1 00:00:00,210 --> 00:00:05,168 Now we are going to proceed with the more technical discussion and derivation of 2 00:00:05,168 --> 00:00:10,510 Feynman's ideas the Feynman path integral. And I should mention here from the outset, 3 00:00:10,510 --> 00:00:15,046 that if you are not interested in this technical details, or if you feel that 4 00:00:15,046 --> 00:00:19,726 your mathematical background is not sufficient to follow them very closely, 5 00:00:19,726 --> 00:00:25,774 you may simply skip this and the following segment because the rest of the course is 6 00:00:25,774 --> 00:00:30,178 not going to be heavily dependent on this particular derivation. 7 00:00:30,179 --> 00:00:34,328 But, otherwise, I would encourage you to actually go through this evaluation and 8 00:00:34,328 --> 00:00:37,369 maybe even work it out on your own after the lecture. 9 00:00:37,370 --> 00:00:42,038 Because understanding it would help you develop intuition not only about the path 10 00:00:42,038 --> 00:00:46,655 integral itself, but about quantum mechanic quantum mechanics more generally. 11 00:00:46,655 --> 00:00:52,436 Now the object that we are going to discuss in this part of our segment is the 12 00:00:52,436 --> 00:00:58,471 so-called propagator, which actually plays an important role in various aspects of 13 00:00:58,471 --> 00:01:03,211 quantum theory. But at this level we can view it simply as 14 00:01:03,211 --> 00:01:10,380 an attempt to make quantum mechanics as close as possible to the concepts that we 15 00:01:10,380 --> 00:01:14,790 understand intuitivelly in classical physics. 16 00:01:14,790 --> 00:01:18,870 And these concepts Involves, for example, the notion of a particle localized in a 17 00:01:18,870 --> 00:01:22,490 point in space and the notion of trajectory that the particle follows. 18 00:01:22,490 --> 00:01:27,434 Now, at the operational level, what, it is, it involves, the following 19 00:01:27,434 --> 00:01:31,267 construction. So, let's assume that, at an initial 20 00:01:31,267 --> 00:01:36,792 moment, moment of time equals zero, our quantum particle was localized in the 21 00:01:36,792 --> 00:01:42,605 vicinity of the original point, R sub i. So by localized I mean that its initial 22 00:01:42,605 --> 00:01:48,083 wave function, psi of 0, T equals 0, is equal to R sub i, this architecture/g, 23 00:01:48,083 --> 00:01:54,557 which represents the Eigenvector of the position operator corresponding to this 24 00:01:54,557 --> 00:01:58,140 point. Or in physical terms it means that this 25 00:01:58,140 --> 00:02:03,490 wave function is a very narrow wave packet localized in the vicinity of this R sub i. 26 00:02:03,490 --> 00:02:07,910 Very similar to the wave packet that we saw in the very first lecture in the last 27 00:02:07,910 --> 00:02:12,218 segment of the very first lecture. And, so this initial, state is going to 28 00:02:12,218 --> 00:02:16,439 evolve under the action of the Schrodinger equation, which is the standard 29 00:02:16,439 --> 00:02:20,500 Schrodinger equation, with some kinetic energy and potential energy. 30 00:02:20,500 --> 00:02:25,190 And as we know from the first lecture, so this evolution, quantum evolution will 31 00:02:25,190 --> 00:02:28,324 evolve spreading out of this wave packet with time. 32 00:02:28,325 --> 00:02:33,492 So instead of the localized sort of particle-like entities, going to become 33 00:02:33,492 --> 00:02:39,030 like a cloud surrounding this R sub i, and this is spreading out [inaudible] fashion, 34 00:02:39,030 --> 00:02:44,176 the presence of potential. Now the question that we're going to ask, 35 00:02:44,176 --> 00:02:51,624 and, which brings us to this, notion of a propagator is what part of this particle 36 00:02:51,624 --> 00:02:55,374 will propagate a certain final point, R sub f. 37 00:02:55,374 --> 00:03:01,553 Or, in other words, what part of this spread out wave packet will be located in, 38 00:03:01,553 --> 00:03:05,183 near this point. And this mathematically implies 39 00:03:05,183 --> 00:03:10,439 calculating the overlap between the final wave function, psi of T which, again, is 40 00:03:10,439 --> 00:03:16,256 governed by the Schrodinger equation. And this R sub f, which describes a wave 41 00:03:16,256 --> 00:03:21,900 packet near this, some other final point. So of course I don't want to give you the 42 00:03:21,900 --> 00:03:26,590 impression that the particle literally propotages from one localized state to 43 00:03:26,590 --> 00:03:30,750 another localized state. It does not, it actually spreads out in 44 00:03:30,750 --> 00:03:37,390 the quantum mechanical language but a part of this wave packet will indeed be located 45 00:03:37,390 --> 00:03:41,502 in, near this point, and this is what we want to calculate. 