1 00:00:00,470 --> 00:00:05,510 In this video, I'm going to elaborate a little bit on the physical meaning and 2 00:00:05,510 --> 00:00:08,700 interpretation of operators in quantum mechanics. 3 00:00:08,700 --> 00:00:12,499 And, in particular, we're going to discuss a very important class of problems that 4 00:00:12,499 --> 00:00:17,395 appear throughout quantum mechanics. That is so-called eigenvalue problems. 5 00:00:17,396 --> 00:00:22,962 To see how they, what they are and how they arise, let me first write down the 6 00:00:22,962 --> 00:00:28,514 Time-Dependent Schrodinger equation. We can only go Time-Dependent Schrodinger 7 00:00:28,514 --> 00:00:33,807 equation, but with specifically with the Time-Independent Hamiltonian. 8 00:00:33,808 --> 00:00:38,035 So physically, this means that the potential in which our quantum particle is 9 00:00:38,035 --> 00:00:42,349 moving, is not time dependent. And actually most problems we are going to 10 00:00:42,349 --> 00:00:46,510 see are, fall into this category. Which is a very reasonable assumption, so 11 00:00:46,510 --> 00:00:49,988 that neither with the mass nor the potential are time-dependent. 12 00:00:49,988 --> 00:00:54,707 So, in this case, it turns out there is absolutely no need to deal with this 13 00:00:54,707 --> 00:01:00,359 complicated differential equation, partial differential equation that involves the 14 00:01:00,359 --> 00:01:03,822 time derivative. So, the time dependence can be sort of 15 00:01:03,822 --> 00:01:08,129 factored out of the equation in a very straight forward way, and this is 16 00:01:08,129 --> 00:01:11,847 accomplished by performing separation of variables. 17 00:01:11,848 --> 00:01:16,696 So basically, what we can say is let's look for our solution in the full wave 18 00:01:16,696 --> 00:01:19,608 form. So, the full, the full wave function is 19 00:01:19,608 --> 00:01:24,600 going to be a product of some time-independent piece, psi of r and this 20 00:01:24,600 --> 00:01:28,440 exponential which is sort of left over from the plane wave. 21 00:01:28,440 --> 00:01:34,236 So, this psi of r does not have to be a plane wave if the potential is exists but 22 00:01:34,236 --> 00:01:37,809 it, it is not a plane wave. If there is a potential, but the time 23 00:01:37,809 --> 00:01:43,730 dependence sort of remains the same. And so, if we plug in this uh,[UNKNOWN] 24 00:01:43,730 --> 00:01:50,538 into the Schrodinger equation, so we're going to have in the left-hand side in the 25 00:01:50,538 --> 00:01:56,610 derivative with respect to time psi of r, e to the power minus i energy t over h 26 00:01:56,610 --> 00:01:59,632 bar. So, I should mention that e of course 27 00:01:59,632 --> 00:02:04,682 corresponds to the energy. And so, since you know, this lower case 28 00:02:04,682 --> 00:02:09,647 psi of r is not depend on time, we don't have to differentiate this part. 29 00:02:09,648 --> 00:02:13,043 We're going to differentiate just the exponential. 30 00:02:13,043 --> 00:02:18,660 And just essentially by construction, we're going to get, in the left-hand side 31 00:02:18,660 --> 00:02:24,546 at e psi of r and its exponential. On the other hand, the right-hand side 32 00:02:24,547 --> 00:02:32,946 since there's nothing time-dependent there simply reads the Hamiltonian acting on the 33 00:02:32,947 --> 00:02:38,152 position dependent part. And the, the exponential is also here but 34 00:02:38,152 --> 00:02:42,932 it's sort of irrelevant in the sense that the Hamiltonian doesn't see it. 35 00:02:42,932 --> 00:02:47,458 So, the Hamiltonian only involves the derivative with respect to spatial 36 00:02:47,458 --> 00:02:52,009 coordinates or the potential, which is just multiplication operations. 