1 00:00:00,230 --> 00:00:05,312 In this video, I'm going to complete the most technically demanding part of the 2 00:00:05,312 --> 00:00:10,548 derivation of a path integral of our presentation for the propagator that was 3 00:00:10,548 --> 00:00:16,956 introduced in the previous segment. And here again I show the expression for 4 00:00:16,956 --> 00:00:22,548 this propagator that we derived. So in, in the following I'm going to be 5 00:00:22,548 --> 00:00:27,067 doing my calculations mostly in one dimension for the sake of simplicity. 6 00:00:27,068 --> 00:00:30,986 There is no real reason to go to one dimension so the calculation sort of goes 7 00:00:30,986 --> 00:00:35,171 the same way in two dimensions or three dimensions or 100 dimensions, it's just 8 00:00:35,171 --> 00:00:39,604 going to one dimensions is just simplifies notation so it'll be, we don't have to 9 00:00:39,604 --> 00:00:44,568 deal with vector indices. And also it will allow me to illustrate 10 00:00:44,568 --> 00:00:49,230 certain steps of our derivation in a sort of an intuitive vectorial way. 11 00:00:49,230 --> 00:00:54,730 So, the second comment I would like to make is that what we call the propagator 12 00:00:54,730 --> 00:01:00,634 also appears in the literature under the name of transition amplitude, well, from 13 00:01:00,634 --> 00:01:06,374 an initial point xi to final point xf, in this case, and a very closely related 14 00:01:06,374 --> 00:01:13,113 object is also called the Green function. So if you see these expressions in the 15 00:01:13,113 --> 00:01:19,047 literature, they may be actually very much related to one another, and all refer to 16 00:01:19,047 --> 00:01:23,551 this object [unknown]. Now, the main difficulty in calculating 17 00:01:23,551 --> 00:01:28,519 this object once again, is the presence of this evolution operator, which is an 18 00:01:28,519 --> 00:01:33,140 exponential of [unknown]. And to calculate this exponential of an 19 00:01:33,140 --> 00:01:38,728 operator is a very tricky business. So basically the goal of the remaining 20 00:01:38,728 --> 00:01:43,348 calculation, as we discussed, is going to be to simplify this expression so that it 21 00:01:43,348 --> 00:01:47,208 doesn't really contain any operators. And we deal with sort of regular 22 00:01:47,208 --> 00:01:52,394 mathematical quantities, not operators. And in doing this simplification we're 23 00:01:52,394 --> 00:01:57,213 going to rely on two very useful mathematical, or physical actually, in 24 00:01:57,213 --> 00:02:01,912 this case, two formulas. So, one of them, the first one listed 25 00:02:01,912 --> 00:02:08,660 here, we already have actually discussed in the second lecture last week. 26 00:02:08,660 --> 00:02:12,530 So, this is the so called resolution of the identity operator. 27 00:02:12,530 --> 00:02:18,362 And the fact that it is the identity operator means that so let's see if we 28 00:02:18,362 --> 00:02:23,627 have an arbitrary wave-function of psi, and we have to by this operator on this 29 00:02:23,627 --> 00:02:29,230 wave-function of psi. It is psi, so what we're going to get, is 30 00:02:29,230 --> 00:02:38,539 the full length, x of psi, this matrix elements, times the [unknown] x, dx. 31 00:02:38,540 --> 00:02:43,079 And this is identically equal to the psi itself. 32 00:02:43,080 --> 00:02:48,832 So this is the formula number 1. Another equation which is going to be very 33 00:02:48,832 --> 00:02:52,137 useful, actually, it's a property of the evolution operator. 34 00:02:52,138 --> 00:02:54,400 It actually relies on the very simple observation. 35 00:02:54,400 --> 00:02:59,636 So, the evolution operator itself translates the wave-function from the 36 00:02:59,636 --> 00:03:03,417 initial moment of time. So, this is time and this is, let's say, 37 00:03:03,417 --> 00:03:06,900 time equals 0. So, to the wave-function psi of t, at 38 00:03:06,900 --> 00:03:10,092 time, at the final moment of time t equals t. 39 00:03:10,092 --> 00:03:16,830 So but instead of going directly from 0 to t, we can do that in sort of two steps. 40 00:03:16,830 --> 00:03:22,294 We first can go to, let's say, certain t1, which is between 0 and t. 41 00:03:22,295 --> 00:03:27,524 And then go from t1 to t. So and you know, going from t1 to t at the 42 00:03:27,524 --> 00:03:32,564 response here in the language of this resolution operator is applying this 43 00:03:32,564 --> 00:03:38,004 operator with a h, which depends on t minus t1 to the wave-function that we have 44 00:03:38,004 --> 00:03:41,804 here in the middle. So, on the other hand, the wave-function 45 00:03:41,804 --> 00:03:46,196 in the middle on the pdi of t1 itself can be represented as the action of the 46 00:03:46,196 --> 00:03:50,440 resolution operator U of t1 on the original wave-function of psi of 0. 