1 00:00:00,005 --> 00:00:07,001 Okay so um, some of you might be wondering why should this equation has the 2 00:00:07,001 --> 00:00:12,007 particular form it does. So why does energy play such an important role in, in 3 00:00:12,007 --> 00:00:20,000 the evolution of the quantum system. Okay so. Um, so this video is sort of a brief 4 00:00:20,000 --> 00:00:25,007 video to address this concern, um. It, it's mostly, uh, it's mostly to point you 5 00:00:25,007 --> 00:00:32,005 in some direction if you are interested in pursuing this any further. Okay. So, so 6 00:00:32,005 --> 00:00:38,003 just to recall, um. You, uh, the Hamiltonian of the system is the energy 7 00:00:38,003 --> 00:00:44,007 observable, um. Its eigenstates are the states of definite energy, uh, etcetera. 8 00:00:44,007 --> 00:00:51,006 And Shrodinger's equation says that the evolution of the system is intimately tied to the 9 00:00:51,006 --> 00:00:56,009 Hamiltonian. Right? It's a differential equation which says that, um, which 10 00:00:56,009 --> 00:01:02,007 governs the, the evolution of the system. And it, um, you know, it intimately ties 11 00:01:02,007 --> 00:01:08,007 the change in the state to, to the action of the Hamiltonian on the, on the, on the 12 00:01:08,007 --> 00:01:14,006 state. So where does all this come from? So, it, it turns out that the answer lies 13 00:01:14,006 --> 00:01:21,000 in, in this beautiful connection between symmetry and conserve quantities 14 00:01:21,000 --> 00:01:28,009 in, in, in quantum mechanics. And, this, uh, goes back to work by, uh, Annie . Um, 15 00:01:28,009 --> 00:01:36,000 who is, uh, who is regarded as, uh, as, um, the greatest female mathematician in, 16 00:01:36,000 --> 00:01:43,001 in, in history. The greatest, uh, woman in, in, in mathematics in all of history. 17 00:01:43,001 --> 00:01:51,008 Okay, so, so let's see how we can derive the form of Shrodinger's equation. So the. The first 18 00:01:51,008 --> 00:02:06,005 thing, uh, we, we observe is that from the axiom of unitary evolution. We can derive 19 00:02:06,005 --> 00:02:18,008 with a little bit of work that, that, uh, the time evolution operator must look like 20 00:02:18,008 --> 00:02:36,000 E to the -IMT for some hermitian operator M. Okay, so for some emission. Okay, so 21 00:02:36,000 --> 00:02:44,005 let's take this for granted. So now what they're really trying to understand is 22 00:02:44,005 --> 00:02:51,006 why. Is m equal to the Hamiltonian. Okay, and we don't want to prove that m must be 23 00:02:51,006 --> 00:02:58,002 equal to the Hamiltonian, we want to sort of argue that it must. Right, so, so 24 00:02:58,002 --> 00:03:05,004 we'll, we'll provide a justification which is going to be actually quite compelling. 25 00:03:05,004 --> 00:03:13,002 Okay, so what this relies on is a, is a theorem, which says, if. E is any 26 00:03:13,002 --> 00:03:24,009 observable. Corre sponding to some, some physical quantity so if A measures some 27 00:03:24,009 --> 00:03:38,005 physical quantity that's conserved. Meaning, for example, E corresponds to 28 00:03:38,005 --> 00:03:44,007 momentum or angular momentum or energy. It's some, some quantity that is conserved 29 00:03:44,007 --> 00:04:01,004 over time. Then, a must commute with, with M. Meaning that E times M equal 30 00:04:01,004 --> 00:04:07,004 to M times A. Right? Now I hope you realize that matrices, matrix 31 00:04:07,004 --> 00:04:14,002 multiplication in general, doesn't commute. So, for example, if we have, if 32 00:04:14,002 --> 00:04:24,009 you look at our operators X, the bit-flip operator. And Z, the phase flip operator. 