1 00:00:00,660 --> 00:00:06,072 So now, we will continue putting together the basic conditional principles of 2 00:00:06,072 --> 00:00:11,812 quantum mechanics, and I'm also going to introduce the so-called Dirac notations, 3 00:00:11,812 --> 00:00:16,926 which are mathematical symbols Dirac introduced back in the early days. 4 00:00:16,927 --> 00:00:21,719 And these symbols turned out to be very convenient in doing calculations in 5 00:00:21,719 --> 00:00:26,598 quantum mechanics. So, let me mention actually that as a 6 00:00:26,598 --> 00:00:32,070 practicing theoretical physicist I noticed that an important component of a new, 7 00:00:32,070 --> 00:00:37,504 putting together a new physical theory is to come up with the convenient and elegant 8 00:00:37,504 --> 00:00:41,223 notation. So, these are not some minor unimportant 9 00:00:41,224 --> 00:00:45,373 questions. So, it's actually quite important, and 10 00:00:45,374 --> 00:00:50,405 having awkward notations could make our lives very difficult. 11 00:00:50,405 --> 00:00:54,888 So, Dirac notations certainly satisfy the criterion of being convenient, and also 12 00:00:54,888 --> 00:00:59,830 they are widely used in the literature. So, if you don't know them already, it's a 13 00:00:59,830 --> 00:01:04,660 good idea to familiarize yourselves with the, with them, and this is what we're 14 00:01:04,660 --> 00:01:09,067 going to do today. But before introducing the Dirac notations 15 00:01:09,067 --> 00:01:14,575 and going further with the development of sort of more abstract quantum theory, I 16 00:01:14,575 --> 00:01:19,759 would like to mention a relevant superposition principle in quantum 17 00:01:19,759 --> 00:01:24,150 mechanics. So, this so called superposition principle 18 00:01:24,150 --> 00:01:29,760 simply states that if we have 2 solutions to the Schrodinger equation, psi 1 of r 19 00:01:29,760 --> 00:01:35,030 and t and psi 2 of r and t, they are arbitrary linear combination such as here 20 00:01:35,030 --> 00:01:40,300 with some arbitrary complex constant, c1 and c2, is also a solution to this 21 00:01:40,300 --> 00:01:42,935 equation. So actually, this so-called superposition 22 00:01:42,935 --> 00:01:45,195 principle is not specific to the Schrodinger equation. 23 00:01:45,196 --> 00:01:50,794 The superposition principle holds for any linear differential equation, which the 24 00:01:50,794 --> 00:01:54,450 Schrodinger equation happens to be. So, there are no terms between that. 25 00:01:54,450 --> 00:01:58,716 There are no terms like psi squared psi cubed, etc. 26 00:01:58,716 --> 00:02:02,192 So well, to prove this, one can just simply plug it into the Schrodinger 27 00:02:02,192 --> 00:02:05,085 equation and cancel the corresponding pieces, term by term. 28 00:02:05,086 --> 00:02:12,462 So, what this superposition principle motivates is the notion of so-called 29 00:02:12,462 --> 00:02:20,244 Hilbert space, which is a linear vector space where quantum states sort of live. 30 00:02:20,244 --> 00:02:26,454 To remind you, a linear vector space is a basic concept of Linear Algebra, which 31 00:02:26,454 --> 00:02:32,940 implies a set of elements that we call vectors such that we can define a sum of 32 00:02:32,940 --> 00:02:39,144 any two vectors and a sum of any two vectors itself belongs to this set, to 33 00:02:39,144 --> 00:02:43,006 this vector space. And where we can also define a 34 00:02:43,006 --> 00:02:47,746 multiplication by a constant, either real or complex, depending on the 35 00:02:47,746 --> 00:02:53,434 circumstances, and this sort of stretching of a vector, also gives rise to another 36 00:02:53,434 --> 00:02:57,364 element of the set. So, in other words, this multiplication 37 00:02:57,364 --> 00:03:00,703 and addition of vectors are closed operations. 38 00:03:01,730 --> 00:03:07,559 So, through this superposition principle, we can suspect that whatever the 39 00:03:07,559 --> 00:03:13,388 mathematical objects are the described quantum states, they also in some sense 40 00:03:13,388 --> 00:03:19,234 belong to a closed linear vector space that is what we call the Hilbert space. 41 00:03:19,234 --> 00:03:23,155 But what about the wave-function that we have become already familiar with. 42 00:03:23,156 --> 00:03:30,098 So, this wave-function psi of r should be thought of in this abstract language and 43 00:03:30,098 --> 00:03:35,330 approached as a specific representation of a quantum state. 44 00:03:35,330 --> 00:03:37,784 We're going to discuss the meaning of it a little later. 45 00:03:37,785 --> 00:03:41,650 But here, I just would like to mention that it's much like coordinates of a 46 00:03:41,650 --> 00:03:44,702 vector. So, let's say, if we have in the simplest 47 00:03:44,702 --> 00:03:49,958 example we could have, let's say, vector in two dimensions with some coordinates ax 48 00:03:49,958 --> 00:03:52,484 and ay. So, these particular coordinates are 49 00:03:52,484 --> 00:03:56,256 specific to a choice of coordinate axis. See, we are free to choose another 50 00:03:56,256 --> 00:04:00,018 coordinate system, in which case, the coordinates will change, but the vector 51 00:04:00,018 --> 00:04:03,170 itself in some sense, in some absolute sense will remain the same. 52 00:04:03,170 --> 00:04:08,252 So likewise, in quantum theory, one can think about an existing quantum state, 53 00:04:08,252 --> 00:04:11,780 which is sort of independent of our way to describe it. 54 00:04:11,780 --> 00:04:17,452 And if we choose to do so using the wave-function, it, sort of, corresponds to 55 00:04:17,452 --> 00:04:22,789 the choice of particular basis. But there are circumstances, however, 56 00:04:22,789 --> 00:04:26,624 where we don't want to specify a representation. 