WEBVTT 00:00:00.060 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.800 Commons license. 00:00:03.800 --> 00:00:06.060 Your support will help MIT OpenCourseWare 00:00:06.060 --> 00:00:10.120 continue to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.690 To make a donation or view additional materials 00:00:12.690 --> 00:00:16.590 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.590 --> 00:00:17.297 at ocw.mit.edu. 00:00:23.489 --> 00:00:25.030 PROFESSOR: OK, so I want to start out 00:00:25.030 --> 00:00:27.420 by reviewing a few things and putting 00:00:27.420 --> 00:00:29.620 some machinery together. 00:00:29.620 --> 00:00:31.560 Unfortunately, this thing is sort of stuck. 00:00:31.560 --> 00:00:35.300 We're going to need a later, so I don't know. 00:00:35.300 --> 00:00:37.700 I'll put it up for now. 00:00:37.700 --> 00:00:40.410 So first just a bit of notation. 00:00:40.410 --> 00:00:42.250 This symbol, you should think of it 00:00:42.250 --> 00:00:44.520 like the dot product, or the inner product. 00:00:44.520 --> 00:00:49.160 It's just saying that bracket f g is the integral. 00:00:49.160 --> 00:00:50.450 It's a number that you get. 00:00:50.450 --> 00:00:52.780 So this is a number that you get from the function 00:00:52.780 --> 00:00:55.820 f and the function g by taking f, taking 00:00:55.820 --> 00:00:58.840 its complex conjugate, multiplying it by g, 00:00:58.840 --> 00:01:00.710 and then integrating overall positions. 00:01:00.710 --> 00:01:01.490 All right? 00:01:01.490 --> 00:01:02.740 So it's a way to get a number. 00:01:02.740 --> 00:01:05.310 And you should think about it as the analog for functions 00:01:05.310 --> 00:01:07.530 of the dot product for vectors. 00:01:07.530 --> 00:01:11.490 It's a way to get a number out of two vectors. 00:01:11.490 --> 00:01:15.860 And so, for example, with vectors we could do v dot w, 00:01:15.860 --> 00:01:16.940 and this is some number. 00:01:16.940 --> 00:01:20.712 And it has a nice property that v dot v, 00:01:20.712 --> 00:01:23.170 we can think it as v squared, it's something like a length. 00:01:23.170 --> 00:01:25.480 It's strictly positive, and it's something 00:01:25.480 --> 00:01:26.760 like the length of a vector. 00:01:26.760 --> 00:01:31.320 Similarly, if I take f and take its bracket with f, 00:01:31.320 --> 00:01:34.205 this is equal to the integral dx of f squared, 00:01:34.205 --> 00:01:36.580 and in particular, f could be complex, so f norm squared. 00:01:36.580 --> 00:01:38.000 This is strictly non-negative. 00:01:38.000 --> 00:01:41.550 It could vanish, but it's not negative at a point, 00:01:41.550 --> 00:01:42.740 hence the norm squared. 00:01:42.740 --> 00:01:46.990 So this will be zero if and only if what? 00:01:46.990 --> 00:01:49.010 f is 0, f is the 0 function, right. 00:01:49.010 --> 00:01:51.852 So the same way that if you take a vector, 00:01:51.852 --> 00:01:54.310 and you take its dot product with itself, take it the norm, 00:01:54.310 --> 00:01:56.470 it's 0 if an only if the vector is 0. 00:01:56.470 --> 00:01:59.270 So this beast satisfies a lot of the properties of a dot 00:01:59.270 --> 00:01:59.920 product. 00:01:59.920 --> 00:02:01.980 You should think about it as morally equivalent. 00:02:01.980 --> 00:02:04.190 We'll talk about that in more detail later. 00:02:04.190 --> 00:02:06.670 Second, basic postulate of quantum mechanics, 00:02:06.670 --> 00:02:08.972 to every observable is associated an operator, 00:02:08.972 --> 00:02:11.180 and it's an operator acting on the space of functions 00:02:11.180 --> 00:02:14.340 or on the space of wave functions. 00:02:14.340 --> 00:02:16.249 And to every operator corresponding 00:02:16.249 --> 00:02:17.790 to an observable in quantum mechanics 00:02:17.790 --> 00:02:20.060 are associated a special set of functions called 00:02:20.060 --> 00:02:22.640 the eigenfunctions, such that when the operator acts 00:02:22.640 --> 00:02:24.640 on that function, it gives you the same function 00:02:24.640 --> 00:02:26.560 back times a constant. 00:02:26.560 --> 00:02:29.100 What these functions mean, physically, 00:02:29.100 --> 00:02:32.860 is they are the wave functions describing configurations 00:02:32.860 --> 00:02:35.930 with a definite value of the corresponding observable. 00:02:35.930 --> 00:02:38.240 If I'm in an eigenfunction of position with 00:02:38.240 --> 00:02:41.610 eigenvalue x naught, awesome. 00:02:41.610 --> 00:02:44.740 Thank you, AV person, thank you. 00:02:44.740 --> 00:02:48.742 So if your system is described by a wave function which 00:02:48.742 --> 00:02:50.200 is an eigenfunction of the position 00:02:50.200 --> 00:02:52.470 operator with eigenvalue x naught, 00:02:52.470 --> 00:02:55.084 that means you can be confident that the system is 00:02:55.084 --> 00:02:56.500 in the configuration corresponding 00:02:56.500 --> 00:02:59.111 to having a definite position x naught. 00:02:59.111 --> 00:02:59.610 Right? 00:02:59.610 --> 00:03:02.360 It's not a superposition of different positions. 00:03:02.360 --> 00:03:04.980 It is at x naught. 00:03:04.980 --> 00:03:07.567 Similarly, momentum, momentum has eigenfunctions, 00:03:07.567 --> 00:03:08.900 and we know what these guys are. 00:03:08.900 --> 00:03:12.300 These are the exponentials, e to the iKX's. 00:03:12.300 --> 00:03:14.890 They're the eigenfunctions, and those are the wave functions 00:03:14.890 --> 00:03:18.130 describing states with definite value of the momentum, 00:03:18.130 --> 00:03:19.780 of the associated observable. 00:03:19.780 --> 00:03:26.640 Energy as an operator, energy is described by an operator, which 00:03:26.640 --> 00:03:34.340 has eigenfunctions which I'll call phi sub n, with energy 00:03:34.340 --> 00:03:37.910 as E sub n, those are the eigenvalues. 00:03:37.910 --> 00:03:40.920 And if I tell you that your wave function is the state phi 00:03:40.920 --> 00:03:43.330 sub 2, what that tells you is that the system has 00:03:43.330 --> 00:03:46.540 a definite energy, E sub 2, corresponding 00:03:46.540 --> 00:03:48.883 to that eigenvalue. 00:03:48.883 --> 00:03:51.166 Cool? 00:03:51.166 --> 00:03:53.040 And this is true for any physical observable. 00:03:53.040 --> 00:03:54.350 But these are sort of the basic ones 00:03:54.350 --> 00:03:56.849 that we'll keep focusing on, position, momentum, and energy, 00:03:56.849 --> 00:03:58.930 for the next while. 00:03:58.930 --> 00:04:01.360 Now a nice property about these eigenfunctions 00:04:01.360 --> 00:04:04.740 is that for different eigenvalues, 00:04:04.740 --> 00:04:07.220 the associated wave functions are different functions. 00:04:07.220 --> 00:04:08.320 And what I mean by saying they're different 00:04:08.320 --> 00:04:09.736 functions is that they're actually 00:04:09.736 --> 00:04:12.540 orthogonal functions in the sense of this dot product. 00:04:12.540 --> 00:04:16.050 If I have a state corresponding to be at x 0, 00:04:16.050 --> 00:04:17.579 definite position x 0, that means 00:04:17.579 --> 00:04:20.371 they're in eigenfunction of position with eigenvalue x 0, 00:04:20.371 --> 00:04:21.829 and I have another that corresponds 00:04:21.829 --> 00:04:24.630 to being at x1, an eigenfunction of the position operator 00:04:24.630 --> 00:04:28.240 or the eigenvalue x1, then these wave functions are 00:04:28.240 --> 00:04:29.920 orthogonal to each other. 00:04:29.920 --> 00:04:32.960 And we get 0 if x 0 is not equal to x1. 00:04:32.960 --> 00:04:35.250 Everyone cool with that? 00:04:35.250 --> 00:04:37.520 Now, meanwhile not only are they orthogonal 00:04:37.520 --> 00:04:39.837 but they're normalized in a particular way. 00:04:39.837 --> 00:04:41.670 The inner product gives me a delta function, 00:04:41.670 --> 00:04:44.490 which goes beep once, so that if I integrate against it 00:04:44.490 --> 00:04:45.319 I get a 1. 00:04:45.319 --> 00:04:46.360 Same thing with momentum. 00:04:46.360 --> 00:04:48.734 And you do this, this you're checking on the problem set. 00:04:48.734 --> 00:04:50.930 I don't remember if it was last one or this one. 00:04:50.930 --> 00:04:53.410 And for the energies, energy 1, if I 00:04:53.410 --> 00:04:58.436 know the system is in state energy 1, and let's say e sub n 00:04:58.436 --> 00:05:00.625 and e sub m, those are different states 00:05:00.625 --> 00:05:02.250 if n and m are not equal to each other. 00:05:02.250 --> 00:05:04.867 And this inner product is 0 if n and m are not 00:05:04.867 --> 00:05:06.450 equal to each other and 1 if they are. 00:05:06.450 --> 00:05:08.220 Their properly normalized. 00:05:08.220 --> 00:05:10.240 Everyone cool with that? 00:05:10.240 --> 00:05:11.769 Yeah. 00:05:11.769 --> 00:05:13.768 AUDIENCE: Is it possible that two eigenfunctions 00:05:13.768 --> 00:05:15.514 have the same eigenvalue? 00:05:15.514 --> 00:05:16.430 PROFESSOR: Absolutely. 00:05:16.430 --> 00:05:18.650 It is absolutely possible for two eigenfunctions 00:05:18.650 --> 00:05:19.996 to have the same eigenvalue. 00:05:19.996 --> 00:05:21.120 That is certainly possible. 00:05:21.120 --> 00:05:23.210 AUDIENCE: [INAUDIBLE] 00:05:23.210 --> 00:05:25.257 PROFESSOR: Yeah, good. 00:05:25.257 --> 00:05:26.840 Thank you, this is a good technicality 00:05:26.840 --> 00:05:28.970 that I didn't want to get into, but I'll go and get into it. 00:05:28.970 --> 00:05:30.053 It's a very good question. 00:05:30.053 --> 00:05:31.900 So the question is, is it possible for two 00:05:31.900 --> 00:05:34.620 different eigenfunctions to have the same eigenvalue. 00:05:34.620 --> 00:05:36.640 Could there be two states with the same energy , 00:05:36.640 --> 00:05:38.550 different states, same energy? 00:05:38.550 --> 00:05:40.080 Yeah, that's absolutely possible. 00:05:40.080 --> 00:05:41.280 And we'll run into that. 00:05:41.280 --> 00:05:43.550 And there's nice physics encoded in it. 00:05:43.550 --> 00:05:45.200 But let's think about what that means. 00:05:45.200 --> 00:05:47.400 The subsequent question is well, if that's the case, 00:05:47.400 --> 00:05:49.910 are they really still orthogonal? 00:05:49.910 --> 00:05:51.450 And here's the crucial thing. 00:05:51.450 --> 00:05:55.190 The crucial thing is, let's say I take one function, 00:05:55.190 --> 00:05:58.750 I'll call the function phi 1, consider the function phi 1. 00:05:58.750 --> 00:06:02.330 And let it have energy E1, so that E acting on phi 1 00:06:02.330 --> 00:06:05.020 is equal to E1 phi 1. 00:06:05.020 --> 00:06:08.600 And let there be another function, phi 2, 00:06:08.600 --> 00:06:10.810 such that the energy operator acting on phi 2 00:06:10.810 --> 00:06:14.050 is also equal to E1 phi 2. 00:06:14.050 --> 00:06:16.700 These are said to be degenerate. 00:06:16.700 --> 00:06:18.950 Degenerate doesn't mean you go out and trash your car, 00:06:18.950 --> 00:06:22.510 degenerate that the energies are the same. 00:06:22.510 --> 00:06:23.997 So what does this tell me? 00:06:23.997 --> 00:06:25.080 This tells me a cool fact. 00:06:25.080 --> 00:06:30.390 If I take a wave function phi, and I will call this phi star, 00:06:30.390 --> 00:06:32.560 in honor of Shri Kulkarni, so I've 00:06:32.560 --> 00:06:35.770 got this phi star, which is a linear combination alpha 00:06:35.770 --> 00:06:39.270 phi 1 plus beta phi 2, a linear combination 00:06:39.270 --> 00:06:42.700 of them, a superposition of those two states. 00:06:42.700 --> 00:06:46.060 Is this also an energy eigenfunction? 00:06:46.060 --> 00:06:52.630 Yeah, because if I act on phi star with E, then it's linear, 00:06:52.630 --> 00:06:55.960 so E acting on phi star is E acting on alpha phi 1, 00:06:55.960 --> 00:06:57.370 alpha's a constant, doesn't care. 00:06:57.370 --> 00:06:59.090 Phi 1 gives me an E1. 00:06:59.090 --> 00:07:02.060 Similarly, E acting on phi 2 gives me an E1. 00:07:02.060 --> 00:07:06.490 So if I act with E on this guy, this 00:07:06.490 --> 00:07:09.229 is equal to, from both of these I get an overall factor of E1. 00:07:09.229 --> 00:07:11.270 So notice that we get the same vector back, times 00:07:11.270 --> 00:07:16.260 a constant, a common constant. 00:07:16.260 --> 00:07:18.237 So when we have degenerate eigenfunctions, 00:07:18.237 --> 00:07:20.320 we can take arbitrary linear combinations to them, 00:07:20.320 --> 00:07:22.700 get another degenerate eigenfunction. 00:07:22.700 --> 00:07:23.522 Cool? 00:07:23.522 --> 00:07:25.230 So this is like, imagine I have a vector, 00:07:25.230 --> 00:07:26.460 and I have another vector. 00:07:26.460 --> 00:07:28.380 And they share the property that they're both eigenfunctions 00:07:28.380 --> 00:07:29.240 of some operator. 00:07:29.240 --> 00:07:30.948 That means any linear combination of them 00:07:30.948 --> 00:07:32.284 is also, right? 00:07:32.284 --> 00:07:33.950 So there's a whole vector space, there's 00:07:33.950 --> 00:07:36.850 a whole space of possible functions 00:07:36.850 --> 00:07:38.704 that all have the same eigenvalue. 00:07:38.704 --> 00:07:40.870 So now you say, well, look, are these two orthogonal 00:07:40.870 --> 00:07:41.360 to each other? 