1 00:00:00,180 --> 00:00:05,741 In the previous video, we went over a rather technical mathematical derivation 2 00:00:05,741 --> 00:00:11,302 of the continuity equation, and this derivation involved in a rather essential 3 00:00:11,302 --> 00:00:18,416 way manipulations with various operators. The appearance of operators is actually 4 00:00:18,416 --> 00:00:23,177 not specific to the particular problem of continuity equation but they appear 5 00:00:23,177 --> 00:00:28,007 actually throughout quantum mechanics and play a very important role generally in 6 00:00:28,007 --> 00:00:30,772 the mathematical formulas of quantum physics. 7 00:00:30,772 --> 00:00:36,028 In this short video, I would like to, make a few general comments about operators and 8 00:00:36,028 --> 00:00:39,105 relate them to quantum observable quantity. 9 00:00:39,106 --> 00:00:46,062 So, in order for me to trace the origin of the operator objects in quantum theory, 10 00:00:46,062 --> 00:00:53,289 let us go back a little bit and remind ourselves the way the Schrodinger equation 11 00:00:53,289 --> 00:00:56,226 was derived. So I use derive here in quotes because 12 00:00:56,226 --> 00:01:00,146 there is no really rigorous way to derive the Schrodinger equation as a fundamental 13 00:01:00,146 --> 00:01:04,560 equation of quantum theory. But we sort of followed the logic that the 14 00:01:04,560 --> 00:01:07,582 early practioneers, such as Schrodinger used. 15 00:01:07,583 --> 00:01:12,210 And the first piece of information which was essential was the fact, the 16 00:01:12,210 --> 00:01:17,110 experimental fact that quantum electrons and other quantum particles may exhibit 17 00:01:17,110 --> 00:01:21,784 wave-like properties. And in, in order to explain this we sort 18 00:01:21,784 --> 00:01:27,511 of introduced, in that profession, a plane-wave function which would explain 19 00:01:27,511 --> 00:01:30,994 the date in session. And in this wave function we also 20 00:01:30,994 --> 00:01:36,244 introduced the, the momentum and the energy of the electron which, according to 21 00:01:36,244 --> 00:01:41,344 another piece of information that is known from the theory, must be related to one 22 00:01:41,344 --> 00:01:46,594 another using this, this version relation So basically kinetic energy of a free 23 00:01:46,594 --> 00:01:49,657 particle mv squared over 2 or p squared over 2m. 24 00:01:49,658 --> 00:01:56,758 Now the equation itself was constructed in order to ensure that the plane wave 25 00:01:56,758 --> 00:02:03,258 finction would come out as its solution and this expression relation would be 26 00:02:03,258 --> 00:02:08,644 respective and so this is what equation actually look like. 27 00:02:08,645 --> 00:02:14,928 And it evolved the second derivative with respect to the coordinate and the first 28 00:02:14,928 --> 00:02:19,812 derivative with respect to tau. Now the next step and this is really the 29 00:02:19,812 --> 00:02:25,320 first time we sort of thought of introducing an operator was to write the 30 00:02:25,320 --> 00:02:29,962 equation in different way. So we had the sort of universal left hand 31 00:02:29,962 --> 00:02:34,715 side which is just i h bar. The derivate with respect to time and the 32 00:02:34,715 --> 00:02:40,324 right hand side was an, sort of energy operator, the [unknown] where we associate 33 00:02:40,324 --> 00:02:45,232 it with the momentum that would have appeared in the classical theory. 34 00:02:45,232 --> 00:02:53,614 This guy, which is minus i h bar gradient Which indeed is an operator that converts, 35 00:02:53,614 --> 00:02:59,611 well that acts on the wave function. And the last step in deriving the gain 36 00:02:59,611 --> 00:03:05,605 includes the Schrodinger equation was to generalize this energy operator, well 37 00:03:05,605 --> 00:03:10,951 called Hamiltonian from being the free energy operator which is just p squared 38 00:03:10,951 --> 00:03:16,378 over 2m to kinetic energy plus potential energy, and this is the fundamental 39 00:03:16,378 --> 00:03:22,129 Schrodinger equation, which remarkably describes all of the relativistic quantum 40 00:03:22,129 --> 00:03:27,880 physics, including and describes atomic spectra and well even many particle 41 00:03:27,880 --> 00:03:31,880 systems, theory of metals, theory of superconductivity. 