46 00:03:41,503 --> 00:03:46,180 And this overlap is the propagator we're going to focus on. 47 00:03:46,180 --> 00:03:51,960 Now another question we sort of may ask is how will the particle get there? 48 00:03:51,960 --> 00:03:57,444 How will the particle go from the initial point to this partially final point? 49 00:03:57,445 --> 00:04:02,334 So this question, strictly speaking, doesn't really make sense in the context 50 00:04:02,334 --> 00:04:06,674 of the standard quantum theory based on the Schrdinger equation, because there is 51 00:04:06,674 --> 00:04:11,324 really no notion of the trajectory, which implies the ability to measure momentum 52 00:04:11,324 --> 00:04:15,352 and coordinate at the same time, which is not possible in quantum mechanics. 53 00:04:15,353 --> 00:04:21,992 But, as we will see, in this different formulation, by Feynman, so this question 54 00:04:21,992 --> 00:04:26,660 sort of acquires a meaning, and going a bit ahead of ourselves. 55 00:04:26,661 --> 00:04:31,704 Sort of letting me sort of, reiterate or advertise again that what we're going to 56 00:04:31,704 --> 00:04:37,597 find Is that this particle goes from this initial point to a final point for, like, 57 00:04:37,597 --> 00:04:41,330 all possible trajectories that we can possibly imagine. 58 00:04:41,330 --> 00:04:44,720 So here, if you lose a trajectory that I plot. 59 00:04:44,720 --> 00:04:52,507 Now to, to move, to move on, to actually calculate this overlap, this propagator. 60 00:04:52,508 --> 00:04:58,689 Clearly, what we need to find is well, the final wave function, psi of T. 61 00:04:58,690 --> 00:05:02,842 We know the initial condition. We, we sort of know what, the final state 62 00:05:02,842 --> 00:05:05,944 we want to get. Now we need to, well, actually solve the 63 00:05:05,944 --> 00:05:09,457 Shrodinger equation. Which, in general, is very complicated. 64 00:05:09,458 --> 00:05:13,862 But there is a formal solution we can, write, using, the so called evolution 65 00:05:13,862 --> 00:05:18,840 operator. So we're interested in solving the general 66 00:05:18,840 --> 00:05:25,333 Schrodinger equation here with some initial condition at t equals 0. 67 00:05:25,333 --> 00:05:29,934 So assume that we know the wave function at the initial moment of time. 68 00:05:29,934 --> 00:05:34,216 So, this of course is a completely general formulation which is not specific to the 69 00:05:34,216 --> 00:05:38,746 derivation of any path integrals. But, well in our case, the initial 70 00:05:38,746 --> 00:05:45,349 condition we have chosen to be this localized wave packet near a certain point 71 00:05:45,349 --> 00:05:48,278 R-sub-i. So in any case well to solve the 72 00:05:48,278 --> 00:05:53,978 Schrdinger equation this time-dependent Schrdinger equation basically means to 73 00:05:53,978 --> 00:05:57,686 find the wave function as a function of time, psi of t. 74 00:05:57,687 --> 00:06:05,042 And the evolution operation that I just mentioned, is formally relates the initial 75 00:06:05,042 --> 00:06:09,095 condition to the final wave function, psi of T. 76 00:06:09,095 --> 00:06:14,806 So, it's action is exactly in some sense rotation from psi of 0 to psi of T. 77 00:06:14,806 --> 00:06:20,019 I use the word rotation having in mind this geometric picture that we. 78 00:06:20,020 --> 00:06:25,846 Introduced last week to motivate the direct notations you know, this kept 79 00:06:25,846 --> 00:06:31,668 vector is psi of t which in sums, are meant to represent a sort of vector, 80 00:06:31,668 --> 00:06:37,654 abstract vector in a linear vector space corresponding to the state of the physical 81 00:06:37,654 --> 00:06:43,513 system. Now if we did have a sort of guard variety 82 00:06:43,513 --> 00:06:50,361 a linear vector space as our [inaudible] space which normally is very 83 00:06:50,361 --> 00:06:57,744 multidimensional so we cannot really draw the axis, but if we had, if we mention for 84 00:06:57,744 --> 00:07:04,913 a second that we have such free axis and there is a state psi, oops, psi 0, which 85 00:07:04,913 --> 00:07:09,697 represents our initial state. So, the norm of this vector is so the 86 00:07:09,697 --> 00:07:14,722 absolute value squared of this vector sort of corresponds to the total probability of 87 00:07:14,722 --> 00:07:19,274 finding our quantum state. Our quantum particle in a certain state 88 00:07:19,274 --> 00:07:24,238 and well, from the Borne interpretation we can say that this probability is equal to 89 00:07:24,238 --> 00:07:26,431 one. We will find a particle in which they are 90 00:07:26,431 --> 00:07:29,934 in some state. So the norm of this vector should be 91 00:07:29,934 --> 00:07:33,614 preserved as we perform a quantum evolution. 