00:07:41.360 --> 00:07:41.640 No. 00:07:41.640 --> 00:07:41.820 These two? 00:07:41.820 --> 00:07:42.240 No. 00:07:42.240 --> 00:07:43.140 But here's the thing. 00:07:43.140 --> 00:07:45.306 If you have a vector space, if you have a the space, 00:07:45.306 --> 00:07:48.030 you can always find orthogonal guys and a basis 00:07:48.030 --> 00:07:50.190 for that space, yes? 00:07:50.190 --> 00:07:53.470 So while it's not true that the eigenfunctions are always 00:07:53.470 --> 00:07:56.460 orthogonal, it is true-- 00:07:56.460 --> 00:07:59.920 we will not prove this, but we will discuss the proof of it 00:07:59.920 --> 00:08:01.770 later by pulling the mathematician out 00:08:01.770 --> 00:08:03.930 of the closet-- 00:08:03.930 --> 00:08:06.370 the proof will say that it is possible to find 00:08:06.370 --> 00:08:09.620 a set of eigenfunctions which are orthogonal in precisely 00:08:09.620 --> 00:08:12.380 this fashion, even if there are degeneracies. 00:08:12.380 --> 00:08:12.880 OK? 00:08:12.880 --> 00:08:14.960 That theorem is called the spectral theorem. 00:08:14.960 --> 00:08:16.724 And we'll discuss it later. 00:08:16.724 --> 00:08:18.140 So it is always possible to do so. 00:08:18.140 --> 00:08:20.460 But you must be alert that there may be degeneracies. 00:08:20.460 --> 00:08:22.140 There aren't always degeneracies. 00:08:22.140 --> 00:08:24.040 In fact, degeneracies are very special. 00:08:24.040 --> 00:08:27.840 Why should two numbers happen to be the same? 00:08:27.840 --> 00:08:29.840 Something has to be forcing them to be the same. 00:08:29.840 --> 00:08:31.715 That's going to be an important theme for us. 00:08:31.715 --> 00:08:32.966 But it certainly is possible. 00:08:32.966 --> 00:08:33.549 Good question. 00:08:33.549 --> 00:08:36.085 Other questions? 00:08:36.085 --> 00:08:36.585 Yeah. 00:08:36.585 --> 00:08:42.340 AUDIENCE: [INAUDIBLE] 00:08:42.340 --> 00:08:46.110 PROFESSOR: Yeah, so using the triangular brackets-- 00:08:46.110 --> 00:08:49.140 so there's another notation for the same thing, which is f g, 00:08:49.140 --> 00:08:51.351 but this carries some slightly different weight. 00:08:51.351 --> 00:08:53.600 It mean something slightly-- you'll see this in books, 00:08:53.600 --> 00:08:55.500 and this means something very similar to this. 00:08:55.500 --> 00:08:56.680 But I'm not going to use this notation. 00:08:56.680 --> 00:08:58.010 It's called Dirac notation. 00:08:58.010 --> 00:08:59.360 We'll talk about it later in the semester, 00:08:59.360 --> 00:09:00.640 but we're not going to talk about it just yet. 00:09:00.640 --> 00:09:02.490 But when you see this, effectively it 00:09:02.490 --> 00:09:05.565 means the same thing as this. 00:09:05.565 --> 00:09:07.260 This is sort of like dialect. 00:09:07.260 --> 00:09:10.590 You know, it's like French and Quebecois. 00:09:10.590 --> 00:09:12.290 Other questions? 00:09:12.290 --> 00:09:14.190 My wife's Canadian. 00:09:14.190 --> 00:09:15.750 Other questions? 00:09:15.750 --> 00:09:17.760 OK. 00:09:17.760 --> 00:09:19.200 So given this fact, given the fact 00:09:19.200 --> 00:09:20.574 that we can associate observables 00:09:20.574 --> 00:09:23.210 to operators, operators come with special functions, 00:09:23.210 --> 00:09:25.460 the eigenfunctions, those eigenfunctions corresponding 00:09:25.460 --> 00:09:27.410 to have a definite value of the observable, 00:09:27.410 --> 00:09:29.320 and they're orthonormal. 00:09:29.320 --> 00:09:30.940 This tells us, and this is really 00:09:30.940 --> 00:09:32.523 the statement of the spectral theorem, 00:09:32.523 --> 00:09:36.340 that any function can be expanded in a basis of states 00:09:36.340 --> 00:09:38.550 with definite values of some observable. 00:09:38.550 --> 00:09:40.200 So for example, consider position. 00:09:40.200 --> 00:09:41.720 I claim that any wave function can 00:09:41.720 --> 00:09:44.660 be expanded as a superposition of states 00:09:44.660 --> 00:09:46.190 with definite position. 00:09:46.190 --> 00:09:48.510 So here's an arbitrary function, here's 00:09:48.510 --> 00:09:52.140 this set of states with definite position, the delta functions. 00:09:52.140 --> 00:09:55.290 And I can write any function as a superposition 00:09:55.290 --> 00:09:57.960 with some coefficients of states with definite position, 00:09:57.960 --> 00:10:01.814 integrating over all possible positions, x0. 00:10:01.814 --> 00:10:03.480 And this is also sort of trivially true, 00:10:03.480 --> 00:10:04.910 because what's this integral? 00:10:04.910 --> 00:10:07.650 Well, it's an integral, dx0 over all possible positions 00:10:07.650 --> 00:10:09.170 of this delta function. 00:10:09.170 --> 00:10:12.090 But we're evaluating at x, so this is 0 00:10:12.090 --> 00:10:13.550 unless x is equal to x0. 00:10:13.550 --> 00:10:15.125 So I can just put in x instead of x0, 00:10:15.125 --> 00:10:16.670 and that gives me psi of x. 00:10:16.670 --> 00:10:18.760 Sort of tautological We can do the same thing 00:10:18.760 --> 00:10:20.442 for momentum eigenfunctions. 00:10:20.442 --> 00:10:22.150 I claim that any function can be expanded 00:10:22.150 --> 00:10:24.340 in a superposition of momentum eigenfunctions, where 00:10:24.340 --> 00:10:27.160 I sum over all possible values in the momentum 00:10:27.160 --> 00:10:30.120 with some weight. 00:10:30.120 --> 00:10:31.670 This psi tilde of K is just telling 00:10:31.670 --> 00:10:34.480 me how much amplitude there is at that wave number. 00:10:34.480 --> 00:10:35.120 Cool? 00:10:35.120 --> 00:10:37.660 But this is the Fourier theorem, it's a Fourier expansion. 00:10:37.660 --> 00:10:40.864 So purely mathematically, we know that this is true. 00:10:40.864 --> 00:10:42.530 But there's also the physical statement. 00:10:42.530 --> 00:10:45.050 Any state can be expressed as a superposition of states 00:10:45.050 --> 00:10:47.950 with definite momentum. 00:10:47.950 --> 00:10:50.300 There's a math in here, but there's also physics in it. 00:10:50.300 --> 00:10:52.370 Finally, this is less obvious from a mathematical point 00:10:52.370 --> 00:10:54.036 of view, because I haven't even told you 00:10:54.036 --> 00:10:56.560 what energy is, any wave function can be expanded 00:10:56.560 --> 00:10:58.360 in states with definite energy. 00:10:58.360 --> 00:11:02.110 So this is a state, my state En, with definite energy, 00:11:02.110 --> 00:11:05.200 with some coefficient summed over all possible values 00:11:05.200 --> 00:11:07.790 of the energy. 00:11:07.790 --> 00:11:11.750 Given any physical observable, any physical observable, 00:11:11.750 --> 00:11:13.860 momentum, position, angular momentum, 00:11:13.860 --> 00:11:16.769 whatever, given any physical observable, 00:11:16.769 --> 00:11:18.310 a given wave function can be expanded 00:11:18.310 --> 00:11:21.920 as some superposition of having definite values of that. 00:11:21.920 --> 00:11:25.950 Will it in general have definite values of the observable? 00:11:25.950 --> 00:11:28.500 Well a general state be an energy eigenfunction? 00:11:28.500 --> 00:11:29.420 No. 00:11:29.420 --> 00:11:35.210 But any state is a superposition of energy eigenfunctions. 00:11:35.210 --> 00:11:38.240 Will a random state have definite position? 00:11:38.240 --> 00:11:38.870 Certainly not. 00:11:38.870 --> 00:11:41.010 You could have this wave function. 00:11:41.010 --> 00:11:43.110 Superposition. 00:11:43.110 --> 00:11:45.210 Yeah. 00:11:45.210 --> 00:11:50.710 AUDIENCE: Why is the energy special such 00:11:50.710 --> 00:11:54.680 that you can make an arbitrary state with a countable number 00:11:54.680 --> 00:11:56.954 of energy eigenfunctions rather than having 00:11:56.954 --> 00:11:58.120 to do a continuous spectrum? 00:11:58.120 --> 00:11:58.890 PROFESSOR: Excellent question. 00:11:58.890 --> 00:12:00.340 So I'm going to phrase that slightly differently. 00:12:00.340 --> 00:12:01.280 It's an excellent question, and we'll 00:12:01.280 --> 00:12:03.460 come to that at the end of today's lecture. 00:12:03.460 --> 00:12:06.190 So the question is, those are integrals, that 00:12:06.190 --> 00:12:08.115 is a sum over discrete things. 00:12:08.115 --> 00:12:08.615 Why? 00:12:08.615 --> 00:12:12.012 Why is the possible values of the position continuous, 00:12:12.012 --> 00:12:14.470 possible values of momentum continuous, and possible values 00:12:14.470 --> 00:12:16.690 of energy discrete? 00:12:16.690 --> 00:12:20.050 The answer to this will become apparent 00:12:20.050 --> 00:12:22.152 over the course of your next few problem sets. 00:12:22.152 --> 00:12:23.610 You have to do some problems to get 00:12:23.610 --> 00:12:25.190 your fingers dirty to really understand this. 00:12:25.190 --> 00:12:26.648 But here's the statement, and we'll 00:12:26.648 --> 00:12:30.700 see the first version of this at the end of today's lecture. 00:12:30.700 --> 00:12:33.420 Sometimes the allowed energies of a system, the energy 00:12:33.420 --> 00:12:35.240 eigenvalues, are discrete. 00:12:35.240 --> 00:12:37.170 Sometimes they are continuous. 00:12:37.170 --> 00:12:41.056 They will be discrete when you have bound states, states that 00:12:41.056 --> 00:12:42.930 are trapped in some region and aren't allowed 00:12:42.930 --> 00:12:44.410 to get arbitrarily far away. 00:12:44.410 --> 00:12:48.160 They'll be continuous when you have states that 00:12:48.160 --> 00:12:50.720 can get arbitrarily far away. 00:12:50.720 --> 00:12:54.090 Sometimes the momentum will be allowed to be discrete values, 00:12:54.090 --> 00:12:56.930 sometimes it will be allowed to be continuous values. 00:12:56.930 --> 00:12:59.416 And we'll see exactly why subsequently. 00:12:59.416 --> 00:13:00.790 But the thing I want to emphasize 00:13:00.790 --> 00:13:03.720 is that I'm writing this to emphasize that it's possible 00:13:03.720 --> 00:13:05.920 that each of these can be discrete or continuous. 00:13:05.920 --> 00:13:09.120 The important thing is that once you pick your physical system, 00:13:09.120 --> 00:13:11.572 you ask what are the allowed values of position, what 00:13:11.572 --> 00:13:13.030 are the allowed values of momentum, 00:13:13.030 --> 00:13:15.940 and what are the allowed values of energy. 00:13:15.940 --> 00:13:18.290 And then you sum over all possible values. 00:13:18.290 --> 00:13:21.450 Now, in the examples we looked at yesterday, or last lecture, 00:13:21.450 --> 00:13:23.070 the energy could have been discrete, 00:13:23.070 --> 00:13:25.070 as in the case of the infinite well, 00:13:25.070 --> 00:13:28.680 or continuous, as in the case of the free particle. 00:13:28.680 --> 00:13:30.180 In the case of a continuous particle 00:13:30.180 --> 00:13:31.810 this would have been an integral. 00:13:31.810 --> 00:13:35.890 In the case of the system such as a free particle, where 00:13:35.890 --> 00:13:37.890 the energy could take any of a continuous number 00:13:37.890 --> 00:13:40.424 of possible values, this would be a continuous integral. 00:13:40.424 --> 00:13:41.840 To deal with that, I'm often going 00:13:41.840 --> 00:13:45.660 to use the notation, just shorthand, integral sum. 00:13:45.660 --> 00:13:47.500 Which I know is a horrible bastardization 00:13:47.500 --> 00:13:50.120 of all that's good and just, but on the other hand, 00:13:50.120 --> 00:13:53.000 emphasizes the fact that in some systems 00:13:53.000 --> 00:13:55.200 you will get continuous, in some systems discrete, 00:13:55.200 --> 00:13:57.520 and sometimes you'll have both continuous and discrete. 00:13:57.520 --> 00:14:00.380 For example, in hydrogen, in hydrogen 00:14:00.380 --> 00:14:02.310 we'll find that there are bound states 00:14:02.310 --> 00:14:04.830 where the electron is stuck to the hydrogen nucleus, 00:14:04.830 --> 00:14:05.920 to the proton. 00:14:05.920 --> 00:14:08.510 And there are discrete allowed energy levels 00:14:08.510 --> 00:14:10.170 for that configuration. 00:14:10.170 --> 00:14:12.640 However, once you ionize the hydrogen, 00:14:12.640 --> 00:14:15.290 the electron can add any energy you want. 00:14:15.290 --> 00:14:16.170 It's no longer bound. 00:14:16.170 --> 00:14:18.090 It can just get arbitrarily far away. 00:14:18.090 --> 00:14:21.360 And there are an uncountable infinity, a continuous set 00:14:21.360 --> 00:14:22.744 of possible states. 00:14:22.744 --> 00:14:24.410 So in that situation, we'll find that we 00:14:24.410 --> 00:14:27.230 have both the discrete and continuous series 00:14:27.230 --> 00:14:28.582 of possible states. 00:14:28.582 --> 00:14:29.082 Yeah. 00:14:29.082 --> 00:14:32.420 AUDIENCE: [INAUDIBLE] 00:14:32.420 --> 00:14:34.620 PROFESSOR: Yeah, sure, if you work on a lattice. 00:14:34.620 --> 00:14:36.870 So for example, consider the following quantum system. 00:14:36.870 --> 00:14:38.550 I have an undergraduate. 00:14:38.550 --> 00:14:41.760 And that undergraduate has been placed in 1 of 12 boxes. 00:14:41.760 --> 00:14:42.450 OK? 00:14:42.450 --> 00:14:44.309 Now, what's the state of the undergraduate? 00:14:44.309 --> 00:14:44.850 I don't know. 00:14:44.850 --> 00:14:46.410 Is it a definite position state? 