future time and so on. 362 00:43:42,020 --> 00:43:47,960 Right? That's what it's all about. So at the core of quantum mechanics sits its time evolution equation. 363 00:43:48,290 --> 00:43:53,180 And I'm not going to immediately justify this, I'm just going to write it down. 364 00:43:53,540 --> 00:43:56,570 It's the time. What's the time? 365 00:43:58,370 --> 00:44:03,180 Dependent. Shredding a. 366 00:44:09,690 --> 00:44:21,540 This is the core of the subject. This is where the physics sits and it's high bar what I call the sci fi d t is equal to h upside. 367 00:44:22,680 --> 00:44:28,410 This is why it's because it appears in this central, crucial, vital equation. 368 00:44:29,010 --> 00:44:32,400 The Hamiltonian sits here. That's why the Hamiltonian matters. 369 00:44:32,790 --> 00:44:39,570 Right? Its status in life is unique because it uniquely tells you about the future. 370 00:44:39,720 --> 00:44:43,830 And that's what physics is about. Okay. 371 00:44:44,280 --> 00:44:50,500 And this is the state. Any system. 372 00:44:54,760 --> 00:45:02,650 So it's completely non-negotiable for a state which purports to describe a real physical object. 373 00:45:04,410 --> 00:45:09,570 It has to satisfy this equation. It tells you how the state evolves in time. 374 00:45:09,600 --> 00:45:13,590 It's, of course, a very abstract object at the moment. It won't be telling you much. 375 00:45:14,000 --> 00:45:18,360 And at the moment, I can't connect it. 376 00:45:18,510 --> 00:45:24,620 We will be connecting it very shortly. But just at the moment, I can't connect this for most of you to classical mechanics. 377 00:45:24,630 --> 00:45:34,860 Those of you who've done did the seven short option will recognise this perhaps just a little bit as having something to do with Hamilton's equations. 378 00:45:36,600 --> 00:45:41,459 But we will. So the justification, the physical justification is that this is the dominant equation. 379 00:45:41,460 --> 00:45:47,280 Will, will, will come by and by. But ultimately, there's no way this can be derived from anything you already know. 380 00:45:47,280 --> 00:45:49,470 This cannot be derived out of classical physics. 381 00:45:49,770 --> 00:45:55,770 Classical physics can be derived out of this because classical physics provides an approximation to this. 382 00:45:56,370 --> 00:46:02,579 Right. And the assertion is that nature evolves things according to this equation. 383 00:46:02,580 --> 00:46:08,180 And whether that's true or not can only be determined by experiment. It's got nothing to do with mathematics, and it's got nothing. 384 00:46:08,190 --> 00:46:11,310 And it cannot be justified on the basis of classical physics, ultimately. 385 00:46:11,580 --> 00:46:17,879 But if this is a valid statement, it should it should produce the right Newtonian equations of motion. 386 00:46:17,880 --> 00:46:23,760 I will show you that it does produce the right Newtonian equations of motion, because Newtonian mechanics is an approximation to quantum mechanics. 387 00:46:23,880 --> 00:46:29,170 Right. Okay. 388 00:46:29,330 --> 00:46:32,500 Now, suppose let's. This is. 389 00:46:32,710 --> 00:46:36,610 This is kind of a scary equation, right? So let's. 390 00:46:36,940 --> 00:46:39,960 Let's try and find some circumstance in which we can solve this. 391 00:46:39,970 --> 00:46:44,950 Right. So suppose our system has well-defined energy. 392 00:46:54,120 --> 00:46:59,730 In other words, the state of sci at time t. 393 00:47:00,360 --> 00:47:10,340 Well, the state of upside is equal to d where h e is equal e. 394 00:47:11,700 --> 00:47:16,679 Write a state of well-defined energy has to be an eigen function of the energy operator. 395 00:47:16,680 --> 00:47:22,020 H with eigenvalue e. That's that's what it is. 396 00:47:23,010 --> 00:47:28,110 So let's suppose that we in our system happens to have well-defined energy. 397 00:47:29,580 --> 00:47:43,440 Then it will then it will have to solve this equation and we'll have each by the e by d t is equal to g h e is equal to e e. 398 00:47:44,910 --> 00:47:51,480 So the rate of change of e is simply proportional to E, and we know how to solve that equation. 399 00:47:52,530 --> 00:47:55,560 We spot it just from ordinary, old fashioned calculus. 400 00:47:55,800 --> 00:48:07,770 We spot that this implies that e at time t is equal to e to the minus i e t over bar e of zero. 401 00:48:12,130 --> 00:48:18,010 So I feel I feel entitled to write this down on the basis of just boring classical classical mathematics, 402 00:48:18,010 --> 00:48:24,410 which says that, that if we know that the X by d t, no, I shouldn't do it there. 403 00:48:24,460 --> 00:48:37,750 If I know that the x by d t where x is some variable is equal to x, that implies that x of t is equal to x of nought e to the. 404 00:48:38,310 --> 00:48:46,870 Right. So this result for many a result inspires me to write down that I can now trivially check 405 00:48:46,870 --> 00:48:51,310 by differentiating this right hand side that it satisfies this differential equation. 406 00:48:51,340 --> 00:48:56,649 Right. Because when I because when I differentiate this right hand side, this thing is not a function of time. 407 00:48:56,650 --> 00:49:00,580 It's the it's the value that the state of well-defined energy takes a time t equals nought. 