92 00:07:33,614 --> 00:07:38,690 So in some sense the only thing we can imagine this vector doing is the function 93 00:07:38,690 --> 00:07:43,760 of time under the action of the Schrodinger equation, as its, its rotation 94 00:07:43,760 --> 00:07:47,522 you know, by some angle to a certain new state, psi of T. 95 00:07:47,522 --> 00:07:52,564 And, this rotation, the operator which sort of enforces this rotation is exactly 96 00:07:52,564 --> 00:07:56,123 the solution operator. So, again so here I'm just trying to 97 00:07:56,123 --> 00:08:01,091 represent this in an intuitive way, in general, we're dealing with the generic 98 00:08:01,091 --> 00:08:04,900 quantum mechanic problem. There is no way to draw it, because we 99 00:08:04,900 --> 00:08:09,456 have well, an infinitely dimensional Hubert space, but still, at some level 100 00:08:09,456 --> 00:08:13,884 this picture is preserved. So this evolution operator rotates a 101 00:08:13,884 --> 00:08:19,204 normalized state which describes a particle to a different state, which 102 00:08:19,204 --> 00:08:23,160 describes the same particle. Now it turns out that we can actually, 103 00:08:23,160 --> 00:08:27,180 instead of writing the Schrodinger equation for the wave function, we can as 104 00:08:27,180 --> 00:08:31,070 well just write the same Schrodinger equation for the evolution operator. 105 00:08:31,070 --> 00:08:36,182 The only thing we have to do is just to plug in this equation into the Schrodinger 106 00:08:36,182 --> 00:08:38,960 equation. We're going to see that we have 107 00:08:38,960 --> 00:08:43,389 essentially the identical equation for the evolution operator. 108 00:08:43,390 --> 00:08:49,560 So H U of T. And the initial condition for this 109 00:08:49,560 --> 00:08:57,450 equation Is you observe is equal to 1. Where 1 is just identity operator. 110 00:08:57,450 --> 00:09:01,132 So why? Well it's sort of full as in a very simple 111 00:09:01,132 --> 00:09:04,770 way from it's definition. So if you look at this equation we see 112 00:09:04,770 --> 00:09:08,802 that the evolution operator sort of evolves our initial condition to final 113 00:09:08,802 --> 00:09:11,027 state. But well if physical is there there's 114 00:09:11,027 --> 00:09:13,040 nothing to hold we just stay where we were. 115 00:09:13,040 --> 00:09:17,690 And so therefore psi of 0 is equal to psi of 0, so U of Q is equal to 1, and this is 116 00:09:17,690 --> 00:09:20,815 sort of the initial condition we can enforce. 117 00:09:20,815 --> 00:09:28,799 And one can guess in some sense or just pick a general solution to this matrix or 118 00:09:28,799 --> 00:09:36,172 operator equation, which satisfies both, both the Schrodinger equation itself and 119 00:09:36,172 --> 00:09:43,494 the initial condition by writing u of t, as an exponential of minus i over h bar H 120 00:09:43,495 --> 00:09:48,732 times t. So well, t equals 0, e exponent of 0 is 121 00:09:48,732 --> 00:09:55,464 equal to 1, so it does satisfy the required initial condition. 122 00:09:55,465 --> 00:10:05,996 And if we plug it into this equation, so if we differentiate this sort of ansatz 123 00:10:05,996 --> 00:10:12,485 with respect to time. So well, differentiating the exponential 124 00:10:12,485 --> 00:10:19,040 sort of pulls out this coefficient minus-i over h-bar times h, so minus-i times i is 125 00:10:19,040 --> 00:10:24,645 equal Two one H divided by H is equal to one, so we're just going to have H E to 126 00:10:24,645 --> 00:10:29,825 the power minus I H bar H times T, which indeed is equal to H times U. 127 00:10:29,825 --> 00:10:35,274 Therefore indeed, so we satisfy the required Schrodinger equation. 128 00:10:36,350 --> 00:10:41,518 So an interesting and sort of attractive feature of this evolution operator is that 129 00:10:41,518 --> 00:10:46,686 it solves the Schrodinger equation with all possible initial conditions in one go, 130 00:10:46,686 --> 00:10:51,378 because the evolution operator itself does not actually depend on the initial 131 00:10:51,378 --> 00:10:54,025 condition. So we see that it has a universal initial 132 00:10:54,025 --> 00:10:57,997 condition. And to equal zero, U is equal to one. 133 00:10:57,998 --> 00:11:03,015 So for instance, here in the geometrical interpretation, if we have let's say 134 00:11:03,015 --> 00:11:06,460 different psi of 0, let's say this will be psi of 0. 