00:14:46.410 --> 00:14:47.290 It might be. 00:14:47.290 --> 00:14:49.660 But probably it's a superposition, an arbitrary 00:14:49.660 --> 00:14:51.360 superposition, right? 00:14:51.360 --> 00:14:54.912 Very impressive undergraduates at MIT. 00:14:54.912 --> 00:14:55.745 OK, other questions. 00:14:58.355 --> 00:14:59.491 Yeah. 00:14:59.491 --> 00:15:00.990 AUDIENCE: Do these three [INAUDIBLE] 00:15:00.990 --> 00:15:03.580 hold even if the probability changes over time? 00:15:03.580 --> 00:15:04.170 PROFESSOR: Excellent question. 00:15:04.170 --> 00:15:05.090 We'll come back to that. 00:15:05.090 --> 00:15:06.631 Very good question, leading question. 00:15:06.631 --> 00:15:09.340 OK, so we have this. 00:15:09.340 --> 00:15:12.130 The next thing is that energy eigenfunctions satisfy 00:15:12.130 --> 00:15:15.540 some very special properties. 00:15:15.540 --> 00:15:17.340 And in particular, energy eigenfunctions 00:15:17.340 --> 00:15:20.785 have the property from the Schrodinger equation i h 00:15:20.785 --> 00:15:26.150 bar d t on psi of x and t is equal to the energy operator 00:15:26.150 --> 00:15:28.680 acting on psi of x and t. 00:15:28.680 --> 00:15:30.150 This tells us that if we have psi 00:15:30.150 --> 00:15:37.000 x 0 time t 0 is equal to phi n of x, as we saw last time, 00:15:37.000 --> 00:15:41.080 then the wave function, psi at x at time t 00:15:41.080 --> 00:15:43.100 is equal to phi n of x. 00:15:43.100 --> 00:15:46.200 And it only changes by an overall phase, e to the minus i 00:15:46.200 --> 00:15:50.510 En t over h bar. 00:15:50.510 --> 00:15:52.760 And this ratio En upon h bar will often 00:15:52.760 --> 00:15:56.130 be written omega n is equal to En over h bar. 00:15:56.130 --> 00:15:59.252 This is just the Dupre relations. 00:15:59.252 --> 00:16:01.340 Everyone cool with that? 00:16:01.340 --> 00:16:02.600 So are energy eigenfunctions-- 00:16:07.020 --> 00:16:07.570 how to say. 00:16:07.570 --> 00:16:12.800 No wave function is more morally good than another. 00:16:12.800 --> 00:16:14.727 But some are particularly convenient. 00:16:14.727 --> 00:16:16.560 Energy eigenfunctions have the nice property 00:16:16.560 --> 00:16:17.990 that while they're not in a definite position 00:16:17.990 --> 00:16:20.156 and they don't necessarily have a definite momentum, 00:16:20.156 --> 00:16:22.950 they do evolve over time in a particularly simple way. 00:16:22.950 --> 00:16:25.690 And that and the superposition principle 00:16:25.690 --> 00:16:27.250 allow me to write the following. 00:16:27.250 --> 00:16:34.400 If I know that this is my wave function at psi at x at time 0, 00:16:34.400 --> 00:16:38.230 so let's say in all these cases, this is psi of x at time 0, 00:16:38.230 --> 00:16:40.590 how does this state evolve forward in time? 00:16:44.909 --> 00:16:45.950 It's kind of complicated. 00:16:45.950 --> 00:16:49.060 How does this description, how does psi tilde of k 00:16:49.060 --> 00:16:50.715 evolve forward in time? 00:16:50.715 --> 00:16:51.840 Again, kind of complicated. 00:16:51.840 --> 00:16:54.497 But when expressed in terms of the energy eigenstates, 00:16:54.497 --> 00:16:56.330 the answer to how it evolves forward in time 00:16:56.330 --> 00:16:58.460 is very simple, because I know that this 00:16:58.460 --> 00:17:01.440 is a superposition, a linear combination of states 00:17:01.440 --> 00:17:02.600 with definite energy. 00:17:02.600 --> 00:17:05.079 States with definite energy evolve with a phase. 00:17:05.079 --> 00:17:07.010 And the Schrodinger equation is linear, 00:17:07.010 --> 00:17:08.700 so solutions of the Schrodinger equation 00:17:08.700 --> 00:17:11.680 evolve to become solutions of the Schrodinger equation. 00:17:11.680 --> 00:17:13.950 So how does this state evolve forward in time? 00:17:13.950 --> 00:17:18.476 It evolves forward with a phase, e to the minus i omega n t. 00:17:18.476 --> 00:17:21.560 One for every different terms in this sum. 00:17:21.560 --> 00:17:23.020 Cool? 00:17:23.020 --> 00:17:25.089 So we are going to harp on energy functions, 00:17:25.089 --> 00:17:28.300 not because they're more moral, or more just, or more good, 00:17:28.300 --> 00:17:31.040 but because they're more convenient for solving the time 00:17:31.040 --> 00:17:33.680 evolution problem in quantum mechanics. 00:17:33.680 --> 00:17:37.020 So most of today is going to be about this expansion 00:17:37.020 --> 00:17:40.470 and qualitative features of energy eigenfunctions. 00:17:40.470 --> 00:17:42.690 Cool? 00:17:42.690 --> 00:17:43.510 OK. 00:17:43.510 --> 00:17:45.670 And just to close that out, I just 00:17:45.670 --> 00:17:47.420 want to remind you of a couple of examples 00:17:47.420 --> 00:17:50.022 that we did last time, just get them on board. 00:17:50.022 --> 00:17:51.355 So the first is a free particle. 00:17:54.440 --> 00:17:57.220 So for free particle, we have that our wave functions-- 00:18:00.380 --> 00:18:04.125 well, actually let me not write that down. 00:18:04.125 --> 00:18:06.000 Actually, let me skip over the free particle, 00:18:06.000 --> 00:18:06.999 because it's so trivial. 00:18:06.999 --> 00:18:10.417 Let me just talk about the infinite well. 00:18:10.417 --> 00:18:12.000 So the potential is infinite out here, 00:18:12.000 --> 00:18:13.750 and it's 0 inside the well, and it 00:18:13.750 --> 00:18:18.790 goes from 0 to L. This is just my choice of notation. 00:18:18.790 --> 00:18:22.300 And the energy operator, as usual, 00:18:22.300 --> 00:18:26.165 is p squared upon 2m plus u of x. 00:18:26.165 --> 00:18:27.790 You might say, where did I derive this, 00:18:27.790 --> 00:18:29.000 and the answer is I didn't derive this. 00:18:29.000 --> 00:18:30.030 I just wrote it down. 00:18:30.030 --> 00:18:32.690 It's like force in Newton's equations. 00:18:32.690 --> 00:18:34.730 You just declare some force and you ask, 00:18:34.730 --> 00:18:36.190 what system does is model. 00:18:36.190 --> 00:18:38.120 So here's my system. 00:18:38.120 --> 00:18:41.630 It has what looks like a classical kind of energy, 00:18:41.630 --> 00:18:43.450 except these are all operators. 00:18:43.450 --> 00:18:48.120 And the potential here is this guy, it's 0 between 0 and L, 00:18:48.120 --> 00:18:49.730 and it's infinite elsewhere. 00:18:49.730 --> 00:18:51.740 And as we saw last time, the solutions 00:18:51.740 --> 00:18:53.480 to the energy eigenvalue equation 00:18:53.480 --> 00:18:55.190 are particularly simple. 00:18:55.190 --> 00:18:59.500 Phi sub n of x is equal to root properly 00:18:59.500 --> 00:19:04.620 normalized 2 upon L sine of Kn x, 00:19:04.620 --> 00:19:08.790 where kn is equal to n plus 1 pi, where 00:19:08.790 --> 00:19:11.670 n is an integer upon L. 00:19:11.670 --> 00:19:14.070 And these were chosen to satisfy our boundary conditions, 00:19:14.070 --> 00:19:16.403 that the wave function must vanish here, hence the sine, 00:19:16.403 --> 00:19:19.690 and K was chosen so that it turned over and just hit 0 00:19:19.690 --> 00:19:25.210 as we got to L. And that gave us that the allowed energies were 00:19:25.210 --> 00:19:29.050 discrete, because the En, which you can get by just plugging 00:19:29.050 --> 00:19:31.200 into the energy eigenvalue equation, 00:19:31.200 --> 00:19:34.860 was equal to h bar squared Kn squared upon 2m. 00:19:37.550 --> 00:19:39.370 So this tells us a nice thing. 00:19:39.370 --> 00:19:41.430 First off, in this system, if I take a particle 00:19:41.430 --> 00:19:43.940 and I throw it in here in some arbitrary state 00:19:43.940 --> 00:19:46.070 so that at time t equals zero the wave 00:19:46.070 --> 00:19:54.170 function x 0 is equal to sum over n phi n of x Cn. 00:19:54.170 --> 00:19:56.440 OK? 00:19:56.440 --> 00:19:57.220 Can I do this? 00:19:57.220 --> 00:19:58.845 Can I just pick some arbitrary function 00:19:58.845 --> 00:20:01.320 which is a superposition of energy eigenstates? 00:20:01.320 --> 00:20:02.956 Sure, because any function is. 00:20:02.956 --> 00:20:04.955 Any function can be described as a superposition 00:20:04.955 --> 00:20:06.560 of energy eigenfunctions. 00:20:06.560 --> 00:20:09.350 And if I use the energy eigenfunctions, 00:20:09.350 --> 00:20:11.600 it will automatically satisfy the boundary conditions. 00:20:11.600 --> 00:20:13.190 All good things will happen. 00:20:13.190 --> 00:20:15.170 So this is perfectly fine initial condition. 00:20:15.170 --> 00:20:16.950 What is the system at time t? 00:20:20.986 --> 00:20:22.360 Yeah, we just pick up the phases. 00:20:22.360 --> 00:20:24.100 And what phase is this guy? 00:20:24.100 --> 00:20:28.420 It's this, e to the minus i omega n t. 00:20:28.420 --> 00:20:31.150 And when I write omega n, let me be more explicit about that, 00:20:31.150 --> 00:20:33.820 that's En over h bar. 00:20:33.820 --> 00:20:38.340 So that's h bar Kn squared upon 2m t. 00:20:41.420 --> 00:20:42.550 Cool? 00:20:42.550 --> 00:20:48.740 So there is our solution for arbitrary initial conditions 00:20:48.740 --> 00:20:54.069 to the infinite square well problem in quantum mechanics. 00:20:54.069 --> 00:20:56.610 And you're going to study this in some detail on your problem 00:20:56.610 --> 00:20:58.950 set. 00:20:58.950 --> 00:21:01.920 But just to start with a little bit of intuition, 00:21:01.920 --> 00:21:04.390 let's look at the wave functions and the probability 00:21:04.390 --> 00:21:06.177 distributions for the lowest lying states. 00:21:06.177 --> 00:21:07.760 So for example, let's look at the wave 00:21:07.760 --> 00:21:11.360 function for the ground state, what I will call psi sub 0. 00:21:11.360 --> 00:21:14.959 And this is from 0 to L. And I put these bars here 00:21:14.959 --> 00:21:16.750 not because we're looking at the potential. 00:21:16.750 --> 00:21:23.954 I'm going to be plotting the real part of the wave function. 00:21:23.954 --> 00:21:26.450 But I put these walls here just to emphasize 00:21:26.450 --> 00:21:29.617 that that's where the walls are, at x equals 0 and x equals L. 00:21:29.617 --> 00:21:30.700 So what does it look like? 00:21:30.700 --> 00:21:33.100 Well, the first one is going to sine of Kn x. 00:21:33.100 --> 00:21:34.010 n is 0. 00:21:34.010 --> 00:21:38.804 Kn is going to be pi upon L. So that's again just this guy. 00:21:38.804 --> 00:21:40.470 Now, what's the probability distribution 00:21:40.470 --> 00:21:43.734 associated with psi 0? 00:21:43.734 --> 00:21:45.025 Where do you find the particle? 00:21:47.920 --> 00:21:51.850 So we know that it's just the norm squared of this wave 00:21:51.850 --> 00:21:56.580 function and the norm squared is here at 0, it's 0 00:21:56.580 --> 00:21:58.040 and it rises linearly, because sine 00:21:58.040 --> 00:21:59.580 is linear for small values. 00:21:59.580 --> 00:22:02.980 That makes this quadratic, and a maximum, 00:22:02.980 --> 00:22:04.600 and then quadratic again. 00:22:04.600 --> 00:22:06.710 So there's our probability distribution. 00:22:06.710 --> 00:22:08.014 Now, here's a funny thing. 00:22:08.014 --> 00:22:09.930 Imagine I take a particle, classical particle, 00:22:09.930 --> 00:22:11.607 and I put it in a box. 00:22:11.607 --> 00:22:13.690 And you put it in a box, and you tell it, OK, it's 00:22:13.690 --> 00:22:14.390 got some energy. 00:22:14.390 --> 00:22:16.210 So classically it's got some momentum. 00:22:16.210 --> 00:22:17.834 So it's sort of bouncing back and forth 00:22:17.834 --> 00:22:20.000 and just bounces off the arbitrarily hard walls 00:22:20.000 --> 00:22:20.880 and moves around. 00:22:20.880 --> 00:22:22.880 Where are you most likely to find that particle? 00:22:27.850 --> 00:22:29.530 Where does it spend most of its time? 00:22:33.052 --> 00:22:35.010 It spends the same amount of time at any point. 00:22:35.010 --> 00:22:36.410 It's moving at constant velocity. 00:22:36.410 --> 00:22:39.247 It goes boo, boo, boo, boo, right? 00:22:39.247 --> 00:22:40.830 So what's the probability distribution 00:22:40.830 --> 00:22:44.020 for finding it at any point inside, classically? 00:22:44.020 --> 00:22:45.411 Constant. 00:22:45.411 --> 00:22:47.660 Classically, the probability distribution is constant. 00:22:47.660 --> 00:22:49.576 You're just as likely to find it near the wall 00:22:49.576 --> 00:22:51.615 as not near the wall. 00:22:51.615 --> 00:22:53.740 However, quantum mechanically, for the lowest lying 00:22:53.740 --> 00:22:55.090 state that is clearly not true. 00:22:55.090 --> 00:22:58.310 You're really likely to find it near the wall. 00:22:58.310 --> 00:23:01.060 What's up with that? 00:23:01.060 --> 00:23:03.630 So that's a question that I want to put in your head 00:23:03.630 --> 00:23:05.850 and have you think about. 00:23:05.850 --> 00:23:08.309 You're going to see a similar effect arising over and over. 00:23:08.309 --> 00:23:09.891 And we're going to see at the very end 00:23:09.891 --> 00:23:12.460 that that is directly related, the fact that this goes to 0, 00:23:12.460 --> 00:23:15.370 is directly related, and I'm not kidding, 00:23:15.370 --> 00:23:16.940 to the transparency of diamond. 00:23:22.385 --> 00:23:24.000 OK, I think it was pretty cool. 00:23:26.720 --> 00:23:27.470 They're expensive. 