37 00:02:52,009 --> 00:02:56,923 So, we see that both sides of this equation contain this exponential, and so 38 00:02:56,923 --> 00:03:03,007 we can just cancel this exponential and we are left with let me just switch the order 39 00:03:03,007 --> 00:03:07,922 in which this guys appears. So we are left with a full wave equation, 40 00:03:07,922 --> 00:03:11,440 H[UNKNOWN] psi or r is equal to e times psi of r. 41 00:03:11,440 --> 00:03:16,165 And, this is what is known as Time-Independent Schrodinger equation, a 42 00:03:16,165 --> 00:03:20,752 stationary Schrodinger equation. So, this category problems, where an 43 00:03:20,752 --> 00:03:25,792 operator acting on a vector in some space, reproduces the same vector multiplied by 44 00:03:25,792 --> 00:03:29,221 some number. So this is category of problems in math 45 00:03:29,221 --> 00:03:35,002 are called eigenvalue problems. And in the context of this mathematical 46 00:03:35,002 --> 00:03:42,056 theory psi here represents what's known as eigen vector, let me write it. 47 00:03:42,057 --> 00:03:50,670 And e is a, an eigenvalue of the corresponding operation. 48 00:03:50,670 --> 00:03:55,498 This operation is Hamiltonian. Now, let me tell you from the outset, we 49 00:03:55,498 --> 00:04:00,949 shall discuss it in more details later that the physical significance of these 50 00:04:00,949 --> 00:04:06,242 eigenvalues is in that they determine the possible outcomes of experiment that 51 00:04:06,242 --> 00:04:12,456 actually measure in this case energy. So we know that, for instance in classical 52 00:04:12,456 --> 00:04:16,107 physics, if I were to have some potential well. 53 00:04:16,108 --> 00:04:20,056 So, let's say this is my V of x, and there is some potential. 54 00:04:20,056 --> 00:04:24,250 Well, just one example of problems we're actually going to study. 55 00:04:24,250 --> 00:04:28,346 So, if I have a classical particle, I would be able to put a classical particle 56 00:04:28,346 --> 00:04:33,306 with any energy and this round, we'll just oscillate between the two turning points 57 00:04:33,306 --> 00:04:37,000 if there is no friction. And I can do so for any value of energy. 58 00:04:37,000 --> 00:04:40,712 It turns out that in quantum mechanics, it's not necessarily so. 59 00:04:40,712 --> 00:04:44,976 So, well, it's not so. And the, the range of available energy is 60 00:04:44,976 --> 00:04:48,507 limited in this case. So, what I would have is, is discrete, 61 00:04:48,507 --> 00:04:53,300 so-called discrete spectrum by which I mean a possible solution to this equation. 62 00:04:53,300 --> 00:04:58,448 So this levels, let's say E1 or E2 are going to come out of this eigenvalue 63 00:04:58,448 --> 00:05:02,462 problem measure. So we see that an eigenvalue problem for 64 00:05:02,462 --> 00:05:08,513 the Hamiltonian arises naturally from the original Schrodinger equation. 65 00:05:08,513 --> 00:05:13,960 And as I should becomes a very important independent, in some sense, equation of 66 00:05:13,960 --> 00:05:17,348 quantum theory. And actually, oftentimes, when we say 67 00:05:17,348 --> 00:05:22,040 Schrodinger equation, we actually refers to this H of psi equals[UNKNOWN] psi and 68 00:05:22,040 --> 00:05:25,028 not to the original along with the time derivative. 69 00:05:25,028 --> 00:05:30,044 And furthermore so eigenvalue problems also appear for other operators in quantum 70 00:05:30,044 --> 00:05:32,520 theory. So, it's very important. 71 00:05:32,520 --> 00:05:37,046 And we should know at least a few basic things, basic properties, of say 72 00:05:37,046 --> 00:05:42,617 uh,eigenvalue problems and the operators involved in this eigenvalue problems. 