42 00:03:31,880 --> 00:03:35,832 Everything is contained in a compact way in this remarkable equation. 43 00:03:35,832 --> 00:03:40,776 And it's also sort of remarkable that you can get to it from very few simple 44 00:03:40,776 --> 00:03:46,851 experimental facts, by adding small pieces together and getting to this fundamental 45 00:03:46,851 --> 00:03:50,174 result. Now, but in doing so, we were forced, 46 00:03:50,174 --> 00:03:53,618 indeed. To, introduce, well at least three 47 00:03:53,618 --> 00:03:56,602 operators. So the first one we have seen already many 48 00:03:56,602 --> 00:04:00,949 times, it is the momentum operator minus IH bar, radiant, and by the way I should 49 00:04:00,949 --> 00:04:05,170 mention that the minus sign Is really sort of a convention which is sort of must be 50 00:04:05,170 --> 00:04:09,441 consistent with the convention we choose to describe a plane wave in a sense. 51 00:04:09,442 --> 00:04:13,157 It could have been plus, but this is the convention that we use. 52 00:04:13,158 --> 00:04:20,093 So apart from that we also introduce the potential energy which enters this H hat. 53 00:04:20,094 --> 00:04:25,148 And in this set-up, the potential energy naturally appears as just multiplication 54 00:04:25,148 --> 00:04:27,880 operators. So, we can think about coordinate 55 00:04:27,880 --> 00:04:32,912 operator, multiplication operator that, well there should be actually a vector 56 00:04:32,912 --> 00:04:37,492 here multiplication operator simply multiplies the wave function. 57 00:04:37,492 --> 00:04:43,777 And finally, the kinetic energy itself sort of the first step, was appeared here 58 00:04:43,778 --> 00:04:49,046 and it was the second[INAUDIBLE] with respect to spacial coordinates. 59 00:04:49,046 --> 00:04:54,185 Now it turns out that the appearance of operators was not unique just to this sort 60 00:04:54,185 --> 00:04:58,800 of derivation of the Schrodinger equation, but sort of the popup throughout quantum 61 00:04:58,800 --> 00:05:02,771 theory. Whenever we deal with any analog's of 62 00:05:02,771 --> 00:05:07,330 classical properties. For example, when dealing with the angelo 63 00:05:07,330 --> 00:05:12,988 momentum that classical theory is R cross P, so all we have to do to make it into an 64 00:05:12,988 --> 00:05:18,583 operator, is to put a hat actually on, on top of the corresponding quantities. 65 00:05:18,583 --> 00:05:23,548 So p here would be replaced with minus i, h bar a gradient, according to this, and r 66 00:05:23,548 --> 00:05:26,372 would be just multiplication operator, etc. 67 00:05:26,372 --> 00:05:32,027 So we can always find a proper generalization of classical properties. 68 00:05:32,028 --> 00:05:36,931 So that's great we established that operators do appear in the mathematical 69 00:05:36,931 --> 00:05:41,310 formalism of quantum theory. And from the mathematical point of view it 70 00:05:41,310 --> 00:05:45,248 may be quite interesting. But what physical significance do these 71 00:05:45,248 --> 00:05:50,718 operators really have. Unlike their classical counterparts, which 72 00:05:50,718 --> 00:05:55,688 can be measured directly in a classical experiment, let's say we can measure 73 00:05:55,688 --> 00:06:00,370 momentum, every momentum position, the operators cannot be measured per se. 74 00:06:00,370 --> 00:06:04,840 They are operators. So how do we associate an experimental 75 00:06:04,840 --> 00:06:10,775 observable with them? So in order to answer this question, we're 76 00:06:10,775 --> 00:06:16,130 going to use a sort of method that often times is used in science by theorists 77 00:06:16,130 --> 00:06:21,655 answering the question in a simple, special case and then generalizing the 78 00:06:21,655 --> 00:06:25,473 results to a more general category of object. 