00:23:30.399 --> 00:23:31.940 It's also related to the transparency 00:23:31.940 --> 00:23:34.148 of cubic zirconium, which I guess is less impressive. 00:23:37.020 --> 00:23:39.090 So the first state, again, let's look 00:23:39.090 --> 00:23:43.790 at the real part of psi 1, the first excited state. 00:23:43.790 --> 00:23:46.240 Well, this is now a sine with one extra-- 00:23:46.240 --> 00:23:50.940 with a 2 here, 2 pi, so it goes through 0. 00:23:50.940 --> 00:23:53.740 So the probability distribution associated with psi 1, 00:23:53.740 --> 00:23:56.774 and I should say write this as a function of x, looks 00:23:56.774 --> 00:23:58.190 like, well, again, it's quadratic. 00:23:58.190 --> 00:24:01.310 But it has a 0 again in the middle. 00:24:01.310 --> 00:24:04.250 So it's going to look like-- 00:24:04.250 --> 00:24:07.670 oops, my bad art defeats me. 00:24:07.670 --> 00:24:08.580 OK, there we go. 00:24:08.580 --> 00:24:10.010 So now it's even worse. 00:24:10.010 --> 00:24:12.100 Not only is unlikely to be out here, 00:24:12.100 --> 00:24:14.183 it's also very unlikely to be found in the middle. 00:24:14.183 --> 00:24:16.750 In fact, there is 0 probability you'll find it in the middle. 00:24:16.750 --> 00:24:18.700 That's sort of surprising. 00:24:18.700 --> 00:24:21.290 But you can quickly guess what happens as you 00:24:21.290 --> 00:24:23.770 go to very high energies. 00:24:23.770 --> 00:24:30.010 The real part of psi n let's say 10,000, 10 to the 4, 00:24:30.010 --> 00:24:31.610 what is that going to look like? 00:24:31.610 --> 00:24:33.620 Well, this had no 0s, this had one 0, 00:24:33.620 --> 00:24:35.300 and every time you increase n by 1, 00:24:35.300 --> 00:24:37.570 you're just going to add one more 0 to the sign. 00:24:37.570 --> 00:24:39.570 That's an interesting and suggestive fact. 00:24:39.570 --> 00:24:42.650 So if it's size of 10,000, how many nodes 00:24:42.650 --> 00:24:45.411 are there going to be in the middle of the domain? 00:24:45.411 --> 00:24:45.910 10,000. 00:24:45.910 --> 00:24:48.690 And the amplitude is going to be the same. 00:24:48.690 --> 00:24:51.220 I'm not to be able to do this, but you get the idea. 00:24:51.220 --> 00:24:53.520 And now if I construct the probability distribution, 00:24:53.520 --> 00:24:55.780 what's the probability distribution going to be? 00:24:55.780 --> 00:25:02.030 Probability of the 10,000th psi sub 10 to the 4 of x. 00:25:02.030 --> 00:25:05.830 Well, it's again going to be strictly positive. 00:25:05.830 --> 00:25:10.040 And if you are not able to make measurements on the scale of L 00:25:10.040 --> 00:25:14.340 upon 10,000, but just say like L over 3, because you have 00:25:14.340 --> 00:25:17.331 a thumb and you don't have an infinitely accurate meter, what 00:25:17.331 --> 00:25:17.830 do you see? 00:25:17.830 --> 00:25:20.384 You see effectively a constant probability distribution. 00:25:20.384 --> 00:25:22.050 And actually, I shouldn't draw it there. 00:25:22.050 --> 00:25:23.834 I should draw it through the half, 00:25:23.834 --> 00:25:27.170 because sine squared over 2 averages to one half, 00:25:27.170 --> 00:25:30.500 or, sorry, sine squared averages to one half over many periods. 00:25:30.500 --> 00:25:33.660 So what we see is that the classical probability 00:25:33.660 --> 00:25:38.380 distribution constant does arise when we look 00:25:38.380 --> 00:25:41.590 at very high energy states. 00:25:41.590 --> 00:25:42.880 Cool? 00:25:42.880 --> 00:25:46.084 But it is manifestly not a good description. 00:25:46.084 --> 00:25:48.250 The classical description is not a good description. 00:25:48.250 --> 00:25:51.070 Your intuition is crappy at low energies, 00:25:51.070 --> 00:25:53.570 near the ground state, where quantum effects are dominating, 00:25:53.570 --> 00:25:55.980 because indeed, classically there was no minimum energy. 00:25:55.980 --> 00:25:57.930 Quantum effects have to be dominating there. 00:25:57.930 --> 00:25:59.780 And here we see that even the probability distribution's 00:25:59.780 --> 00:26:01.542 radically different than our intuition. 00:26:01.542 --> 00:26:02.042 Yeah. 00:26:02.042 --> 00:26:12.650 AUDIENCE: [INAUDIBLE] 00:26:12.650 --> 00:26:13.900 PROFESSOR: Keep working on it. 00:26:13.900 --> 00:26:18.820 So I want you all to think about what-- 00:26:18.820 --> 00:26:21.250 you're not, I promise you, unless you've already 00:26:21.250 --> 00:26:22.230 seen some quantum mechanics, you're 00:26:22.230 --> 00:26:24.271 not going to be able to answer this question now. 00:26:24.271 --> 00:26:27.010 But I want you to have it as an uncomfortable little piece 00:26:27.010 --> 00:26:31.420 of sand in the back of your oyster mind-- 00:26:31.420 --> 00:26:38.020 no offense-- what is causing that 0? 00:26:38.020 --> 00:26:39.300 Why are we getting 0? 00:26:39.300 --> 00:26:40.575 And I'll give you a hint. 00:26:40.575 --> 00:26:42.960 In quantum mechanics, anytime something interesting 00:26:42.960 --> 00:26:45.985 happens it's because of superposition and interference. 00:26:49.220 --> 00:26:50.850 All right. 00:26:50.850 --> 00:26:55.044 So with all that said, so any questions now over this story 00:26:55.044 --> 00:26:56.585 about energy eigenfunctions expanding 00:26:56.585 --> 00:27:01.136 in a basis, et cetera, before we get moving? 00:27:01.136 --> 00:27:03.370 No, OK. 00:27:03.370 --> 00:27:05.880 In that case, get out your clickers. 00:27:05.880 --> 00:27:07.620 We're going to test your knowledge. 00:27:17.720 --> 00:27:20.836 Channel 41, for those of you who have to adjust it. 00:27:20.836 --> 00:27:32.380 [CHATTER] 00:27:32.380 --> 00:27:32.880 Wow. 00:27:37.100 --> 00:27:38.100 That's kind of worrying. 00:27:42.980 --> 00:27:43.480 Aha. 00:27:58.400 --> 00:28:01.780 OK, ready? 00:28:01.780 --> 00:28:07.010 OK, channel 41, and here we go. 00:28:18.300 --> 00:28:19.789 So go ahead and start now. 00:28:19.789 --> 00:28:21.830 Sorry, there was a little technical glitch there. 00:28:21.830 --> 00:28:24.640 So psi 1 and psi 2 are eigenstates. 00:28:24.640 --> 00:28:27.590 They're non-degenerate, meaning the energies are different. 00:28:27.590 --> 00:28:30.020 Is a superposition psi 1 plus psi 2 also an eigenstate? 00:28:43.398 --> 00:28:45.580 All right, four more seconds. 00:28:48.610 --> 00:28:49.560 All right. 00:28:49.560 --> 00:28:52.910 I want everyone to turn to the person next to you 00:28:52.910 --> 00:28:54.050 and discuss this. 00:28:54.050 --> 00:28:56.538 You've got about 30 seconds to discuss, or a minute. 00:28:56.538 --> 00:29:39.780 [CHATTER] 00:29:39.780 --> 00:29:41.940 All right. 00:29:41.940 --> 00:29:45.230 I want everyone, now that you've got an answer, click again, 00:29:45.230 --> 00:29:47.300 put in your current best guess. 00:29:50.564 --> 00:29:51.230 Oh, wait, sorry. 00:29:51.230 --> 00:29:54.180 For some reason I have to start over again. 00:29:54.180 --> 00:29:56.080 OK, now click. 00:30:02.690 --> 00:30:05.090 This is the best. 00:30:05.090 --> 00:30:08.180 I'm such a convert to clickers, this is just fantastic. 00:30:08.180 --> 00:30:11.750 So you guys went from, so roughly you all 00:30:11.750 --> 00:30:20.480 went from about 30, 60, 10, to what are we now? 00:30:20.480 --> 00:30:26.990 8, 82, and 10. 00:30:26.990 --> 00:30:29.230 So it sounds like you guys are predicting answer b. 00:30:29.230 --> 00:30:30.200 And the answer is-- 00:30:33.587 --> 00:30:34.420 I like the suspense. 00:30:34.420 --> 00:30:37.170 There we go. 00:30:37.170 --> 00:30:38.150 B, good. 00:30:38.150 --> 00:30:40.160 So here's a quick question. 00:30:42.890 --> 00:30:44.020 So why? 00:30:44.020 --> 00:30:49.910 And the reason why is that if we have E on psi 1 plus psi 2, 00:30:49.910 --> 00:30:56.540 this is equal to E on psi 1 plus E on psi 2, operator, operator, 00:30:56.540 --> 00:30:59.200 operator, but this is equal to E 1 00:30:59.200 --> 00:31:05.380 psi 1 E 2 psi 2, which if E1 and E2 are not equal, 00:31:05.380 --> 00:31:09.070 which is not equal to E times psi 1 plus psi 2. 00:31:09.070 --> 00:31:10.350 Right? 00:31:10.350 --> 00:31:14.740 Not equal to E anything times psi 1 plus psi 2. 00:31:14.740 --> 00:31:16.280 And it needs to be, in order to be 00:31:16.280 --> 00:31:18.300 an eigenfunction, an eigenfunction of the energy 00:31:18.300 --> 00:31:20.960 operator. 00:31:20.960 --> 00:31:22.538 Yeah. 00:31:22.538 --> 00:31:24.162 AUDIENCE: So I was thinking about this, 00:31:24.162 --> 00:31:26.037 if this was kind of a silly random case where 00:31:26.037 --> 00:31:27.140 one of the energies is 0. 00:31:27.140 --> 00:31:29.640 Does this only happen if you have something that's infinite? 00:31:29.640 --> 00:31:31.360 PROFESSOR: Yeah, that's a really good question. 00:31:31.360 --> 00:31:33.340 So first off, how do you measure an energy? 00:31:36.240 --> 00:31:38.790 Do you ever measure an energy? 00:31:38.790 --> 00:31:41.365 Do you ever measure a voltage, the actual value 00:31:41.365 --> 00:31:43.490 of the scalar potential, the electromagnetic scalar 00:31:43.490 --> 00:31:43.670 potential? 00:31:43.670 --> 00:31:44.169 No. 00:31:44.169 --> 00:31:46.259 You measure a difference. 00:31:46.259 --> 00:31:47.550 Do you ever measure the energy? 00:31:47.550 --> 00:31:49.296 No, you measure a difference in energy. 00:31:49.296 --> 00:31:51.670 So the absolute value of energy is sort of a silly thing. 00:31:51.670 --> 00:31:54.440 But we always talk about it as if it's not. 00:31:54.440 --> 00:31:55.680 We say, that's got energy 14. 00:31:55.680 --> 00:31:58.200 It's a little bit suspicious. 00:31:58.200 --> 00:32:00.480 So to answer your question, there's 00:32:00.480 --> 00:32:02.290 nothing hallowed about the number 0, 00:32:02.290 --> 00:32:04.282 although we will often refer to zero energy 00:32:04.282 --> 00:32:05.490 with a very specific meaning. 00:32:05.490 --> 00:32:06.980 What we really mean in that case is 00:32:06.980 --> 00:32:09.420 the value of the potential energy at infinity. 00:32:09.420 --> 00:32:11.220 So when I say energy, usually what I mean 00:32:11.220 --> 00:32:12.790 is relative to the value at infinity. 00:32:12.790 --> 00:32:14.415 So then let me ask your question again. 00:32:14.415 --> 00:32:16.510 Your question is it possible to have energy 0? 00:32:16.510 --> 00:32:17.802 Absolutely, and we'll see that. 00:32:17.802 --> 00:32:20.093 And it's actually going to be really interesting what's 00:32:20.093 --> 00:32:21.910 true of states with energy 0 in that sense. 00:32:21.910 --> 00:32:23.451 Second part of your question, though, 00:32:23.451 --> 00:32:25.690 is how does energy being 0 fit into this? 00:32:25.690 --> 00:32:26.970 Well, does that save us? 00:32:26.970 --> 00:32:29.210 Suppose one of the energies is 0. 00:32:29.210 --> 00:32:31.690 Then that says E on psi 1 plus psi 2 is equal to, 00:32:31.690 --> 00:32:34.790 let's say E2 is 0. 00:32:34.790 --> 00:32:35.950 Well, that term is gone. 00:32:35.950 --> 00:32:37.074 So there's just the one E1. 00:32:37.074 --> 00:32:38.657 Are we in energy eigenstate? 00:32:38.657 --> 00:32:40.240 No, because it's still not of the form 00:32:40.240 --> 00:32:42.290 E times psi 1 plus psi 2. 00:32:42.290 --> 00:32:44.740 So it doesn't save us, but it's an interesting question 00:32:44.740 --> 00:32:46.200 for the future. 00:32:46.200 --> 00:32:48.290 All right. 00:32:48.290 --> 00:32:50.935 Next question, four parts. 00:33:07.100 --> 00:33:10.250 So the question says x and p commute to i h bar. 00:33:10.250 --> 00:33:11.470 We've shown this. 00:33:11.470 --> 00:33:14.580 Is p x equal to i h bar, and is ip plus cx 00:33:14.580 --> 00:33:15.910 the same as cx plus ip? 00:33:29.920 --> 00:33:33.740 If you're really unsure you can ask the person next to you, 00:33:33.740 --> 00:33:37.170 but you don't have to. 00:33:37.170 --> 00:33:41.290 OK, so this is looking good. 00:33:41.290 --> 00:33:44.060 Everyone have an answer in? 00:33:44.060 --> 00:33:44.560 No? 00:33:49.770 --> 00:33:55.310 Five, four, three, two, one, OK, good. 00:33:55.310 --> 00:34:03.242 So the answer is C, which most of you got, but not everyone. 00:34:03.242 --> 00:34:05.200 A bunch of you put D. So let's talk through it. 00:34:05.200 --> 00:34:10.639 So remember what the definition of the commutator is. 00:34:10.639 --> 00:34:15.219 x with p by definition is equal to xp minus px. 00:34:15.219 --> 00:34:18.500 If we change the order here, px is 00:34:18.500 --> 00:34:22.630 equal to minus this, px minus xp. 00:34:22.630 --> 00:34:24.620 It's just the definition of the commutator. 00:34:24.620 --> 00:34:27.800 So on the other hand, if you add things, does 7 plus 6 00:34:27.800 --> 00:34:28.940 equal 6 plus 7? 00:34:28.940 --> 00:34:29.440 Yeah. 00:34:29.440 --> 00:34:31.116 Well, of course 6 times 7 is 7 times 6. 00:34:31.116 --> 00:34:32.699 So that's not a terribly good analogy. 00:34:36.880 --> 00:34:39.880 Does the order of addition of operators matter? 00:34:39.880 --> 00:34:42.540 No. 00:34:42.540 --> 00:34:44.639 Yeah. 00:34:44.639 --> 00:34:45.630 Yeah, exactly. 00:34:45.630 --> 00:34:46.130 Exactly. 00:34:46.130 --> 00:34:47.130 So it's slightly sneaky. 