47 00:03:50,440 --> 00:03:55,984 So, therefore what we derive here is that the evolution operator U of t can be 48 00:03:55,984 --> 00:04:01,222 written as the product of two evolution operators U of t minus t1 times U of t1. 49 00:04:01,222 --> 00:04:08,122 Now so the particularly simple way of splitting the time interval into two, 50 00:04:08,122 --> 00:04:12,087 let's, in this case, is just dividing it in half. 51 00:04:12,087 --> 00:04:18,711 So, let's say again, so we have this time axis of 0 t, and we just divide it into 52 00:04:18,711 --> 00:04:24,390 two pieces, so from 0 to t2 and t2 to t. And so therefore, we have the expression 53 00:04:24,390 --> 00:04:29,110 for the evolution operator U of t, is equal to U of t over 2 times U of t over 54 00:04:29,110 --> 00:04:33,101 2. So now if we're going to use the well, if 55 00:04:33,101 --> 00:04:39,309 we're going to real, if we're going to focus on the actual propagator, which is 56 00:04:39,309 --> 00:04:46,002 the matrix element of this evolution operator between the initial point xi and 57 00:04:46,002 --> 00:04:51,628 the final point xf, what we can do, we can, well we can write it as xf U of t 58 00:04:51,628 --> 00:04:57,739 over 2, U of t over 2, xi and insert here the resolution of the identity that we 59 00:04:57,739 --> 00:05:03,318 discussed on the previous slide. So, if we do so, we get to this 60 00:05:03,318 --> 00:05:12,504 expression, which essentially splits the original propagator into two propagators. 61 00:05:12,504 --> 00:05:19,703 One goes from xi to x, and the other goes from point x to the final point x sub f. 62 00:05:19,703 --> 00:05:24,624 This splitting of the regional propagator into two allows a very useful and 63 00:05:24,624 --> 00:05:30,552 intuitive, illustration that I'm going to present now which will actually bring us 64 00:05:30,552 --> 00:05:36,457 closer to the notion of a path integral. So, let's say this is my coordinate x and 65 00:05:36,457 --> 00:05:41,906 this is time t. So and this is my 0 and final time t. 66 00:05:41,906 --> 00:05:51,573 This is going to be my xi and xf. And so, what I'm interested in is 67 00:05:51,574 --> 00:05:59,399 calculating the propagator of going from the initial moment of time, xi, to this xf 68 00:05:59,399 --> 00:06:03,191 at t equals t. So, this is essentially what I'm 69 00:06:03,191 --> 00:06:09,236 calculating, and this sort of solid line connecting the two points sort of 70 00:06:09,236 --> 00:06:15,609 corresponds to my G of xi, xf, t. Now, this equation tells me that I can 71 00:06:15,609 --> 00:06:20,845 take, well, the point in the middle. So, this is going to be my t over 2 and 72 00:06:20,845 --> 00:06:25,005 what I have to do in order to calculate this propagator, I have to consider all 73 00:06:25,005 --> 00:06:29,935 possible coordinates in between. So, let's say, I will go from xi to x and 74 00:06:29,935 --> 00:06:34,436 from x to xf. This is going to be one path. 75 00:06:34,436 --> 00:06:41,244 Another path is going to be to go to a different x, like as so, etc., etc. 76 00:06:41,244 --> 00:06:46,878 So, this solid lines are going to be sort of representing you know, the first and 77 00:06:46,878 --> 00:06:53,121 the second propagator in this interval. And well, obviously this procedure of 78 00:06:53,121 --> 00:06:59,358 splitting the original time, time interval and position interval into two can be 79 00:06:59,358 --> 00:07:01,950 continued. So, what I can do, I can keep on 80 00:07:01,950 --> 00:07:06,920 truncating my, my time interval into smaller and smaller pieces, and what I'm, 81 00:07:06,920 --> 00:07:11,610 what I will continue doing so I will have more and more intermediate coordinates 82 00:07:11,610 --> 00:07:16,580 that are going to appear, and therefore, I will generate all possible trajectories 83 00:07:16,580 --> 00:07:21,152 that are going to cover the entire space. And this is how path integral or integral 84 00:07:21,152 --> 00:07:25,844 over trajectories really appears, but at this, at the level of this illustration, 85 00:07:25,844 --> 00:07:30,196 this path integral, this picture is not very useful for sort of practical 86 00:07:30,196 --> 00:07:33,508 purposes. To make some progress here, so let me sort 87 00:07:33,508 --> 00:07:38,188 of formalize what I just said and what I said is simply the fact that, let me just 88 00:07:38,188 --> 00:07:43,084 split, instead of splitting my time interval into 2 pieces, let it, let's, let 89 00:07:43,084 --> 00:07:47,310 me split it into n pieces and so an n here is going to be very, very large. 90 00:07:47,310 --> 00:07:54,017 So, that t over n, the each individual at the time interval is extremely small. 