408 00:49:00,580 --> 00:49:08,350 So it has no time derivative. So the time derivative comes merely from this, which is a totally boring exponential of a bunch of real numbers. 409 00:49:08,390 --> 00:49:12,400 Well, with part from the AI. All right, so we know how to differentiate this. 410 00:49:12,700 --> 00:49:16,959 So it's easy to evaluate the time derivative of this, and it's trivial to check that. 411 00:49:16,960 --> 00:49:20,980 Then it's that. Then e satisfies this equation. So what does this tell us? 412 00:49:23,230 --> 00:49:31,570 This is a very important result. It tells us that the time evolution of states of well-defined energy is really dead trivial. 413 00:49:32,530 --> 00:49:34,209 They basically don't change. 414 00:49:34,210 --> 00:49:41,950 All that happens is the phase goes around in increments at a constant rate, each over each bar with a frequency of reach bar, 415 00:49:41,950 --> 00:49:49,120 which is, of course, incredibly for typical systems like this is incredibly large because bar is so small, it's on the bottom there. 416 00:49:49,120 --> 00:49:51,550 So this frequency is stupendous for an object like that. 417 00:49:52,120 --> 00:49:57,400 So this thing has some energy and its wave function is zooming around at some hysterical rate. 418 00:49:58,330 --> 00:49:59,140 That's all that's happening. 419 00:50:02,720 --> 00:50:13,110 The beautiful thing is that this enables us to solve the general problem, because if I have if I have a CI, I want to solve this. 420 00:50:13,110 --> 00:50:18,950 So I've got now some system that's not in a state of well-defined energy. And we'll see that real systems never are in states of well-defined energy. 421 00:50:19,280 --> 00:50:27,860 But then I can surely write this as a linear combination with coefficients that depend on time of states of well-defined energy. 422 00:50:29,420 --> 00:50:34,370 Right? These are a complete set of states because they're. Yeah, we've been through this. 423 00:50:34,370 --> 00:50:35,360 This is just boring, right? 424 00:50:36,020 --> 00:50:46,280 So I can put I simply put this and that's this expression, this expansion into both sites of my time dependent Schrodinger equation. 425 00:50:46,520 --> 00:50:55,879 And we discover that, that we discover that h bar the up side by d t is equal to h bar brackets. 426 00:50:55,880 --> 00:51:10,130 We have to differentiate this stuff so it's a and dot e and t plus a and times the time derivative of 427 00:51:10,130 --> 00:51:28,130 this times key and by d t was that equal to that's equal to on this side h enter the sum and e n. 428 00:51:32,360 --> 00:51:35,750 I've missed some of her. And indeed I have. I missed out a son over. Thank you. 429 00:51:36,110 --> 00:51:40,580 Just about here. I'm kind of conscious of that horrible clock. 430 00:51:45,590 --> 00:51:52,270 And. But, uh. Well, okay, why don't we just write this? 431 00:51:52,280 --> 00:52:00,110 Why don't we just write carry this on and write this as a sum over n of a and h e n. 432 00:52:02,460 --> 00:52:06,150 But. This term. 433 00:52:06,150 --> 00:52:22,740 This term here cancels this term here i hpr and so far d n by d t is h e n so these terms all cancel those terms leading to the conclusion. 434 00:52:27,040 --> 00:52:31,030 So when I when I look at this stuff is equal to this stuff. 435 00:52:31,660 --> 00:52:37,330 I've cancelled this to the right side now. So there's nothing. And the left side has this stuff has a dot. 436 00:52:37,690 --> 00:52:52,630 So I've got the conclusion that the sum of N of a and dot e and of t equals nought bra through with an e i of t. 437 00:52:52,840 --> 00:52:56,950 And that leads to the conclusion that I. Dot equals nought. 438 00:53:00,210 --> 00:53:03,210 So the AI's. The AI a constant. 439 00:53:04,050 --> 00:53:07,110 So we have a solution. This enables us to. 440 00:53:07,440 --> 00:53:09,990 To write down the solution to the general problem. 441 00:53:10,230 --> 00:53:22,090 We have that upside of t is equal to the sum of some constants and which you can determine from the initial conditions times e. 442 00:53:22,110 --> 00:53:28,020 N of t. But I can explicitly write that out because I know how this thing evolves in time. 443 00:53:28,320 --> 00:53:34,860 This is the sum and of nought e to the minus i e and of t e n. 444 00:53:34,860 --> 00:53:38,910 T of ball times. N of nought. 445 00:53:40,170 --> 00:53:44,310 So once. So this is the really, really this is a fabulously important equation. 446 00:53:44,700 --> 00:53:48,930 So this part of it is needs to be purchased the back of the retina. 447 00:53:49,440 --> 00:53:54,450 And it it's the key to everything because it tell and what it tells us is once we know what these states, 448 00:53:54,450 --> 00:54:03,500 the well-defined energy are and the approved energies, we can trivially evolve in time the dynamical state of our system and predict the future. 449 00:54:03,510 --> 00:54:12,690 We have everything. That's it. So a large part, a huge part of this subject revolves around finding what these states of well-defined energy are, 450 00:54:12,870 --> 00:54:16,020 because they have this enormous predictive power. 451 00:54:16,030 --> 00:54:20,219 They are the miracle. They are sort of one to drug. 452 00:54:20,220 --> 00:54:25,140 They solve the problem, they do it. So we'll talk some more about them tomorrow.