00:34:47.130 --> 00:34:49.840 OK, next question. 00:34:49.840 --> 00:34:51.170 OK, this one has five. 00:34:54.860 --> 00:34:56.670 f and g are both wave functions. 00:34:56.670 --> 00:34:58.090 c is a constant. 00:34:58.090 --> 00:35:03.150 Then if we take the inner product c times f with g, 00:35:03.150 --> 00:35:05.160 this is equal to what? 00:35:21.000 --> 00:35:24.240 Three, two, one, OK. 00:35:24.240 --> 00:35:25.720 So the answer is-- 00:35:25.720 --> 00:35:27.610 so this one definitely discuss. 00:35:27.610 --> 00:35:29.778 Discuss with the person next to you. 00:35:29.778 --> 00:35:55.520 [CHATTER] 00:35:55.520 --> 00:35:56.280 All right. 00:36:01.440 --> 00:36:05.910 OK, go ahead and enter your guess again, 00:36:05.910 --> 00:36:08.490 or your answer again, let it not be a guess. 00:36:11.220 --> 00:36:12.855 OK, 10 seconds. 00:36:15.450 --> 00:36:15.950 Wow. 00:36:15.950 --> 00:36:16.630 OK, fantastic. 00:36:16.630 --> 00:36:17.630 That works like a champ. 00:36:17.630 --> 00:36:19.040 So what's the answer? 00:36:19.040 --> 00:36:20.970 Yes, complex conjugation. 00:36:20.970 --> 00:36:21.970 Don't screw that one up. 00:36:21.970 --> 00:36:24.110 It's very easy to forget, but it matters a lot. 00:36:28.297 --> 00:36:29.380 Cursor keeps disappearing. 00:36:29.380 --> 00:36:32.360 OK, next one. 00:36:32.360 --> 00:36:40.120 A wave function has been expressed 00:36:40.120 --> 00:36:41.570 as a sum of energy eigenfunctions. 00:36:41.570 --> 00:36:45.445 Here I'm calling them mu rather than phi, but same thing. 00:36:45.445 --> 00:36:47.070 Compared to the original wave function, 00:36:47.070 --> 00:36:49.920 the set of coefficients, given that we're using the energy 00:36:49.920 --> 00:36:52.730 basis, the set of coefficients contains more or less 00:36:52.730 --> 00:36:55.390 the same information, or it can't be determined. 00:37:00.650 --> 00:37:02.360 OK, five seconds. 00:37:06.990 --> 00:37:07.780 All right. 00:37:07.780 --> 00:37:11.660 And the answer is C, great. 00:37:11.660 --> 00:37:13.390 OK, next one. 00:37:16.429 --> 00:37:17.720 So right now we're normalizing. 00:37:22.540 --> 00:37:23.470 OK. 00:37:23.470 --> 00:37:25.820 All stationary states, or all energy eigenstates, 00:37:25.820 --> 00:37:28.570 have the form that spatial and time dependence 00:37:28.570 --> 00:37:30.760 is the spatial dependence, the energy eigenfunction, 00:37:30.760 --> 00:37:35.110 times a phase, so that the norm squared is time independent. 00:37:35.110 --> 00:37:37.290 Consider the sum of two non-degenerate energy 00:37:37.290 --> 00:37:40.280 eigenstates psi 1 and psi 2. 00:37:40.280 --> 00:37:42.990 Non-degenerate means they have different energy. 00:37:45.540 --> 00:37:47.140 Is the wave function stationary? 00:37:47.140 --> 00:37:48.840 Is the probability distribution time 00:37:48.840 --> 00:37:50.420 independent or is it time dependent? 00:38:00.600 --> 00:38:02.980 This one's not trivial. 00:38:02.980 --> 00:38:03.530 Oh, shoot. 00:38:03.530 --> 00:38:04.730 I forgot to get it started. 00:38:04.730 --> 00:38:06.399 Sorry. 00:38:06.399 --> 00:38:08.940 It's particularly non-trivial if you can't enter your answer. 00:38:08.940 --> 00:38:09.210 Right. 00:38:09.210 --> 00:38:10.626 So go ahead and enter your answer. 00:38:17.438 --> 00:38:19.420 Whoo, yeah. 00:38:19.420 --> 00:38:20.715 This one always kills people. 00:38:31.315 --> 00:38:32.190 No chatting just yet. 00:38:32.190 --> 00:38:33.565 Test yourself, not your neighbor. 00:38:38.427 --> 00:38:40.010 It's fine to look deep into your soul, 00:38:40.010 --> 00:38:41.980 but don't look deep into the soul of the person sitting next 00:38:41.980 --> 00:38:42.480 to you. 00:38:46.660 --> 00:38:47.590 All right. 00:38:47.590 --> 00:38:51.840 So at this point, chat with your neighbor. 00:38:51.840 --> 00:38:54.086 Let me just give you some presage. 00:38:54.086 --> 00:38:57.720 The parallel strategy's probably not so good, because about half 00:38:57.720 --> 00:39:00.273 of you got it right, and about half of you got it wrong. 00:39:00.273 --> 00:40:07.070 [CHATTER] 00:40:07.070 --> 00:40:07.950 All right. 00:40:07.950 --> 00:40:09.375 Let's vote again. 00:40:11.980 --> 00:40:15.720 And hold on, starting now. 00:40:15.720 --> 00:40:16.480 OK, vote again. 00:40:16.480 --> 00:40:18.063 You've got 10 seconds to enter a vote. 00:40:24.520 --> 00:40:25.710 Wow. 00:40:25.710 --> 00:40:28.110 OK, two seconds. 00:40:28.110 --> 00:40:29.020 Good. 00:40:29.020 --> 00:40:36.940 So the distribution on this one went from 30, 50, 00:40:36.940 --> 00:40:47.290 20 initially, to now it is 10, 80, and 10. 00:40:47.290 --> 00:40:51.030 Amazingly, you guys got worse. 00:40:51.030 --> 00:40:55.620 The answer is C. And I want you to discuss with each other 00:40:55.620 --> 00:40:57.174 why it's C. 00:40:57.174 --> 00:41:27.840 [CHATTER] 00:41:27.840 --> 00:41:29.420 All right. 00:41:29.420 --> 00:41:32.060 OK. 00:41:32.060 --> 00:41:33.830 So let me talk you through it. 00:41:33.830 --> 00:41:36.230 So the wave function, we've said psi of x and t 00:41:36.230 --> 00:41:45.865 is equal to phi 1 at x, e to the minus i omega 1 t plus phi 00:41:45.865 --> 00:41:50.940 2 of x e to the minus i omega 2 t. 00:41:50.940 --> 00:41:52.670 So great, we take the norm squared. 00:41:52.670 --> 00:41:55.342 What's the probability to find it at x at time t. 00:41:55.342 --> 00:41:56.800 The probability density is the norm 00:41:56.800 --> 00:41:59.260 squared of this guy, psi squared, which 00:41:59.260 --> 00:42:04.590 is equal to phi 1 complex conjugate e to the plus i omega 00:42:04.590 --> 00:42:12.200 1 t plus phi 2 complex conjugate e to the plus i omega 2t times 00:42:12.200 --> 00:42:16.210 the thing itself phi 1 of x e to the minus i 00:42:16.210 --> 00:42:23.320 omega 1 t plus phi 2 of x e to the minus i omega 2t, right? 00:42:23.320 --> 00:42:25.870 So this has four terms. 00:42:25.870 --> 00:42:28.470 The first term is psi 1 norm squared. 00:42:28.470 --> 00:42:30.650 The phases cancel, right? 00:42:30.650 --> 00:42:33.340 You're going to see this happen a billion times in 804. 00:42:33.340 --> 00:42:35.850 The first term is going to be phi 1 norm squared. 00:42:35.850 --> 00:42:37.970 There's another term, which is phi 2 norm squared. 00:42:37.970 --> 00:42:40.345 Again the phases exactly cancel, even the minus i omega 2 00:42:40.345 --> 00:42:42.080 t to the plus i omega 2 t. 00:42:42.080 --> 00:42:44.470 Plus phi 2 squared. 00:42:44.470 --> 00:42:47.280 But then there are two cross terms, the interference terms. 00:42:47.280 --> 00:42:55.180 Plus phi 1 complex conjugate phi 2 e to the i omega 1 t 00:42:55.180 --> 00:43:00.670 e to the plus i omega 1 t, i omega 1 t, and e to the minus 00:43:00.670 --> 00:43:01.869 i omega 2t, minus omega 2. 00:43:01.869 --> 00:43:04.160 So we have a cross-term which depends on the difference 00:43:04.160 --> 00:43:05.194 in frequencies. 00:43:05.194 --> 00:43:07.110 Frequencies are like energies modulo on h-bar, 00:43:07.110 --> 00:43:09.240 so it's a difference in energies. 00:43:09.240 --> 00:43:10.740 And then there's another term, which 00:43:10.740 --> 00:43:12.670 is the complex conjugate of this guy, 00:43:12.670 --> 00:43:15.950 phi 2 star times phi 1 phi 2 complex conjugate phi 1 00:43:15.950 --> 00:43:17.825 and the phases are also the complex conjugate 00:43:17.825 --> 00:43:28.350 e to the minus i omega 1 minus omega 2 t of x of x of x of x. 00:43:28.350 --> 00:43:31.880 So is there time dependence in this, in principle? 00:43:31.880 --> 00:43:33.890 Absolutely, from the interference terms. 00:43:33.890 --> 00:43:35.500 Were we not in the superposition, 00:43:35.500 --> 00:43:37.470 we would not have interference terms. 00:43:37.470 --> 00:43:40.330 Time dependence comes from interference, when we expand 00:43:40.330 --> 00:43:41.600 in energy eigenfunctions. 00:43:41.600 --> 00:43:43.187 Cool? 00:43:43.187 --> 00:43:44.270 However, can these vanish? 00:43:44.270 --> 00:43:44.769 When? 00:43:48.450 --> 00:43:50.460 Sorry, say again? 00:43:50.460 --> 00:43:54.577 Great, so when omega 1 equals omega 2, what happens? 00:43:54.577 --> 00:43:55.660 Time dependence goes away. 00:43:55.660 --> 00:44:00.750 But omega 1 is e 1 over h bar, omega 2 is e 2 over h bar, 00:44:00.750 --> 00:44:03.595 and we started out by saying these are non-degenerate. 00:44:03.595 --> 00:44:05.970 So if they're non-degenerate, the energies are different, 00:44:05.970 --> 00:44:08.361 the frequencies are different, so that doesn't help us. 00:44:08.361 --> 00:44:09.860 How do we kill this time dependence? 00:44:12.860 --> 00:44:13.399 Yes. 00:44:13.399 --> 00:44:15.190 If the two functions aren't just orthogonal 00:44:15.190 --> 00:44:17.150 in a functional sense, but if we have the following. 00:44:17.150 --> 00:44:18.410 Suppose phi 1 is like this. 00:44:18.410 --> 00:44:21.430 It's 0 everywhere except for in some lump that's phi 1, 00:44:21.430 --> 00:44:25.230 and phi 2 is 0 everywhere except here. 00:44:25.230 --> 00:44:27.670 Then anywhere that phi 1 is non-zero, phi 2 is zero. 00:44:27.670 --> 00:44:30.790 And anywhere where phi 2 is non-zero, phi 1 is zero. 00:44:30.790 --> 00:44:33.940 So this can point-wise vanish. 00:44:33.940 --> 00:44:37.250 Do you expect this to happen generically? 00:44:37.250 --> 00:44:39.850 Does it happen for the energy eigenfunctions 00:44:39.850 --> 00:44:41.270 in the infinite square well? 00:44:44.710 --> 00:44:45.360 Sine waves? 00:44:47.981 --> 00:44:48.480 No. 00:44:48.480 --> 00:44:50.100 They have zero at isolated points, 00:44:50.100 --> 00:44:53.019 but they're non-zero generically. 00:44:53.019 --> 00:44:54.310 Yeah, so it doesn't work there. 00:44:54.310 --> 00:44:55.875 What about for the free particle? 00:44:55.875 --> 00:44:57.250 Well, those are just plain waves. 00:44:57.250 --> 00:44:58.680 Does that ever happen? 00:44:58.680 --> 00:44:59.610 No. 00:44:59.610 --> 00:45:02.540 OK, so this is an incredibly special case. 00:45:02.540 --> 00:45:04.040 We'll actually see it in one problem 00:45:04.040 --> 00:45:05.485 on a problem set later on. 00:45:05.485 --> 00:45:07.130 It's a very special case. 00:45:07.130 --> 00:45:09.935 So technically, the answer is C. And I 00:45:09.935 --> 00:45:11.570 want you guys to keep your minds open 00:45:11.570 --> 00:45:15.310 on these sorts of questions, when does a spatial dependence 00:45:15.310 --> 00:45:17.370 matter and when are there interference terms. 00:45:17.370 --> 00:45:18.260 Those are two different questions, 00:45:18.260 --> 00:45:19.718 and I want you to tease them apart. 00:45:19.718 --> 00:45:21.490 OK? 00:45:21.490 --> 00:45:22.590 Cool? 00:45:22.590 --> 00:45:23.784 Yeah? 00:45:23.784 --> 00:45:26.462 AUDIENCE: Is a valid way to think about this 00:45:26.462 --> 00:45:34.594 to think that you're fixing the initial [INAUDIBLE] 00:45:34.594 --> 00:45:36.760 PROFESSOR: That's a very good way to think about it. 00:45:36.760 --> 00:45:39.150 That's exactly right. 00:45:39.150 --> 00:45:41.082 That's a very, very good question. 00:45:41.082 --> 00:45:43.040 Let me say that subtly differently, and tell me 00:45:43.040 --> 00:45:44.956 if this agrees with what you were just saying. 00:45:44.956 --> 00:45:47.806 So I can look at this wave function, 00:45:47.806 --> 00:45:49.930 and I already know that the overall phase of a wave 00:45:49.930 --> 00:45:51.150 function doesn't matter. 00:45:51.150 --> 00:45:52.960 That's what it is to say a stationary state is stationary. 00:45:52.960 --> 00:45:54.960 It's got an overall phase that's the only thing, 00:45:54.960 --> 00:45:56.540 norm squared it goes away. 00:45:56.540 --> 00:46:01.950 So I can write this as e to the minus i omega 1 t times phi 00:46:01.950 --> 00:46:08.750 1 of x plus phi 2 of x e to the minus i omega 00:46:08.750 --> 00:46:12.506 2 minus omega 1 t. 00:46:12.506 --> 00:46:13.920 Is that what you mean? 00:46:13.920 --> 00:46:15.170 So that's one way to do this. 00:46:15.170 --> 00:46:16.503 We could also do something else. 00:46:16.503 --> 00:46:22.752 We could do e to the minus i omega 1 plus omega 2 upon 2 t. 00:46:22.752 --> 00:46:24.210 And this is more, I think, what you 00:46:24.210 --> 00:46:26.085 were thinking of, a sort of average frequency 00:46:26.085 --> 00:46:28.070 and then a relative frequency, and then 00:46:28.070 --> 00:46:31.050 the change in the frequencies on these two terms. 00:46:31.050 --> 00:46:31.760 Absolutely. 00:46:31.760 --> 00:46:36.720 So you can organize this in many, many ways. 00:46:36.720 --> 00:46:39.670 But your question gets at a very important point, 00:46:39.670 --> 00:46:41.670 which is that the overall phase doesn't matter. 00:46:41.670 --> 00:46:46.400 But relative phases in a superposition do matter. 00:46:46.400 --> 00:46:48.727 So when does a phase matter in a wave function? 00:46:48.727 --> 00:46:50.560 It does not matter if it's an overall phase. 00:46:50.560 --> 00:46:53.170 But it does matter if it's a relative phase between terms 00:46:53.170 --> 00:46:54.790 in a superposition. 