1 00:00:00,680 --> 00:00:05,600 PROFESSOR: It's a statement about the time dependence 2 00:00:05,600 --> 00:00:08,420 of the expectation values. 3 00:00:08,420 --> 00:00:10,325 It's a pretty fundamental theorem. 4 00:00:13,350 --> 00:00:16,230 So here it goes. 5 00:00:16,230 --> 00:00:23,910 You have d dt of the expectation value 6 00:00:23,910 --> 00:00:26,130 of Q. This is what we want to evaluate. 7 00:00:26,130 --> 00:00:36,810 We Now this would be d dt of integral psi star of x and t, 8 00:00:36,810 --> 00:00:41,720 Q psi of x and t. 9 00:00:41,720 --> 00:00:44,250 And the d dt acts on the two of them. 10 00:00:44,250 --> 00:00:53,230 So it gives you integral partial psi star dt Q 11 00:00:53,230 --> 00:01:05,950 psi of x and t plus psi star Q partial psi dt. 12 00:01:11,590 --> 00:01:14,065 And this is the integral over the x. 13 00:01:20,060 --> 00:01:23,600 You've seen that kind of stuff. 14 00:01:23,600 --> 00:01:29,030 And what is it? 15 00:01:29,030 --> 00:01:35,980 Well, integral dx, this is this Schrodinger equation, 16 00:01:35,980 --> 00:01:46,655 d psi star dt is i over h bar, H psi star. 17 00:01:50,730 --> 00:01:53,850 From the Schrodinger equation. 18 00:01:53,850 --> 00:01:58,920 Then you have the Q psi of x and t. 19 00:01:58,920 --> 00:02:06,330 On this term, you will have a very similar thing. 20 00:02:06,330 --> 00:02:19,820 Minus i over H bar this time, psi star QH psi of x and t. 21 00:02:19,820 --> 00:02:22,520 So we use the Schrodinger equation 22 00:02:22,520 --> 00:02:26,825 in the form, i d psi dt-- 23 00:02:26,825 --> 00:02:30,470 i H bar d psi dt-- 24 00:02:30,470 --> 00:02:33,780 equal H psi. 25 00:02:33,780 --> 00:02:37,620 I used it twice. 26 00:02:37,620 --> 00:02:42,140 So then, it's actually convenient to multiply here 27 00:02:42,140 --> 00:02:55,600 by i H bar d dt of Q. So I multiplied by i H bar, 28 00:02:55,600 --> 00:03:01,330 and I will cancel the i and the H bar in this term, 29 00:03:01,330 --> 00:03:04,060 minus them this term. 30 00:03:04,060 --> 00:03:20,030 So we'll have d cube x psi star Q H hat psi minus H hat psi 31 00:03:20,030 --> 00:03:22,980 star cube psi. 32 00:03:26,310 --> 00:03:26,820 OK. 33 00:03:26,820 --> 00:03:31,450 Things have simplified very nicely. 34 00:03:31,450 --> 00:03:36,780 And there's just one more thing we can do. 35 00:03:36,780 --> 00:03:42,030 Look, this is the product of Q and H. 36 00:03:42,030 --> 00:03:47,780 But by hermiticity, H in here can 37 00:03:47,780 --> 00:03:52,000 be brought to the other side to act on this wave function. 38 00:03:52,000 --> 00:04:02,950 So this is actually equal to the integral d cube dx psi star 39 00:04:02,950 --> 00:04:07,240 QH hat psi minus-- 40 00:04:07,240 --> 00:04:09,690 the H can go to the other side-- 41 00:04:09,690 --> 00:04:14,439 psi star H cube psi. 42 00:04:17,279 --> 00:04:27,880 But then, what do we see there? 43 00:04:27,880 --> 00:04:31,810 We recognize a commutator. 44 00:04:31,810 --> 00:04:35,350 This commutator is just like we did for x and p, 45 00:04:35,350 --> 00:04:39,280 and we started practicing how to compute them. 46 00:04:39,280 --> 00:04:40,570 They show up here. 47 00:04:40,570 --> 00:04:44,320 And this is maybe one of the reasons commutators are so 48 00:04:44,320 --> 00:04:46,870 important in quantum mechanics. 49 00:04:46,870 --> 00:04:48,785 So what do we have here? 50 00:04:48,785 --> 00:04:56,500 i H bar d dt of the expectation value of Q 51 00:04:56,500 --> 00:05:10,240 is equal to the integral dx of psi star, QH minus HQ psi. 52 00:05:10,240 --> 00:05:11,990 This is all of x and t. 53 00:05:18,370 --> 00:05:29,340 Well, this is nothing else but the commutator of Q and H. 54 00:05:29,340 --> 00:05:39,420 So our final result is that iH bar d dt of the expectation 55 00:05:39,420 --> 00:05:43,560 value of Q is equal to-- 56 00:05:43,560 --> 00:05:44,690 look. 57 00:05:44,690 --> 00:05:48,870 It's the expectation value of the commutator. 58 00:05:48,870 --> 00:05:51,630 Remember, expectation value for an operator-- 59 00:05:51,630 --> 00:05:54,120 the operator is the thing here-- so this 60 00:05:54,120 --> 00:06:07,190 is nothing else than the expectation value of Q with H. 61 00:06:07,190 --> 00:06:10,170 This is actually a pretty important result. 62 00:06:10,170 --> 00:06:16,640 It has all the dynamics of the physics in the observables. 63 00:06:16,640 --> 00:06:21,300 Look, the wave functions used to change in time. 64 00:06:21,300 --> 00:06:26,820 Due to their change in time, the expectation values 65 00:06:26,820 --> 00:06:29,610 of the operators change in time. 66 00:06:29,610 --> 00:06:32,970 Because this integral can't depend on time. 67 00:06:32,970 --> 00:06:36,480 But here what you have succeeded is 68 00:06:36,480 --> 00:06:41,430 to represent the change in time of the expectation value-- 69 00:06:41,430 --> 00:06:43,350 the change in time of the position 70 00:06:43,350 --> 00:06:46,020 that you expect you find your particle-- 71 00:06:46,020 --> 00:06:50,190 in terms of the expectation value of a commutator 72 00:06:50,190 --> 00:06:51,960 with a Hamiltonian. 73 00:06:51,960 --> 00:06:57,720 So if some quantity commutes with a Hamiltonian, 74 00:06:57,720 --> 00:07:01,440 its expectation value will not change in time. 75 00:07:01,440 --> 00:07:07,540 If you have a Hamiltonian, say with a free particle, 76 00:07:07,540 --> 00:07:11,130 well, the momentum commutes with this. 77 00:07:11,130 --> 00:07:13,950 Therefore the expected value of the momentum, 78 00:07:13,950 --> 00:07:17,910 you already know, since the momentum commutes 79 00:07:17,910 --> 00:07:19,410 with H. This is 0. 80 00:07:19,410 --> 00:07:21,600 The expected value of this is 0. 81 00:07:21,600 --> 00:07:25,200 And the expected value of the momentum will not change, 82 00:07:25,200 --> 00:07:26,860 will be conserved. 83 00:07:26,860 --> 00:07:30,450 So conservation laws in quantum mechanics 84 00:07:30,450 --> 00:07:34,860 have to do with things that commute with the Hamiltonian. 85 00:07:34,860 --> 00:07:38,690 And it's an idea we're going to develop on and on.