33 00:04:26,002 --> 00:04:37,003 Right? Then what's, what's X times Z? Well, so its a, zero one, one zero times, 34 00:04:37,003 --> 00:04:51,009 one minus one zero, zero, which is what you get zero here. Minus one. One. Zero. 35 00:04:51,009 --> 00:05:04,006 And what's z times x? It's one minus one, zero, zero. Times zero, one, one, zero. 36 00:05:04,006 --> 00:05:13,005 Which is what? Zero. One. Minus 1-0. Right which are, these are not equal. So in 37 00:05:13,005 --> 00:05:19,009 general, if you're given two general operators, they will not commute. If A 38 00:05:19,009 --> 00:05:27,003 corresponds to any conserve quantity, then it must commute with the time of, you, you 39 00:05:27,003 --> 00:05:33,003 know, this, this [inaudible] operator M which occurs in, in this unitary 40 00:05:33,003 --> 00:05:39,008 evolution. Now, lets this see why, why does this follow, so lets sketch out a 41 00:05:39,008 --> 00:05:51,005 proof. So. So lets, lets let psi be the state of the system at, st some time. And 42 00:05:51,005 --> 00:06:07,005 lets I prime. B equal to psi, B the state which is, which is E to the minus IMT. 43 00:06:07,005 --> 00:06:18,000 Psi will be the state after an infinitesimal interval of time T. So, now 44 00:06:18,000 --> 00:06:23,009 what does it mean to say that A is a conserved quantity. A is conserved 45 00:06:25,006 --> 00:06:36,007 quantity means that, that if you look at, the expected value of A under 46 00:06:36,007 --> 00:06:49,009 psi. So psi A psi, this must be equal to psi prime, a psi prime. Okay, but what's, 47 00:06:49,009 --> 00:07:00,000 what's psi prime is psi , psi , psi prime will, psi prime is U psi , so it's, its U 48 00:07:00,000 --> 00:07:14,000 psi. And then A U dagger psi. Okay, but this, this, this equation hold for all 49 00:07:14,000 --> 00:07:23,005 psi . Since this hold for all psi , this implies that in fact A must be equal to U 50 00:07:23,005 --> 00:07:39,004 dagger A U. Right? And so. U-dagger au is, is of course just um, um, e to the imt a 51 00:07:39,006 --> 00:07:49,009 to minus imt, which if you, you, if you expand out with a Taylor expansion, this, 52 00:07:49,009 --> 00:07:58,009 this, is, this is to a first order equal to one plus IMT, times A, times one minus 53 00:07:58,009 --> 00:08:07,004 IMT. Remembering that T is very small, so higher order terms don't, don't really, 54 00:08:07,004 --> 00:08:24,000 don't really matter here. And this further is about A + I T times M A minus 55 00:08:24,000 --> 00:08:34,006 A M. And so since A is equal to this, it, it tells us that this must equal to zero. 56 00:08:34,006 --> 00:08:41,008 Right here I'm ignoring the, or the t squared term, which comes from this 57 00:08:41,008 --> 00:08:48,007 product here because t is turning to zero. And so, now, if this, if this is zero, 58 00:08:48,007 --> 00:08:56,003 then that tells us that ma=am, which is, is, which is, really worth saying that a 59 00:08:56,003 --> 00:09:03,005 must commute with m. Okay, okay, so what have we learned so far? What we've learned 60 00:09:03,005 --> 00:09:11,007 is that first, first what, what we said is that just from the fact that we have 61 00:09:11,007 --> 00:09:15,006 unitary evolution. It follows with a little bit of work which we won't go in 62 00:09:15,006 --> 00:09:26,002 to, that U must be of the form E to the minus I M T. That M is emission and 63 00:09:26,002 --> 00:09:38,007 the second thing we know is that if E is conserved then E must commute with M. 64 00:09:39,008 --> 00:09:45,004 Okay . So now, there are various quantities that might be conserved. There 65 00:09:45,004 --> 00:09:50,005 are things like momentum or angular momentum . But, one can argue that these 66 00:09:50,005 --> 00:09:56,006 are accidental con, conserved quantities. The, you know this are conservation 67 00:09:56,006 --> 00:10:02,004 relations which may or may not hold. So for example conservation