00:46:54.790 --> 00:46:55.870 Cool? 00:46:55.870 --> 00:46:57.510 Very good question. 00:46:57.510 --> 00:47:00.330 Other questions? 00:47:00.330 --> 00:47:03.550 If not, then I have some. 00:47:03.550 --> 00:47:07.930 So, consider a system which is in the state-- 00:47:07.930 --> 00:47:10.227 so I should give you five-- 00:47:10.227 --> 00:47:12.810 system is in a state which is a linear combination of n equals 00:47:12.810 --> 00:47:14.280 1 and n equals 2 eigenstates. 00:47:16.956 --> 00:47:18.580 What's the probability that measurement 00:47:18.580 --> 00:47:20.082 will give us energy E1? 00:47:20.082 --> 00:47:21.373 And it's in this superposition. 00:47:28.390 --> 00:47:29.215 OK, five seconds. 00:47:33.780 --> 00:47:34.660 OK, fantastic. 00:47:34.660 --> 00:47:36.450 What's the answer? 00:47:36.450 --> 00:47:37.497 Yes, C, great. 00:47:37.497 --> 00:47:38.580 OK, everyone got that one. 00:47:38.580 --> 00:47:42.100 So one's a slightly more interesting question. 00:47:45.580 --> 00:47:55.210 Suppose I have an infinite well with width L. 00:47:55.210 --> 00:47:57.150 How does the energy, the ground state energy, 00:47:57.150 --> 00:48:01.175 compare to that of a system with a wider well? 00:48:16.070 --> 00:48:21.335 So L versus a larger L. OK, four seconds. 00:48:25.140 --> 00:48:28.940 OK, quickly discuss amongst yourselves, like 10 seconds. 00:48:28.940 --> 00:48:45.160 [CHATTER] 00:48:45.160 --> 00:48:47.300 All right. 00:48:47.300 --> 00:48:48.185 Now click again. 00:48:51.493 --> 00:48:51.993 Yeah. 00:48:56.080 --> 00:48:57.520 All right. 00:48:57.520 --> 00:48:58.260 Five seconds. 00:48:58.260 --> 00:49:02.270 One, two, three, four, five, great. 00:49:02.270 --> 00:49:08.470 OK, the answer is A. OK, great, because the energy 00:49:08.470 --> 00:49:11.530 of the infinite well goes like K squared. 00:49:11.530 --> 00:49:18.974 K goes like 1 over L. So the energy is, if we make it wider, 00:49:18.974 --> 00:49:21.140 the energy if we make it wider is going to be lower. 00:49:27.260 --> 00:49:28.725 And last couple of questions. 00:49:31.510 --> 00:49:34.900 OK, so t equals 0. 00:49:34.900 --> 00:49:38.350 Could the wave function for an electron in an infinite square 00:49:38.350 --> 00:49:44.880 well of width a, rather than L, be A sine squared of pi x 00:49:44.880 --> 00:49:48.680 upon a, where A is suitably chosen to be normalized? 00:50:10.460 --> 00:50:13.870 All right, you've got about five seconds left. 00:50:17.550 --> 00:50:21.420 And OK, we are at chance. 00:50:21.420 --> 00:50:24.710 We are at even odds, and the answer 00:50:24.710 --> 00:50:28.380 is not a superposition of A and B, 00:50:28.380 --> 00:50:31.130 so I encourage you to discuss with the people around you. 00:50:31.130 --> 00:50:34.567 [CHATTER] 00:50:51.731 --> 00:50:52.230 Great. 00:50:52.230 --> 00:50:54.400 What properties had it better satisfy in order 00:50:54.400 --> 00:50:56.774 to be a viable wave function? 00:50:56.774 --> 00:50:58.440 What properties should the wave function 00:50:58.440 --> 00:51:00.391 have so that it's reasonable? 00:51:00.391 --> 00:51:00.890 Yeah. 00:51:00.890 --> 00:51:02.080 Is it zero at the ends? 00:51:02.080 --> 00:51:03.001 Yeah. 00:51:03.001 --> 00:51:03.500 Good. 00:51:03.500 --> 00:51:04.841 Is it smooth? 00:51:04.841 --> 00:51:07.100 Yeah. 00:51:07.100 --> 00:51:07.830 Exactly. 00:51:07.830 --> 00:51:09.691 And so you can write it as a superposition. 00:51:09.691 --> 00:51:10.190 Excellent. 00:51:10.190 --> 00:51:11.692 So the answer is? 00:51:11.692 --> 00:51:12.192 Yeah. 00:51:17.516 --> 00:51:18.015 All right. 00:51:20.780 --> 00:51:21.750 Vote again. 00:51:25.170 --> 00:51:26.794 OK, I might have missed a few people. 00:51:26.794 --> 00:51:27.710 So go ahead and start. 00:51:31.800 --> 00:51:33.185 OK, five more seconds. 00:51:37.230 --> 00:51:37.950 All right. 00:51:37.950 --> 00:51:43.820 So we went from 50-50 to 77-23. 00:51:43.820 --> 00:51:44.820 That's pretty good. 00:51:44.820 --> 00:51:47.260 What's the answer? 00:51:47.260 --> 00:51:47.890 A. Why? 00:51:51.710 --> 00:51:53.892 Is this an energy eigenstate? 00:51:53.892 --> 00:51:55.160 No. 00:51:55.160 --> 00:51:56.960 Does that matter? 00:51:56.960 --> 00:51:57.620 No. 00:51:57.620 --> 00:51:59.710 What properties had this wave function better 00:51:59.710 --> 00:52:05.460 satisfy to be a reasonable wave function in this potential? 00:52:05.460 --> 00:52:06.660 Say again? 00:52:06.660 --> 00:52:08.020 It's got to vanish at the walls. 00:52:08.020 --> 00:52:09.853 It's got to satisfy the boundary conditions. 00:52:09.853 --> 00:52:12.290 What else must be true of this wave function? 00:52:12.290 --> 00:52:13.010 Normalizable. 00:52:13.010 --> 00:52:14.240 Is it normalizable? 00:52:14.240 --> 00:52:14.880 Yeah. 00:52:14.880 --> 00:52:16.400 What else? 00:52:16.400 --> 00:52:16.900 Continuous. 00:52:16.900 --> 00:52:18.524 It better not have any discontinuities. 00:52:18.524 --> 00:52:19.471 Is it continuous? 00:52:19.471 --> 00:52:19.970 Great. 00:52:19.970 --> 00:52:20.470 OK. 00:52:20.470 --> 00:52:23.691 Is there any reason that this is a stupid wave function? 00:52:23.691 --> 00:52:24.190 No. 00:52:24.190 --> 00:52:25.500 It's perfectly reasonable. 00:52:25.500 --> 00:52:29.930 It's not an energy eigenfunction, but-- 00:52:29.930 --> 00:52:31.330 Yeah, cool? 00:52:31.330 --> 00:52:32.150 Yeah. 00:52:32.150 --> 00:52:34.252 AUDIENCE: This is sort of like a math question. 00:52:34.252 --> 00:52:35.768 So to write that at a superposition, 00:52:35.768 --> 00:52:38.636 you have to write it like basically a Fourier sign 00:52:38.636 --> 00:52:39.592 series? 00:52:39.592 --> 00:52:42.940 Isn't the [INAUDIBLE] function even, though? 00:52:42.940 --> 00:52:46.426 PROFESSOR: On this domain, that and the sines are even. 00:52:46.426 --> 00:52:48.925 So this is actually odd, but we're only looking at it from 0 00:52:48.925 --> 00:52:53.480 to L. So, I mean that half of it. 00:52:53.480 --> 00:52:56.080 The sines are odd, but we're only looking at the first peak. 00:52:56.080 --> 00:52:56.780 So you could just as well have written 00:52:56.780 --> 00:52:59.420 that as cosine of the midpoint plus the distance 00:52:59.420 --> 00:53:01.595 from the midpoint. 00:53:01.595 --> 00:53:03.470 Actually, let me say that again, because it's 00:53:03.470 --> 00:53:05.511 a much better question I just give it shrift for. 00:53:10.050 --> 00:53:11.010 So here's the question. 00:53:11.010 --> 00:53:15.359 The question is, look, so sine is an odd function, 00:53:15.359 --> 00:53:16.900 but sine squared is an even function. 00:53:16.900 --> 00:53:19.670 So how can you expand sine squared, an even function, 00:53:19.670 --> 00:53:21.780 in terms of sines, an odd function? 00:53:21.780 --> 00:53:23.860 But think about this physically. 00:53:23.860 --> 00:53:30.570 Here's sine squared in our domain, and here's sine. 00:53:30.570 --> 00:53:31.810 Now what do you mean by even? 00:53:31.810 --> 00:53:33.905 Usually by even we mean reflection around zero. 00:53:33.905 --> 00:53:35.780 But I could just as well have said reflection 00:53:35.780 --> 00:53:36.770 around the origin. 00:53:36.770 --> 00:53:39.030 This potential is symmetric. 00:53:39.030 --> 00:53:41.745 And the energy eigenfunctions are symmetric about the origin. 00:53:41.745 --> 00:53:44.120 They're not symmetric about reflection around this point. 00:53:44.120 --> 00:53:46.380 But they are symmetric about reflection around this point. 00:53:46.380 --> 00:53:48.210 That's a particularly natural place to call it 0. 00:53:48.210 --> 00:53:49.930 So I was calling them sine because I was calling this 0, 00:53:49.930 --> 00:53:51.304 but I could have called it cosine 00:53:51.304 --> 00:53:55.470 if I called this 0, for the same Kx. 00:53:55.470 --> 00:53:58.570 And indeed, can we expand this sine 00:53:58.570 --> 00:54:01.570 squared function in terms of a basis of these sines 00:54:01.570 --> 00:54:02.610 on the domain 0 to L? 00:54:02.610 --> 00:54:04.030 Absolutely. 00:54:04.030 --> 00:54:07.220 Very good question. 00:54:07.220 --> 00:54:09.355 And lastly, last clicker question. 00:54:12.700 --> 00:54:13.460 Oops. 00:54:13.460 --> 00:54:16.030 Whatever. 00:54:16.030 --> 00:54:17.210 OK. 00:54:17.210 --> 00:54:20.600 At t equals 0, a particle is described by the wave function 00:54:20.600 --> 00:54:21.130 we just saw. 00:54:23.461 --> 00:54:25.710 Which of the following is true about the wave function 00:54:25.710 --> 00:54:26.600 at subsequent times? 00:54:37.650 --> 00:54:38.210 5 seconds. 00:54:42.776 --> 00:54:43.730 Whew. 00:54:43.730 --> 00:54:44.650 Oh, OK. 00:54:44.650 --> 00:54:47.360 In the last few seconds we had an explosive burst for A, B, 00:54:47.360 --> 00:54:49.870 and C. So our current distribution 00:54:49.870 --> 00:54:54.960 is 8, 16, 10, and 67, sounds like 67 is popular. 00:54:54.960 --> 00:54:58.630 Discuss quickly, very quickly, with the person next to you. 00:54:58.630 --> 00:55:02.510 [CHATTER] 00:55:13.670 --> 00:55:17.093 OK, and vote again. 00:55:21.580 --> 00:55:22.980 OK, five seconds. 00:55:22.980 --> 00:55:24.180 Get your last vote in. 00:55:28.240 --> 00:55:29.300 All right. 00:55:29.300 --> 00:55:33.472 And the answer is D. Yay. 00:55:37.289 --> 00:55:38.580 So let's think about the logic. 00:55:38.580 --> 00:55:39.950 Let's go through the logic here. 00:55:39.950 --> 00:55:42.360 So as was pointed out by a student up here earlier, 00:55:42.360 --> 00:55:46.070 the wave function sine squared of pi x 00:55:46.070 --> 00:55:49.100 can be expanded in terms of the energy eigenfunction. 00:55:49.100 --> 00:55:51.150 Any reasonable function can be expanded 00:55:51.150 --> 00:55:53.800 in terms of a superposition of definite energy states 00:55:53.800 --> 00:55:55.600 of energy eigenfunctions. 00:55:55.600 --> 00:55:58.490 So that means we can write psi at some time 00:55:58.490 --> 00:56:04.060 as a superposition Cn sine of n pi x upon a e to the minus i e 00:56:04.060 --> 00:56:09.450 n t upon h bar, since those are, in fact, the eigenfunctions. 00:56:09.450 --> 00:56:10.200 So we can do that. 00:56:10.200 --> 00:56:11.877 Now, when we look at the time evolution, 00:56:11.877 --> 00:56:13.710 we know that each term in that superposition 00:56:13.710 --> 00:56:15.450 evolves with a phase. 00:56:15.450 --> 00:56:19.220 The overall wave function does not evolve with a phase. 00:56:19.220 --> 00:56:21.360 It is not an energy eigenstate. 00:56:21.360 --> 00:56:23.460 There are going to be interference terms due 00:56:23.460 --> 00:56:26.207 to the fact that it's a superposition. 00:56:26.207 --> 00:56:28.540 So its probability distribution is not time-independent. 00:56:28.540 --> 00:56:29.640 It is a superposition. 00:56:29.640 --> 00:56:33.430 And so the wave function doesn't rotate by an overall phase. 00:56:33.430 --> 00:56:37.030 However, we can solve the Schrodinger equation, 00:56:37.030 --> 00:56:37.810 as we did before. 00:56:37.810 --> 00:56:40.440 The wave function is expanded at time 0 00:56:40.440 --> 00:56:43.544 as the energy eigenfunctions times some set of coefficients. 00:56:43.544 --> 00:56:44.960 And the time evolution corresponds 00:56:44.960 --> 00:56:48.230 to adding two each independent term in the superposition 00:56:48.230 --> 00:56:52.120 the appropriate phase for that energy eigenstate. 00:56:52.120 --> 00:56:54.040 Cool? 00:56:54.040 --> 00:56:54.540 All right. 00:56:54.540 --> 00:57:03.220 So the answer is D. And that's it for the clicker questions. 00:57:03.220 --> 00:57:07.050 OK, so any questions on the clicker questions so far? 00:57:07.050 --> 00:57:09.050 OK, those are going to be posted on the web site 00:57:09.050 --> 00:57:12.190 so you can go over them. 00:57:12.190 --> 00:57:14.250 And now back to energy eigenfunctions. 00:57:24.770 --> 00:57:27.730 So what I want to talk about now is the qualitative behavior 00:57:27.730 --> 00:57:29.140 of energy eigenfunctions. 00:57:29.140 --> 00:57:31.460 Suppose I know I have an energy eigenfunction. 00:57:31.460 --> 00:57:33.955 What can I say generally about its structure? 00:57:38.870 --> 00:57:41.175 So let me ask the question, qualitative behavior. 00:57:47.270 --> 00:57:49.670 So suppose someone hands you a potential U of x. 00:57:49.670 --> 00:57:51.990 Someone hands you some potential, U of x, 00:57:51.990 --> 00:57:54.900 and says, look, I've got this potential. 00:57:54.900 --> 00:57:57.580 Maybe I'll draw it for you. 00:57:57.580 --> 00:57:59.850 It's got some wiggles, and then a big wiggle, 00:57:59.850 --> 00:58:01.880 and then it's got a big wiggle, and then-- 00:58:01.880 --> 00:58:03.195 do I want to do that? 00:58:03.195 --> 00:58:04.275 Yeah, let's do that. 00:58:04.275 --> 00:58:08.089 Then a big wiggle, and something like this. 00:58:08.089 --> 00:58:09.630 And someone shows you this potential. 00:58:09.630 --> 00:58:11.921 And they say, look, what are the energy eigenfunctions? 00:58:13.770 --> 00:58:17.510 Well, OK, free particle was easy. 00:58:17.510 --> 00:58:19.070 The infinite square well was easy. 00:58:19.070 --> 00:58:21.207 We could solve that analytically. 