73 00:05:42,618 --> 00:05:47,280 So we don't have time now to go into comprehensive theory of this. 74 00:05:47,280 --> 00:05:51,744 Mathematical theories could be, in principle, a subject of a year long 75 00:05:51,744 --> 00:05:54,932 course. So therefore, I will just emphasize a few 76 00:05:54,932 --> 00:06:00,782 things that are really crucial for the future, and also introduced some notations 77 00:06:00,782 --> 00:06:05,260 and terms that are important. So, one such important thing is the notion 78 00:06:05,260 --> 00:06:09,269 of a Hermitian operator, that is defined by the[UNKNOWN] relation. 79 00:06:09,269 --> 00:06:14,133 So, let's imagine we have an operator A acting in a space of functions psi of r. 80 00:06:14,133 --> 00:06:18,951 So I'm not going to define precisely mathematically what this space is, but 81 00:06:18,951 --> 00:06:23,696 I'm, you know, we should just think about wave functions that describe the 82 00:06:23,696 --> 00:06:28,942 properties of our quantum particles. So one can define Hermitian-adjoint 83 00:06:28,942 --> 00:06:33,030 operator, A dagger to the operator A by the following relation. 84 00:06:33,030 --> 00:06:38,923 So, the left-hand side of this equation is essentially identical to what we saw in 85 00:06:38,923 --> 00:06:44,318 the previous video when we defined an average or an expectation value of an 86 00:06:44,318 --> 00:06:47,958 operator A. It's an integral of an entire space of psi 87 00:06:47,958 --> 00:06:51,585 star. Let's see how a wave function times the 88 00:06:51,585 --> 00:06:56,677 action of our operator A on psi. And the operator A dagger sort of, is 89 00:06:56,678 --> 00:07:01,870 moves from the right to the left. And now acts on the function which is 90 00:07:01,870 --> 00:07:05,998 ah,[UNKNOWN]. And so, if the two expressions are equal 91 00:07:05,998 --> 00:07:10,522 for any psi of r, then we have found the Hermitian-adjoint operator. 92 00:07:11,800 --> 00:07:17,056 And an important class of operators that appear in quantum mechanics are called 93 00:07:17,056 --> 00:07:20,770 Hermitian operators. And they are characterized by the fact 94 00:07:20,770 --> 00:07:24,560 that their Hermitian-adjoint is equal to the operator itself. 95 00:07:24,560 --> 00:07:28,546 So the, they are called Hermitian. The importance of the Hermitian operators 96 00:07:28,546 --> 00:07:32,328 manifests itself, if we look at their properties, the properties of their 97 00:07:32,328 --> 00:07:36,382 eigenvalue problem. So in general, if we have an eigenvalue 98 00:07:36,382 --> 00:07:42,094 problem for a Hermitian operator, there is no guarantee that its eigenvalues are 99 00:07:42,094 --> 00:07:46,070 going to be real. So in the context of quantum physics, this 100 00:07:46,070 --> 00:07:50,900 would lead to a bit of a problem because well, let's see if we have a wave function 101 00:07:50,900 --> 00:07:53,769 which is described by, let's say, psi sub a. 102 00:07:53,770 --> 00:07:58,360 And we want to take, to measure the expectation value of this, the quantity 103 00:07:58,360 --> 00:08:02,964 corresponding to this operator A. So, we will calculate this average value 104 00:08:02,964 --> 00:08:07,449 and we're going to get, if A is complex, we're going to get an expectation value 105 00:08:07,449 --> 00:08:10,886 which is complex. And it's sort of, in most case, it doesn't 106 00:08:10,886 --> 00:08:13,556 make sense. So, for example, if we measure energy of a 107 00:08:13,556 --> 00:08:16,990 particle, we certainly don't want to get i the measuring constant i. 108 00:08:16,990 --> 00:08:22,030 We want to get a reasonable number that has sort of it's counter part in classical 109 00:08:22,030 --> 00:08:25,454 physics, which does not normally account these number. 