135 00:11:06,460 --> 00:11:12,196 Well prime. So the evolution of this other initial 136 00:11:12,196 --> 00:11:18,307 condition in this sort of Hilbert space would be a rotation by the same angle 137 00:11:18,307 --> 00:11:24,929 around the, the same axis of the different side of T it would say, side T prime here. 138 00:11:24,929 --> 00:11:29,686 And but the evolution operator that would enforce this rotation would be exactly the 139 00:11:29,686 --> 00:11:33,337 same as before. So it would be, it will be the exactly the 140 00:11:33,337 --> 00:11:36,957 same u of T. So in our case and in particular, so we're 141 00:11:36,957 --> 00:11:42,039 interested in knowing how not this particular initial condition evolves with 142 00:11:42,039 --> 00:11:47,121 time, and therefore the wave function would be equal to e to the power minus i 143 00:11:47,121 --> 00:11:51,400 over e Hamiltonian times time acting on this initial condition. 144 00:11:51,400 --> 00:11:59,655 So this is sort of a exact and universal solution of the Schrodinger equation, 145 00:11:59,655 --> 00:12:03,780 whatever it is. With, with this initial condition of 146 00:12:03,780 --> 00:12:07,280 localized particles in certain region in space. 147 00:12:07,280 --> 00:12:13,000 Now let us go back to the main question that we posed in the beginning of this 148 00:12:13,000 --> 00:12:19,688 segment, namely the question about the overlap between the wave function psi of 149 00:12:19,688 --> 00:12:27,100 t, sort of propagated from the vicinity of the point R sub i, and the wave back at 150 00:12:27,100 --> 00:12:31,414 localized in the vicinity of the point R sub f. 151 00:12:31,414 --> 00:12:38,660 So, what we found in the previous slide is a formal exact expression for this psi of 152 00:12:38,660 --> 00:12:41,662 t. So psi of t is the action of the evolution 153 00:12:41,662 --> 00:12:46,664 operation, the initial condition, R sub i. And therefore, if we put two and two 154 00:12:46,664 --> 00:12:52,931 together, the expression. Or the, propagator is going to be the 155 00:12:52,931 --> 00:13:00,424 fully matrix element is going to be Rf, here we're going to have the solution 156 00:13:00,424 --> 00:13:06,884 operator, ht, r sub i. So this guy is simply the psi of t. 157 00:13:06,885 --> 00:13:12,490 And so we get, the expression that we're actually going to use in the next segment 158 00:13:12,490 --> 00:13:17,736 to derive the Feynman path integral. So a Feynman path integral is another 159 00:13:17,736 --> 00:13:23,750 representation of this metric's element between Rf and Ri of this operator U of T. 160 00:13:23,750 --> 00:13:29,780 So at this stage you may ask me why would we bother to look for any other expression 161 00:13:29,780 --> 00:13:35,699 apart from the one we just derived, which looks pretty compact and simple. 162 00:13:35,700 --> 00:13:39,976 So the answer to this is that this simplicity is deceptive. 163 00:13:39,976 --> 00:13:45,740 Because the main object here, this evolution operator involves exponentiating 164 00:13:45,740 --> 00:13:50,525 an operator, the Camille 2 onion/g. And to actually calculate such an 165 00:13:50,525 --> 00:13:54,453 exponential of an operator, exponential of a matrix. 166 00:13:54,453 --> 00:14:01,373 In a rather complicated exercise and in general even, we define the exponential 167 00:14:01,373 --> 00:14:07,652 open operator, is a tricky business, so as a side comment here, let me remind you 168 00:14:07,652 --> 00:14:14,204 that for an average operator, a matrix x, in our case this act as this minus i over 169 00:14:14,204 --> 00:14:20,847 h bar Ht, so its exponential is defined by the Taylor series, so it's 1 plus x plus x 170 00:14:20,847 --> 00:14:25,632 squared over 2 plus x cubed over 3 factorial, plus etc. 171 00:14:25,632 --> 00:14:32,555 And so to calculate this series with matrix as operator is rather complicated. 172 00:14:32,555 --> 00:14:38,120 And so the Feynman approach, as we will see, essentially circumvents the need to 173 00:14:38,120 --> 00:14:44,124 calculate this complicated exponential and truncates the series, right here, and 174 00:14:44,124 --> 00:14:49,575 basically the end result as we will see will be, an expression that doesn't have 175 00:14:49,575 --> 00:14:54,238 any operators whatsoever. And not only that the final result also 176 00:14:54,238 --> 00:14:58,516 will have a very clear intuitive interpretation that we already, sort of, 177 00:14:58,516 --> 00:15:01,214 advertised in the beginning of this lecture.