00:58:21.207 --> 00:58:23.290 The next involved solving a differential equation. 00:58:23.290 --> 00:58:25.360 So what differential equation is this going to lead us to? 00:58:25.360 --> 00:58:27.570 Well, we know that the energy eigenvalue equation 00:58:27.570 --> 00:58:32.070 is minus h bar squared upon 2 m phi 00:58:32.070 --> 00:58:38.570 prime prime of x plus U of x phi x, so that's the energy 00:58:38.570 --> 00:58:41.620 operator acting on phi, is equal to, 00:58:41.620 --> 00:58:47.949 saying that it's an energy eigenfunction, phi sub E, 00:58:47.949 --> 00:58:49.740 says that it's equal to the energy operator 00:58:49.740 --> 00:58:52.670 acting on this eigenfunction is just a constant E phi sub 00:58:52.670 --> 00:58:54.870 E of x. 00:58:54.870 --> 00:58:56.680 And I'm going to work at moment in time, 00:58:56.680 --> 00:59:00.420 so we're going to drop all the t dependence for the moment. 00:59:00.420 --> 00:59:01.920 So this is the differential equation 00:59:01.920 --> 00:59:07.072 we need to solve where U of x is this god-awful function. 00:59:07.072 --> 00:59:08.780 Do you think it's very likely that you're 00:59:08.780 --> 00:59:11.670 going to be able to solve this analytically? 00:59:11.670 --> 00:59:13.700 Probably not. 00:59:13.700 --> 00:59:18.761 However, some basic ideas will help 00:59:18.761 --> 00:59:21.010 you get an intuition for what the wave function should 00:59:21.010 --> 00:59:22.240 look like. 00:59:22.240 --> 00:59:25.910 And I cannot overstate the importance of being able 00:59:25.910 --> 00:59:29.560 to eyeball a system and guess the qualitative features of its 00:59:29.560 --> 00:59:32.030 wave functions, because that intuition, 00:59:32.030 --> 00:59:34.590 that ability to estimate, is going to contain an awful lot 00:59:34.590 --> 00:59:35.340 of physics. 00:59:35.340 --> 00:59:36.790 So let's try to extract it. 00:59:36.790 --> 00:59:38.930 So in order to do so, I want to start 00:59:38.930 --> 00:59:40.945 by massaging this equation into a form which 00:59:40.945 --> 00:59:42.070 is particularly convenient. 00:59:44.790 --> 00:59:47.910 So in particular, I'm going to write this equation as phi sub 00:59:47.910 --> 00:59:49.449 E prime prime. 00:59:49.449 --> 00:59:51.740 So what I'm going to do is I'm going to take this term, 00:59:51.740 --> 00:59:53.615 I'm going to notice this has two derivatives, 00:59:53.615 --> 00:59:56.026 this has no derivatives, this has no derivatives. 00:59:56.026 --> 00:59:59.130 And I'm going to move this term over here and combine 00:59:59.130 --> 01:00:01.600 these terms into E minus U of x, and I'm 01:00:01.600 --> 01:00:05.320 going to divide each side by 2m upon h bar squared with a minus 01:00:05.320 --> 01:00:08.260 sign, giving me that phi prime prime of E 01:00:08.260 --> 01:00:14.420 of x upon phi E of x dividing through by this phi E 01:00:14.420 --> 01:00:18.250 is equal to minus 2m over h bar squared. 01:00:21.682 --> 01:00:23.140 And let's just get our signs right. 01:00:23.140 --> 01:00:24.560 We've got the minus from here, so this 01:00:24.560 --> 01:00:25.960 is going to be E minus U of x. 01:00:33.931 --> 01:00:35.680 So you might look at that and think, well, 01:00:35.680 --> 01:00:40.010 why is that any better than what I've just written down. 01:00:40.010 --> 01:00:42.724 But what is the second derivative of function? 01:00:42.724 --> 01:00:44.890 It's telling you not its slope, but it's telling you 01:00:44.890 --> 01:00:46.550 how the slope changes. 01:00:46.550 --> 01:00:49.588 It's telling about the curvature of the function. 01:00:49.588 --> 01:00:52.046 And what this is telling me is something very, very useful. 01:00:54.179 --> 01:00:55.970 So for example, let's look at the function. 01:00:55.970 --> 01:00:57.360 Let's assume that the function is real, 01:00:57.360 --> 01:00:58.901 although we know in general it's not. 01:00:58.901 --> 01:01:02.100 Let's assume that the function is real for simplicity. 01:01:02.100 --> 01:01:05.910 So we're going to plot the real part of phi 01:01:05.910 --> 01:01:07.210 in the vertical axis. 01:01:07.210 --> 01:01:08.630 And this is x. 01:01:08.630 --> 01:01:15.130 Suppose the real part of phi is positive at some point. 01:01:15.130 --> 01:01:17.070 Phi prime prime, if it's positive, 01:01:17.070 --> 01:01:19.297 tells us that not only is the slope positive, 01:01:19.297 --> 01:01:20.130 but it's increasing. 01:01:20.130 --> 01:01:21.600 Or it doesn't tell us anything about the slope, 01:01:21.600 --> 01:01:23.080 but it tells us that whatever the slope, it's increasing. 01:01:23.080 --> 01:01:25.650 If it's negative, the slope is increasing as we increase x. 01:01:25.650 --> 01:01:28.210 If it's positive, it's increasing as we increase x. 01:01:28.210 --> 01:01:31.450 So it's telling us that the wave function looks like this, 01:01:31.450 --> 01:01:34.880 locally, something like that. 01:01:34.880 --> 01:01:39.040 If phi is negative, if phi is negative, 01:01:39.040 --> 01:01:41.254 then if this quantity is positive, 01:01:41.254 --> 01:01:42.920 then phi prime prime has to be negative. 01:01:42.920 --> 01:01:45.150 But negative is curving down. 01:01:50.210 --> 01:01:54.780 So if this quantity, which I will call the curvature, 01:01:54.780 --> 01:02:01.260 if this quantity is positive, it curves away from the axis. 01:02:01.260 --> 01:02:07.100 So this is phi prime prime over phi greater than 0. 01:02:07.100 --> 01:02:09.700 If this quantity is positive, the function 01:02:09.700 --> 01:02:11.050 curves away from the axis. 01:02:11.050 --> 01:02:12.440 Cool? 01:02:12.440 --> 01:02:16.270 If this quantity is negative, phi prime prime upon 01:02:16.270 --> 01:02:19.490 phi less than 0, exactly the opposite. 01:02:19.490 --> 01:02:20.920 This has to be negative. 01:02:20.920 --> 01:02:23.480 If phi is positive, then phi prime prime has to be negative. 01:02:23.480 --> 01:02:24.907 It has to be curving down. 01:02:27.710 --> 01:02:30.290 And similarly, if phi is negative, 01:02:30.290 --> 01:02:33.170 then phi prime prime has to be positive, 01:02:33.170 --> 01:02:34.500 and it has to curve up. 01:02:34.500 --> 01:02:37.840 So if this quantity is positive, if the curvature is positive, 01:02:37.840 --> 01:02:39.440 it curves away from the axis. 01:02:39.440 --> 01:02:41.950 If the curvature is negative, if this quantity is negative, 01:02:41.950 --> 01:02:44.120 it curves towards the axis. 01:02:44.120 --> 01:02:47.590 So what does that tell you about solutions when the curvature is 01:02:47.590 --> 01:02:49.206 positive or negative? 01:02:49.206 --> 01:02:50.330 It tells you the following. 01:02:52.930 --> 01:02:54.980 It tells you that, imagine we have 01:02:54.980 --> 01:02:59.450 a function where phi prime prime over phi is constant. 01:02:59.450 --> 01:03:03.050 And in particular, let's let phi prime prime over phi 01:03:03.050 --> 01:03:06.760 be a constant, which is positive. 01:03:06.760 --> 01:03:10.671 And I'll call that positive constant kappa squared. 01:03:10.671 --> 01:03:12.170 And to emphasize that it's positive, 01:03:12.170 --> 01:03:14.174 I'm going to call it kappa squared. 01:03:14.174 --> 01:03:15.090 It's a positive thing. 01:03:15.090 --> 01:03:16.980 It's a real number squared. 01:03:16.980 --> 01:03:18.635 What does the solution look like? 01:03:22.510 --> 01:03:24.120 Well, this quantity is positive. 01:03:24.120 --> 01:03:25.661 It's always going to be curving away. 01:03:25.661 --> 01:03:28.287 So we have solutions that look like this or solutions 01:03:28.287 --> 01:03:29.120 that look like this. 01:03:29.120 --> 01:03:29.990 Can it ever be 0? 01:03:32.907 --> 01:03:34.740 Yeah, sure, it could be an inflection point. 01:03:34.740 --> 01:03:36.698 So for example, here the curvature is positive, 01:03:36.698 --> 01:03:40.170 but at this point the curvature has to switch to be like this. 01:03:40.170 --> 01:03:41.960 What functions are of this form? 01:03:44.752 --> 01:03:45.960 Let me give you another hint. 01:03:45.960 --> 01:03:46.550 Here's one. 01:03:46.550 --> 01:03:49.400 Is this curvature positive? 01:03:49.400 --> 01:03:50.057 Yes. 01:03:50.057 --> 01:03:50.890 What about this one? 01:03:53.470 --> 01:03:53.970 Yup. 01:03:53.970 --> 01:03:56.220 Those are all positive curvature. 01:03:56.220 --> 01:03:57.910 And these are exponentials. 01:03:57.910 --> 01:03:59.930 And the solution to this differential equation 01:03:59.930 --> 01:04:06.670 is e to the plus kappa x or e to the minus kappa x. 01:04:06.670 --> 01:04:08.940 And an arbitrary solution of this equation 01:04:08.940 --> 01:04:12.130 is a superposition A e to the kappa x plus B 01:04:12.130 --> 01:04:14.346 e to the minus kappa x. 01:04:14.346 --> 01:04:16.244 Everyone cool with that? 01:04:16.244 --> 01:04:17.660 When this quantity is positive, we 01:04:17.660 --> 01:04:19.390 get growing and collapsing exponentials. 01:04:21.890 --> 01:04:22.390 Yeah? 01:04:25.610 --> 01:04:29.990 On the other hand, if phi prime prime over phi 01:04:29.990 --> 01:04:34.010 is a negative number, i.e. minus what I'll call k 01:04:34.010 --> 01:04:44.010 squared, then the curvature has to be negative. 01:04:44.010 --> 01:04:47.460 And what functions have everywhere negative curvature? 01:04:47.460 --> 01:04:49.169 Sinusoidals. 01:04:49.169 --> 01:04:49.669 Cool? 01:04:53.540 --> 01:05:00.445 And the general solution is A e to the i K x plus B 01:05:00.445 --> 01:05:03.226 e to the minus i K x. 01:05:03.226 --> 01:05:06.710 So that differential equation, also known as sine and cosine. 01:05:09.170 --> 01:05:09.670 Cool? 01:05:12.540 --> 01:05:19.765 So putting that together with our original function, 01:05:19.765 --> 01:05:21.140 let's bring this up. 01:05:24.110 --> 01:05:26.680 So we want to think about the wave functions here. 01:05:26.680 --> 01:05:29.014 But in order to think about the energy eigenstates, 01:05:29.014 --> 01:05:30.305 we need to decide on an energy. 01:05:32.850 --> 01:05:35.794 We need to pick an energy, because you 01:05:35.794 --> 01:05:37.960 can't find the solution without figuring the energy. 01:05:37.960 --> 01:05:38.950 But notice something nice here. 01:05:38.950 --> 01:05:40.260 So suppose the energy is e. 01:05:40.260 --> 01:05:42.360 And let me just draw E. This is a constant. 01:05:42.360 --> 01:05:43.187 The energy is this. 01:05:43.187 --> 01:05:45.520 So this is the value of E. Here we're drawing potential. 01:05:45.520 --> 01:05:48.489 But this is the value of the energy, which is a constant. 01:05:48.489 --> 01:05:49.280 It's just a number. 01:05:53.680 --> 01:05:57.200 If you had a classical particle moving in this potential, 01:05:57.200 --> 01:05:58.760 what would happen? 01:05:58.760 --> 01:05:59.650 It would roll around. 01:05:59.650 --> 01:06:01.360 So for example, let's say you gave it this energy 01:06:01.360 --> 01:06:02.210 by putting it here. 01:06:02.210 --> 01:06:04.168 And think of this as a gravitational potential. 01:06:04.168 --> 01:06:06.157 You put it here, you let go, and it falls down. 01:06:06.157 --> 01:06:07.990 And it'll keep rolling until it gets up here 01:06:07.990 --> 01:06:10.230 to the classical turning point. 01:06:10.230 --> 01:06:12.080 And at that point, its kinetic energy 01:06:12.080 --> 01:06:13.710 must be 0, because its potential energy 01:06:13.710 --> 01:06:15.410 is its total energy, at which point 01:06:15.410 --> 01:06:17.670 it will turn around and fall back. 01:06:17.670 --> 01:06:18.359 Yes? 01:06:18.359 --> 01:06:20.150 If you take your ball, and you put it here, 01:06:20.150 --> 01:06:21.525 and you let it roll, does it ever 01:06:21.525 --> 01:06:23.960 get here, to this position? 01:06:23.960 --> 01:06:25.990 No, because it doesn't have enough energy. 01:06:25.990 --> 01:06:29.620 Classically, this is a forbidden position. 01:06:29.620 --> 01:06:31.790 So given an energy and given a potential, 01:06:31.790 --> 01:06:38.200 we can break the system up into classically allowed zones 01:06:38.200 --> 01:06:39.760 and classically forbidden zones. 01:06:45.170 --> 01:06:47.120 Cool? 01:06:47.120 --> 01:06:49.210 Now, in a classically allowed zone, 01:06:49.210 --> 01:06:53.670 the energy is greater than the potential. 01:06:53.670 --> 01:06:55.170 And in a classically forbidden zone, 01:06:55.170 --> 01:06:57.300 the energy is less than the potential. 01:07:00.530 --> 01:07:03.230 Everyone cool with that? 01:07:03.230 --> 01:07:06.080 But this tells us something really nice. 01:07:06.080 --> 01:07:09.429 If the energy is greater than the potential, 01:07:09.429 --> 01:07:10.970 what do you know about the curvature? 01:07:15.050 --> 01:07:15.550 Yeah. 01:07:15.550 --> 01:07:17.175 If we're in a classically allowed zone, 01:07:17.175 --> 01:07:20.150 so the energy is greater than the potential, 01:07:20.150 --> 01:07:22.860 then this quantity is positive, there's a minus sign here, 01:07:22.860 --> 01:07:23.920 so this is negative. 01:07:23.920 --> 01:07:25.760 So the curvature is negative. 