110 00:08:25,455 --> 00:08:31,049 So what I'm saying here is that if we want to describe physically reasonable 111 00:08:31,049 --> 00:08:36,122 quantities, so we want these operators to give rise to real eignevalues. 112 00:08:36,122 --> 00:08:39,670 So, this would make sense. And the Hermitian operators have this 113 00:08:39,670 --> 00:08:45,146 property, exactly, this property so the eigenvalues of the Hermitian operators are 114 00:08:45,146 --> 00:08:49,702 all real. So, another crucial important property of 115 00:08:49,702 --> 00:08:56,234 the eigenvalue problem for a Hermitian operator is that its eigenvectors, these 116 00:08:56,234 --> 00:09:02,819 solutions psi sub a of r to this equation form a basis in an appropriate space. 117 00:09:02,820 --> 00:09:06,120 So that is the space where our functions leave. 118 00:09:06,120 --> 00:09:10,024 So for instance, in the context of quantum physics, these are going to be wave 119 00:09:10,024 --> 00:09:12,758 functions. And of course, I should mention that the 120 00:09:12,758 --> 00:09:16,154 statements I'm making are not mathematically rigorous statements. 121 00:09:16,154 --> 00:09:20,733 So, if you want to see a precise definition or a theorem, so there is, of 122 00:09:20,733 --> 00:09:24,730 course very rich mathematical literature on the subject. 123 00:09:24,730 --> 00:09:27,582 And I would encourage you to look into this, but at this stage, I just want to 124 00:09:27,582 --> 00:09:31,364 give you an idea about the properties. And so, the property that I'm presenting 125 00:09:31,364 --> 00:09:35,720 here is that essentially, if you have any relevant function, psi of r, which has 126 00:09:35,720 --> 00:09:40,340 nothing to do with our problem and this function sort of lives in this space where 127 00:09:40,340 --> 00:09:44,932 our operator acts. So then, a set of all eigenvectors of this 128 00:09:44,932 --> 00:09:49,805 operator, which in principle, can be either a discrete or continuous. 129 00:09:49,805 --> 00:09:52,040 In which case, I would have written an integral, even both. 130 00:09:52,040 --> 00:09:56,844 So, but in any case, any such wave function can be represented as a linear 131 00:09:56,844 --> 00:10:00,761 combination of the eigenfunctions of the operator A. 132 00:10:00,762 --> 00:10:05,187 And this has a very consequences as a set in quantum physics. 133 00:10:05,188 --> 00:10:10,030 Now going closer to quantum physics and further away from sort of mathematical 134 00:10:10,030 --> 00:10:14,860 statements, let me just summarize things that really are important in the context 135 00:10:14,860 --> 00:10:18,160 of quantum mechanics. So, the first statement here is that 136 00:10:18,160 --> 00:10:22,320 physical observables, essentially things that we want to measure in quantum 137 00:10:22,320 --> 00:10:26,608 mechanics, are described by the Hermitian of operators, sometimes also called 138 00:10:26,608 --> 00:10:30,331 self-adjoint operators. So, I should mention that there exists a 139 00:10:30,331 --> 00:10:35,021 subtle difference, actually a mathematical difference between the Hermitian and 140 00:10:35,021 --> 00:10:38,569 self-adjoint operators. But it involves rather complicated math. 141 00:10:38,569 --> 00:10:42,229 And in the framework of this course, we can just consider Hermitians and 142 00:10:42,229 --> 00:10:46,886 self-adjoint to be the same thing. But if among you there's a math aficionado 143 00:10:46,886 --> 00:10:51,996 who wants to know the difference you know, ask this question in the discussion forum 144 00:10:51,996 --> 00:10:57,602 and I will be happy to elaborate on this. But anyway, so if we have identified a 145 00:10:57,602 --> 00:11:03,377 quantity of interest, our observable A, and identified an operator associated with 146 00:11:03,377 --> 00:11:07,098 this observable. So eigen, then, then we are, we, we need 147 00:11:07,098 --> 00:11:09,917 to solve the eigenvalue problem for this operator. 