of momentum well 68 00:10:02,004 --> 00:10:08,003 you know it may or may not, momentum may or may not be conserved depending upon, 69 00:10:08,003 --> 00:10:13,005 for example if you, if you have a change in potential energy then momentum is not 70 00:10:13,005 --> 00:10:20,007 conserved. Okay? Energy is always conserved and remember we are working 71 00:10:20,007 --> 00:10:29,003 non-relativisticly, so, so energy is always conserved. Okay, so, so, so there, 72 00:10:29,003 --> 00:10:34,005 there must be some intrinsic reason so, so what we, what we, are saying is there must 73 00:10:34,005 --> 00:10:49,005 be some intrinsic reason why M, and H commute. And what we are going to reason is that, 74 00:10:49,005 --> 00:10:57,008 in fact, the most natural intrinsic reason why H and M commute is that M is some 75 00:10:57,008 --> 00:11:06,008 multiple of H. Right? So H equal to, say. Right, this is, this is what they're 76 00:11:06,008 --> 00:11:10,002 really claiming, that there's some constant, which turns out to be h-bar such 77 00:11:10,002 --> 00:11:16,000 that our energy operator, the Hamiltonian h is h-bar times m, where m is the 78 00:11:16,000 --> 00:11:20,007 operator which, which describes the evolution of the system, and that's, 79 00:11:20,007 --> 00:11:26,004 that's what they're claiming. Okay, so, maybe, maybe, let me actually go through 80 00:11:26,004 --> 00:11:30,004 it quickly. Uh, you can read about it in the notes if you're interested, but um, 81 00:11:30,004 --> 00:11:37,002 let's, let's just go with it. Uh, um, so, so the rough form of reasoning is that, 82 00:11:37,002 --> 00:11:44,005 well, in order for M and H to commute intrinsically, well, you know, it, uh, 83 00:11:44,005 --> 00:11:51,003 the, the one way we can imagine it is if, if, uh, if H is actually some function of 84 00:11:51,003 --> 00:12:00,008 M, some polynomial in M, general. And then, the next thing to show is that. In 85 00:12:00,008 --> 00:12:06,005 fact, there are other physical considerations. So this is, you know, this 86 00:12:06,005 --> 00:12:13,003 was to make sure that H and M actually commute. But the next, the next thing to 87 00:12:13,003 --> 00:12:21,003 show is that, in fact, there are physical considerations which imply that if H is 88 00:12:21,003 --> 00:12:36,009 some function of M, this function must necessarily be linear. Okay and that, 89 00:12:36,009 --> 00:12:43,002 that has to do with saying. Well, suppose that our system consists of two pieces, 90 00:12:43,002 --> 00:12:52,001 two disjoint pieces, one and two. So, this has an associated omission operator m1. 91 00:12:52,001 --> 00:12:59,002 This has an associated omission operator m2. This has a Hamiltonian h1. This has a Hamiltonian 92 00:12:59,002 --> 00:13:09,002 h2. Well then the, then the energy of the, of the composite system, if our, if 93 00:13:09,002 --> 00:13:15,004 our system consists of these, these two disjoined pieces, the energy is clearly H1 94 00:13:15,004 --> 00:13:22,005 plus H2. But on the other hand, the operator associated with this, the hermitian 95 00:13:22,005 --> 00:13:33,002 operator must also be M1 plus M2. And so, H must be = to F of M1 plus HM2. But we are 96 00:13:33,002 --> 00:13:40,009 claiming H, but, but on the other hand H one is, is F of M one, H two is F of M 97 00:13:40,009 --> 00:13:50,001 two, and so F of M one plus M two equal to f of m1, less f of m2. And so that 98 00:13:50,001 --> 00:13:55,004 tells us that f must be a linear function. Okay, this, this is maybe too much for 99 00:13:55,004 --> 00:14:00,007 most of you, but, I just added it in for those of you who are interested. Um, it's, 100 00:14:00,007 --> 00:14:06,005 it's certainly, uh, not in any way required for, anything that follows. Okay.