91 00:07:54,018 --> 00:07:59,497 And so, each of this small time interval was the evolution operator on each of this 92 00:07:59,497 --> 00:08:04,425 time intervals will response to this exponential, either the bar i or H bar H t 93 00:08:04,425 --> 00:08:07,908 over N. So this t, t over n, I can make as small 94 00:08:07,908 --> 00:08:11,374 as I want. So, and if I make it really, really small, 95 00:08:11,374 --> 00:08:16,588 so, instead of writing the full Taylor series for this exponential, I can write 96 00:08:16,588 --> 00:08:20,500 it as simply two terms, one minus the first linear term. 97 00:08:20,500 --> 00:08:26,322 And just, I try and take my [unknown] right here and so basically, approximate 98 00:08:26,322 --> 00:08:32,208 my evolution operator in each of this dimension intervals as, as so. 99 00:08:32,208 --> 00:08:37,554 So, as a matter of fact, you can say, that you know, I shouldn't have done this 100 00:08:37,554 --> 00:08:42,284 deviation. So, because at the end of the day, what I 101 00:08:42,284 --> 00:08:49,874 have actually derived is that either the power minus i over h bar, h Hamiltonian 102 00:08:49,874 --> 00:08:56,584 times t is equal to 1 minus I over H bar H t over N to the power N, where N goes to 103 00:08:56,584 --> 00:08:59,372 infinity. And this is actually, and I'm tryin' to 104 00:08:59,372 --> 00:09:02,988 give definition of exponential. And this is absolutely correct, this is 105 00:09:02,988 --> 00:09:07,674 all there is, so we're basically just writing instead of writing the complicated 106 00:09:07,674 --> 00:09:12,096 nonlinear function exponential of an operator that's a complicated function, 107 00:09:12,096 --> 00:09:16,254 we're just going to write it as a product of function which are linear in our 108 00:09:16,254 --> 00:09:20,023 [inaudible] doing it. And as we, as we do so in between each of 109 00:09:20,023 --> 00:09:24,511 these terms in this product, I'm going to insert my favorite resolution of the 110 00:09:24,511 --> 00:09:28,586 identity. So, this dx, x, x. 111 00:09:28,586 --> 00:09:33,582 And well, since I have many x's now, so I'll have to label them differently, so 112 00:09:33,582 --> 00:09:39,900 there's going to be x1, x2, x3, which will sort of correspond to this intermediate 113 00:09:39,900 --> 00:09:43,116 coordinates that I have in, in my projections. 114 00:09:43,117 --> 00:09:49,152 Now, if we put everything together, we get this complicated-looking expression for 115 00:09:49,152 --> 00:09:54,847 the propagator, which is obtained by, well, sort of sandwich, sandwiching this 116 00:09:54,847 --> 00:09:58,603 product of evolution operators between xi and xf. 117 00:09:58,603 --> 00:10:05,351 And here, I'm also going to be using the notation x0 is the initial one and x sub N 118 00:10:05,351 --> 00:10:10,301 is going to be the final one. So, this will allow me to write the, this 119 00:10:10,301 --> 00:10:15,202 propagator in a more compact way. And also as I mentioned so in between of 120 00:10:15,202 --> 00:10:20,710 each of these sort of small evolution operators for a small time, t over N or 121 00:10:20,710 --> 00:10:24,310 delta t. So, I insert this resolution identity but 122 00:10:24,310 --> 00:10:29,268 I have to sep distinguish different instances where I have to do so and 123 00:10:29,268 --> 00:10:32,327 therefore, I have so many different integrals. 124 00:10:32,327 --> 00:10:37,437 Now the reason again on doing this is because instead of dealing with this 125 00:10:37,437 --> 00:10:42,547 complicated exponential, I now can deal with this linear matrix elements of, of 126 00:10:42,547 --> 00:10:47,289 the Hamiltonian, which can be calculated in a rather straightforward way. 127 00:10:48,590 --> 00:10:53,903 So just to summarize what, what remains to be calculated if you just look at this 128 00:10:53,903 --> 00:10:58,677 expression is, well simply the matrix element between x of k plus 1 and x of k 129 00:10:58,677 --> 00:11:03,684 and the matrix element of the Hamiltonian. So, all these guys are, you know, i, H 130 00:11:03,684 --> 00:11:08,223 bar, m and tr are numbers, so they don't really act on this on this function. 131 00:11:08,224 --> 00:11:13,630 And the Hamiltonian, itself, of course is, is just basically the operator [unknown] 132 00:11:13,630 --> 00:11:18,886 wanting to the energy [unknown] because the kinetic energy part and the potential 133 00:11:18,886 --> 00:11:22,816 energy part. And so, apart from this matrix element, I 134 00:11:22,816 --> 00:11:29,046 have to calculate two other matrix elements, which essentially are the last 135 00:11:29,046 --> 00:11:35,902 pieces of the puzzle that requires to be completed in order to get this [unknown]. 136 00:11:35,902 --> 00:11:40,782 And so, and the remaining goal of this video, I'm going to be calculating 137 00:11:40,782 --> 00:11:42,303 basically these guys.