79 00:06:25,474 --> 00:06:31,022 So and the special case that we're going to consider is the measurement of a 80 00:06:31,022 --> 00:06:36,339 coordinate of a quantum particle. So, the Born rule tells us that the, the 81 00:06:36,339 --> 00:06:41,230 absolute value of the wave function squared uses the probability distribution 82 00:06:41,230 --> 00:06:44,645 function. Of different positions of a quantum 83 00:06:44,645 --> 00:06:48,516 particle. Now in general, if we have a probability 84 00:06:48,516 --> 00:06:54,245 distribution function let's say F capital F of x of some random variable x. 85 00:06:54,246 --> 00:07:00,079 So if we want to find, let's say the mean value the expectation value of this x What 86 00:07:00,079 --> 00:07:05,342 we're going to do, is we're going to average X, integrate X over all possible 87 00:07:05,343 --> 00:07:09,787 values weighted with this probability of distribution. 88 00:07:09,788 --> 00:07:14,360 So by, in full analogy with this, we can define the mean value of a, a 89 00:07:14,360 --> 00:07:19,224 three-dimensional coordinate by integrating the wave function, absolute 90 00:07:19,224 --> 00:07:24,822 value of wave function squared. With the coordinate over the volume. 91 00:07:24,822 --> 00:07:30,754 Over the basically all three-dimensional space, there's a square here. 92 00:07:30,754 --> 00:07:34,718 So and this is an actual definition of the mean position. 93 00:07:34,718 --> 00:07:38,943 So, we can rewrite this definition as so, because psi, absoluste value of psi 94 00:07:38,943 --> 00:07:43,750 squared of course. You simply si star times si of R. 95 00:07:43,750 --> 00:07:51,298 So now comes this generalization of this definition of the expectation value of a 96 00:07:51,298 --> 00:07:57,860 coordinate to a general case of an arbitrary quantum mechanical operator. 97 00:07:57,860 --> 00:08:03,640 So lets say we have a classical property X lets say momentum, angular momentum 98 00:08:03,640 --> 00:08:08,970 coordinate energy, whatever. And let's assume we know the quantum 99 00:08:08,970 --> 00:08:12,713 mechanical operator associated with this, some x hat. 100 00:08:12,713 --> 00:08:18,552 So the expectation value of this quantum observable X is going to be calculated in 101 00:08:18,552 --> 00:08:24,789 full analogy with this equation by simply replacing the coordinate operator in this 102 00:08:24,789 --> 00:08:28,140 equation with the operator X, whatever it is. 103 00:08:28,140 --> 00:08:35,212 So for example if x happens to be so let's say, if x happens to be a momentum, then 104 00:08:35,212 --> 00:08:41,556 we're going to simply write it the expectation value of momentum is going to 105 00:08:41,556 --> 00:08:47,010 be an integral of v star minus i h bar gradient of psi, et cetera. 106 00:08:47,010 --> 00:08:50,979 So it turns out that this conjecture actually works. 107 00:08:50,980 --> 00:08:57,337 In that it is consistent with the experimental data and the experimental 108 00:08:57,337 --> 00:09:03,798 data implies that if we, let's say we have and example of quantum systems, each 109 00:09:03,798 --> 00:09:10,350 described by wave function psi of r, and if we measure, this quantum, some property 110 00:09:10,350 --> 00:09:15,127 x of this quantum systems we're going to get different results. 111 00:09:15,128 --> 00:09:20,444 But the average value of this measurement is going to be consistent with this 112 00:09:20,444 --> 00:09:25,502 definition. With this definition of, the expectation 113 00:09:25,502 --> 00:09:29,876 value. And, so, it becomes a very important basic 114 00:09:29,876 --> 00:09:34,617 principle of quantum mechanics, that I read right here. 115 00:09:34,617 --> 00:09:39,420 First of all, that physical observables are associated with what I didn't discuss, 116 00:09:39,420 --> 00:09:44,486 self-adjoint, operators, acting on the wave function, with the expectation value 117 00:09:44,486 --> 00:09:45,900 defined as above.