01:07:28.920 --> 01:07:31.680 Remember, curvature is negative means 01:07:31.680 --> 01:07:35.000 that we curve towards the axis. 01:07:35.000 --> 01:07:37.015 So in a classically allowed region, 01:07:37.015 --> 01:07:38.640 the wave function should be sinusoidal. 01:07:43.840 --> 01:07:46.050 What about in the classically forbidden regions? 01:07:46.050 --> 01:07:47.280 In the classically forbidden regions, 01:07:47.280 --> 01:07:48.890 the energy is less than the potential. 01:07:48.890 --> 01:07:50.848 That means in magnitude this is less than this, 01:07:50.848 --> 01:07:53.270 this is a negative number, minus sign, 01:07:53.270 --> 01:07:55.365 the curvature is going to be minus 01:07:55.365 --> 01:07:59.050 times a minus is a positive, so the curvature's positive. 01:08:02.260 --> 01:08:05.190 So the solutions are either growing exponentials 01:08:05.190 --> 01:08:10.000 or shrinking exponentials or superpositions of them. 01:08:10.000 --> 01:08:11.810 Everyone cool with that? 01:08:11.810 --> 01:08:14.340 So let's think about a simple example. 01:08:14.340 --> 01:08:16.452 Let's work through this in a simple example. 01:08:16.452 --> 01:08:18.575 And let me give you a little bit more board space. 01:08:22.430 --> 01:08:26.122 Simple example would be a potential that looks like this. 01:08:26.122 --> 01:08:27.580 And let's just suppose that we want 01:08:27.580 --> 01:08:33.760 to find an energy eigenfunction with energy that's E. Well, 01:08:33.760 --> 01:08:36.250 this is a classically allowed zone, 01:08:36.250 --> 01:08:39.939 and these are the classically forbidden regions. 01:08:39.939 --> 01:08:43.370 Now I want to ask, what does the wave function look like? 01:08:43.370 --> 01:08:46.630 And I don't want to draw it on top of the energy diagram, 01:08:46.630 --> 01:08:49.037 because wave function is not an energy. 01:08:49.037 --> 01:08:50.620 Wave function is a different quantity, 01:08:50.620 --> 01:08:52.370 because it's got different axes and I want 01:08:52.370 --> 01:08:54.279 it drawn on a different plot. 01:08:54.279 --> 01:08:57.960 So but as a function of x-- 01:08:57.960 --> 01:09:00.935 so just to get the positions straight, 01:09:00.935 --> 01:09:02.560 these are the bounds of the classically 01:09:02.560 --> 01:09:04.899 allowed and forbidden regions. 01:09:04.899 --> 01:09:07.140 What do we expect? 01:09:07.140 --> 01:09:09.830 Well, we expect that it's going to be sinusoidal in here. 01:09:13.160 --> 01:09:15.770 We expect that it's going to be exponential growing 01:09:15.770 --> 01:09:20.399 or converging out here, exp. 01:09:20.399 --> 01:09:25.100 But one last important thing is that not only is the curvature 01:09:25.100 --> 01:09:28.270 negative in here in these classically allowed regions, 01:09:28.270 --> 01:09:30.080 but the magnitude of the curvature, 01:09:30.080 --> 01:09:32.870 how rapidly it's turning over, how big that second derivative 01:09:32.870 --> 01:09:35.040 is, depends on the difference between the energy 01:09:35.040 --> 01:09:35.865 and the potential. 01:09:35.865 --> 01:09:38.240 The greater the difference, the more rapid the curvature, 01:09:38.240 --> 01:09:40.500 the more rapid the turning over and fluctuation. 01:09:40.500 --> 01:09:43.450 If the differences between the potential and the true energy, 01:09:43.450 --> 01:09:46.720 the total energy, is small, then the curvature is very small. 01:09:46.720 --> 01:09:49.990 So the derivative changes very gradually. 01:09:49.990 --> 01:09:51.660 What does that tell us? 01:09:51.660 --> 01:09:54.194 That tells us that in here the wave function is oscillating 01:09:54.194 --> 01:09:56.110 rapidly, because the curvature, the difference 01:09:56.110 --> 01:09:58.320 between the energy and the potential is large, 01:09:58.320 --> 01:10:01.690 and so the wave function is oscillating rapidly. 01:10:01.690 --> 01:10:04.200 As we get out towards the classical turning points, 01:10:04.200 --> 01:10:07.640 the wave function will be oscillating less rapidly. 01:10:07.640 --> 01:10:09.865 The slope will be changing more gradually. 01:10:09.865 --> 01:10:11.810 And as a consequence, two things happen. 01:10:11.810 --> 01:10:13.768 Let me actually draw this slightly differently. 01:10:17.647 --> 01:10:19.230 So as a consequence two things happen. 01:10:19.230 --> 01:10:21.130 One is the wavelength gets longer, 01:10:21.130 --> 01:10:22.847 because the curvature is smaller. 01:10:22.847 --> 01:10:24.680 And the second is the amplitude gets larger, 01:10:24.680 --> 01:10:27.280 because it keeps on having a positive slope for longer 01:10:27.280 --> 01:10:31.520 and longer, and it takes longer to curve back down. 01:10:31.520 --> 01:10:34.260 So here we have rapid oscillations. 01:10:34.260 --> 01:10:41.430 And then the oscillations get longer and longer wavelength, 01:10:41.430 --> 01:10:43.892 until we get out to the classical turning point. 01:10:43.892 --> 01:10:45.225 And at this point, what happens? 01:10:47.704 --> 01:10:49.120 Yeah, it's got to be [INAUDIBLE].. 01:10:49.120 --> 01:10:51.860 Now, here we have some sine, and some superpositions 01:10:51.860 --> 01:10:53.647 of sine and cosines, exponentials. 01:10:53.647 --> 01:10:55.730 And in particular, it arrives here with some slope 01:10:55.730 --> 01:10:57.490 and with some value. 01:10:57.490 --> 01:11:00.140 We know this side we've got to get exponentials. 01:11:00.140 --> 01:11:02.330 And so this sum of sines and cosines at this point 01:11:02.330 --> 01:11:06.400 must match the sum of exponentials. 01:11:06.400 --> 01:11:08.282 How must it do so? 01:11:08.282 --> 01:11:10.490 What must be true of the wave function at this point? 01:11:13.880 --> 01:11:16.090 Can it be discontinuous? 01:11:16.090 --> 01:11:17.850 Can its derivative be discontinuous? 01:11:17.850 --> 01:11:18.350 No. 01:11:18.350 --> 01:11:21.800 So the value and the derivative must be continuous. 01:11:21.800 --> 01:11:24.690 So that tells us precisely which linear combination 01:11:24.690 --> 01:11:29.089 of positive growing and shrinking exponentials we get. 01:11:29.089 --> 01:11:30.630 So we'll get some linear combination, 01:11:30.630 --> 01:11:32.360 which may do this for awhile. 01:11:32.360 --> 01:11:33.860 But since it's got some contribution 01:11:33.860 --> 01:11:36.490 of positive exponential, it'll just grow exponentially off 01:11:36.490 --> 01:11:37.982 to infinity. 01:11:37.982 --> 01:11:39.940 And as the energy gets further and further away 01:11:39.940 --> 01:11:42.780 from the potential, now in their negative sine, what 01:11:42.780 --> 01:11:45.444 happens to the rate of growth? 01:11:45.444 --> 01:11:46.610 It gets more and more rapid. 01:11:46.610 --> 01:11:48.540 So this just diverges more and more rapidly. 01:11:48.540 --> 01:11:51.554 Similarly, out here we have to match the slope. 01:11:51.554 --> 01:11:52.970 And we know that the curvature has 01:11:52.970 --> 01:11:55.530 to be now positive, so it has to do this. 01:11:59.120 --> 01:12:00.040 So two questions. 01:12:00.040 --> 01:12:03.820 First off, is this sketch of the wave function 01:12:03.820 --> 01:12:05.560 a reasonable sketch, given what we 01:12:05.560 --> 01:12:08.570 know about curvature and this potential of a wave function 01:12:08.570 --> 01:12:09.280 with that energy? 01:12:13.007 --> 01:12:14.840 Are there ways in which it's a bad estimate? 01:12:17.798 --> 01:12:19.590 AUDIENCE: [INAUDIBLE] 01:12:19.590 --> 01:12:21.685 PROFESSOR: OK, excellent. 01:12:21.685 --> 01:12:24.600 AUDIENCE: On the right side, could it have crossed zero? 01:12:24.600 --> 01:12:25.720 PROFESSOR: Absolutely, it could have crossed zero. 01:12:25.720 --> 01:12:26.920 So I may have drawn this badly. 01:12:26.920 --> 01:12:28.060 It turned out it was a little subtle. 01:12:28.060 --> 01:12:28.740 It's not obvious. 01:12:28.740 --> 01:12:30.340 Maybe it actually punched all the way through zero, 01:12:30.340 --> 01:12:31.660 and then it diverged down negative. 01:12:31.660 --> 01:12:32.785 That's absolutely positive. 01:12:32.785 --> 01:12:35.075 So that was one of the quibbles you could have. 01:12:35.075 --> 01:12:36.450 Another quibble you could have is 01:12:36.450 --> 01:12:39.200 that it looks like I have constant wavelength in here. 01:12:39.200 --> 01:12:41.010 But the potential's actually changing. 01:12:41.010 --> 01:12:43.070 And what you should chalk this up to, 01:12:43.070 --> 01:12:46.299 if you'll pardon the pun, is my artistic skills are limited. 01:12:46.299 --> 01:12:48.340 So this is always going to be sort of inescapable 01:12:48.340 --> 01:12:50.505 when you qualitatively draw something. 01:12:50.505 --> 01:12:53.130 On a test, I'm not going to bag you points on things like that. 01:12:53.130 --> 01:12:54.584 That's what I want to emphasize. 01:12:54.584 --> 01:12:56.250 But the second thing, is there something 01:12:56.250 --> 01:12:57.622 bad about this wave function? 01:12:57.622 --> 01:12:58.830 Yes, you've already named it. 01:12:58.830 --> 01:13:01.090 What's bad about this wave function? 01:13:01.090 --> 01:13:03.150 It's badly non-normalizable. 01:13:03.150 --> 01:13:06.820 It diverges off to infinity out here and out here. 01:13:06.820 --> 01:13:10.110 What does that tell you? 01:13:10.110 --> 01:13:10.860 It's not physical. 01:13:10.860 --> 01:13:11.170 Good. 01:13:11.170 --> 01:13:13.045 What else does it tell you about this system? 01:13:16.110 --> 01:13:18.220 Sorry? 01:13:18.220 --> 01:13:18.870 Excellent. 01:13:18.870 --> 01:13:21.670 Is this an allowable energy? 01:13:21.670 --> 01:13:22.840 No. 01:13:22.840 --> 01:13:25.130 If the wave function has this energy, 01:13:25.130 --> 01:13:28.340 it is impossible to make it continuous, 01:13:28.340 --> 01:13:30.010 assuming that I drew it correctly, 01:13:30.010 --> 01:13:31.530 and have it converge. 01:13:31.530 --> 01:13:33.640 Is this wave function allowable? 01:13:33.640 --> 01:13:36.180 No, because it does not satisfy our boundary conditions. 01:13:36.180 --> 01:13:38.763 Our boundary conditions are that the wave function must vanish 01:13:38.763 --> 01:13:41.710 out here and it must vanish out here at infinity 01:13:41.710 --> 01:13:43.785 in order to be normalizable. 01:13:43.785 --> 01:13:44.410 Here we failed. 01:13:47.590 --> 01:13:49.610 Now, you can imagine that-- so let's decrease 01:13:49.610 --> 01:13:50.610 the energy a little bit. 01:13:50.610 --> 01:13:53.760 If we decrease the energy, our trial energy just a little 01:13:53.760 --> 01:13:55.200 tiny bit, what happens? 01:13:55.200 --> 01:13:58.570 Well, that's going to decrease the curvature in here. 01:13:58.570 --> 01:14:01.260 We decrease, we bring the energy in just a little tiny bit. 01:14:01.260 --> 01:14:03.550 That means this is a little bit smaller. 01:14:03.550 --> 01:14:05.130 The potential stays the same. 01:14:05.130 --> 01:14:06.760 So the curvature in the allowed region 01:14:06.760 --> 01:14:09.010 is just a little tiny bit smaller. 01:14:09.010 --> 01:14:10.600 And meanwhile, the allowed region 01:14:10.600 --> 01:14:12.649 has got just a little bit thinner. 01:14:12.649 --> 01:14:14.940 And what that will do is the curvature's a little less, 01:14:14.940 --> 01:14:17.500 the region's a little less, so now we have-- 01:14:28.130 --> 01:14:30.885 Sorry, I get excited. 01:14:30.885 --> 01:14:33.010 And if we tweak the energy, what's going to happen? 01:14:33.010 --> 01:14:35.176 Well, it's going to arrive here a little bit sooner. 01:14:38.220 --> 01:14:40.490 And let's imagine something like this. 01:14:40.490 --> 01:14:45.250 And if we chose the energy just right, 01:14:45.250 --> 01:14:48.120 we would get it to match to a linear combination 01:14:48.120 --> 01:14:50.560 of collapsing and growing exponentials, 01:14:50.560 --> 01:14:53.085 where the contribution from the growing exponential 01:14:53.085 --> 01:14:54.210 in this direction vanishes. 01:14:57.180 --> 01:14:59.320 There's precisely one value of the energy 01:14:59.320 --> 01:15:01.992 that lets me do that with this number of wiggles. 01:15:01.992 --> 01:15:03.950 And so then it goes through and does its thing. 01:15:07.860 --> 01:15:11.570 And we need it to happen on both sides. 01:15:11.570 --> 01:15:14.510 Now if I take that solution, so that it achieves convergence 01:15:14.510 --> 01:15:18.740 out here, and it achieves convergence out here, 01:15:18.740 --> 01:15:21.630 and I take that energy and I increase it by epsilon, 01:15:21.630 --> 01:15:24.450 by just the tiniest little bit, what will happen to this wave 01:15:24.450 --> 01:15:26.057 function? 01:15:26.057 --> 01:15:26.640 It'll diverge. 01:15:26.640 --> 01:15:28.670 It will no longer be normalizable. 01:15:28.670 --> 01:15:31.070 When you have classically forbidden regions, 01:15:31.070 --> 01:15:36.010 are the allowed energies continuous or discrete? 01:15:36.010 --> 01:15:38.820 And that answers a question from earlier in the class. 01:15:38.820 --> 01:15:40.570 And it also is going to be the beginning 01:15:40.570 --> 01:15:43.153 of the answer to the question, why is the spectrum of hydrogen 01:15:43.153 --> 01:15:44.000 discrete. 01:15:44.000 --> 01:15:45.960 See you next time.