148 00:11:09,918 --> 00:11:16,444 And the available eigenvalues, they determine possible values that actually 149 00:11:16,444 --> 00:11:20,750 can be measured in experiment. So, also let me reiterate a very important 150 00:11:20,750 --> 00:11:23,080 fact which was discussed on the previous slide. 151 00:11:23,080 --> 00:11:26,049 And we're going to talk a little bit about this in the next video. 152 00:11:26,050 --> 00:11:32,739 Namely, that these solutions to this eigenvalue problem, this functions psi of 153 00:11:32,739 --> 00:11:37,856 a of r, form a basis in the space where our wave functions leave. 154 00:11:37,856 --> 00:11:42,942 Which implies that any wave function describing the physical state of a system 155 00:11:42,942 --> 00:11:46,946 can be expanded as a linear combination of this eigenvector. 156 00:11:48,010 --> 00:11:53,169 So, the last comment, but not the least important, that I'm going to make relates 157 00:11:53,169 --> 00:11:58,174 to the interpretation of these coefficients, c sub a, the complex numbers 158 00:11:58,174 --> 00:12:02,872 that appear in this expansion. As a matter of fact, actually, this is 159 00:12:02,872 --> 00:12:07,812 probably the most important take home message that I would like you to take out 160 00:12:07,812 --> 00:12:12,282 of this slide. So imagine that we have an ensemble of 161 00:12:12,282 --> 00:12:18,175 identical quantum systems, which exist in exactly the same quantum stage described 162 00:12:18,175 --> 00:12:23,892 by some wave function psi that we expanded in our bases of these eigen states. 163 00:12:23,893 --> 00:12:29,114 And let's imagine that we're going to form, let's say three identical measures 164 00:12:29,114 --> 00:12:33,589 of the identical quantity a. So, we want to measure a in exactly the 165 00:12:33,589 --> 00:12:36,937 same way with exactly the same kinds of systems. 166 00:12:36,938 --> 00:12:40,352 So, the weird thing about quantum mechanics is that even though everything 167 00:12:40,352 --> 00:12:44,024 is the same about these systems, we are actually going to get, strictly speaking, 168 00:12:44,024 --> 00:12:47,461 three different results. So, there is a chance some of them will be 169 00:12:47,461 --> 00:12:49,900 the same bot. There's no guarantee at all. 170 00:12:49,900 --> 00:12:55,375 And the probability of finding a system in a certain state, which is sometimes is 171 00:12:55,375 --> 00:12:59,970 referred to as a collapse of the wave function into certain eigen state. 172 00:12:59,970 --> 00:13:06,928 So, this probability, let me call it w sub a, is equal to the absolute value squared 173 00:13:06,928 --> 00:13:12,465 of the coefficients in this expansion, so this is the probability. 174 00:13:12,465 --> 00:13:17,943 So in this, in this example again, so what, what we can see for sure is, the 175 00:13:17,943 --> 00:13:23,587 only thing we can say for sure is that all these measurements are going to result in 176 00:13:23,587 --> 00:13:28,567 a value of A that belongs to the spectrum, that is, to the set of available 177 00:13:28,567 --> 00:13:33,492 eigenvalues of the operator A. But which particular value is going to be 178 00:13:33,492 --> 00:13:36,588 measured, we don't know. We only can say which is more likely, and 179 00:13:36,588 --> 00:13:40,307 we can quantify this uncertainty. But this is the maximum we can do. 180 00:13:40,308 --> 00:13:45,204 And so, this principle is actually very important in quantum mechanics and we're 181 00:13:45,204 --> 00:13:47,953 going to use it later in the course very often.