57 00:04:26,624 --> 00:04:32,342 In this case, we could have called our vector, state vector as psi with a vector 58 00:04:32,342 --> 00:04:37,550 sign on top which would have been the usual notation in Linear Algebra, but 59 00:04:37,550 --> 00:04:42,842 instead of using this notation, Dirac found useful to use another notation 60 00:04:42,842 --> 00:04:47,124 presented here which is so-called ket vector by Dirac. 61 00:04:47,124 --> 00:04:52,389 And also, there is a bra vector which corresponds essentially to psi dagger and 62 00:04:52,389 --> 00:04:57,264 I, to tell you the truth, there is no profound reason for the choice of this 63 00:04:57,264 --> 00:05:02,412 particular notation so they just turned out to be convenient to a couple metric 64 00:05:02,412 --> 00:05:06,987 elements, etc., we will see. And, not that there is a profound reason 65 00:05:06,987 --> 00:05:12,237 to call them this way, bra and ket vectors basically are called this way because when 66 00:05:12,237 --> 00:05:16,683 put together, they sound like a bracket, which sort of they look like. 67 00:05:16,683 --> 00:05:21,650 So now, let me present a few embarrassingly simple facts from the basic 68 00:05:21,650 --> 00:05:27,010 Linear Algebra that I do expect most of you to know but however, I'm going to, I'm 69 00:05:27,010 --> 00:05:32,530 still going to remind you of those because they're going to be connected to slightly 70 00:05:32,530 --> 00:05:38,050 more complicated structures that arise in the abstract mathematical formulas of 71 00:05:38,050 --> 00:05:43,726 quantum mechanics. So, what I have here is simply a planar 72 00:05:43,726 --> 00:05:49,966 vector in two dimensions, some vector e and if I have a coordinate system, x and 73 00:05:49,966 --> 00:05:55,726 y, with some unit vectors ex and ey, so I can write my vector as a linear 74 00:05:55,726 --> 00:06:03,596 combination of this unit vectors. So, ex here is 1, 0 and ey is 0, 1. 75 00:06:03,597 --> 00:06:10,202 Or I can write it as so. So, the vector and its conjugate. 76 00:06:10,202 --> 00:06:15,244 So, another things I want to emphasize is that if we consider the full length 77 00:06:15,244 --> 00:06:20,204 structure, ex, ex dagger plus ey, ey dagger, on this two, there's a metrics 78 00:06:20,204 --> 00:06:25,886 multiplication of these vectors. So, we're going to have 1, 0, 0, 0 in the 79 00:06:25,886 --> 00:06:33,752 first room plus 0, 0, 0, 1 in the second room, which when put together, becomes an 80 00:06:33,752 --> 00:06:39,231 identity matrix. But very importantly, this result is not 81 00:06:39,231 --> 00:06:44,647 unique to the particular choice of these orthonormal bases. 82 00:06:44,648 --> 00:06:51,800 The result holds for any such orthonormal bases in any dimension and in any linear 83 00:06:51,800 --> 00:06:55,386 vector space. Now, the reason I'm telling you all these 84 00:06:55,386 --> 00:07:00,146 is because in quantum mechanics, so as we just discussed, so the state vectors, they 85 00:07:00,146 --> 00:07:04,974 represent elements in some Hilbert space, in some linear vector space, which we call 86 00:07:04,974 --> 00:07:08,238 Hilbert space. But in order to be able to work with this 87 00:07:08,238 --> 00:07:12,654 vector, just like here, we need introduce a coordinate system so that we have 88 00:07:12,654 --> 00:07:17,412 something concrete to work with. So, in quantum mechanics, we have to 89 00:07:17,412 --> 00:07:20,979 define a basis. Since the Hilbert space is much more 90 00:07:20,979 --> 00:07:26,162 complicated in, generally than this two dimensional vector space, and most often, 91 00:07:26,162 --> 00:07:30,564 we're dealing actually with infinite dimensional Hilbert spaces, so the 92 00:07:30,564 --> 00:07:34,437 structure of the basis is generally more complicated. 93 00:07:34,437 --> 00:07:39,620 But nevertheless, the properties that we just discussed, the simple properties of 94 00:07:39,620 --> 00:07:42,462 Linear Algebra, so are, still remain the same. 95 00:07:42,462 --> 00:07:49,197 So, for example, this identity that we just derived for this basis ex and ey also 96 00:07:49,197 --> 00:07:53,390 is, holds for the basis in a, in the Hilbert space. 97 00:07:53,390 --> 00:07:57,571 So, let's see if we have a set of the x as either a discrete set or a continuum set. 98 00:07:57,572 --> 00:08:03,621 We will see both of them. So the sum of this ket-bra products over 99 00:08:03,621 --> 00:08:08,017 all basis vectors, gives rise to an Identity operation. 100 00:08:08,017 --> 00:08:14,273 And this resolution of identity is going to be extremely useful for us in a number 101 00:08:14,273 --> 00:08:18,981 of derivations that we're going to see throughout the course. 102 00:08:18,981 --> 00:08:22,620 So, for example, for this q, I haven't really specified the physical meaning of 103 00:08:22,620 --> 00:08:26,622 this q, it can be anything. So, one example we're going to see pretty 104 00:08:26,622 --> 00:08:33,072 often is when we're dealing with the integral over volume of product like this, 105 00:08:33,072 --> 00:08:38,724 where this vectors respond to eigenstates of the position operator. 106 00:08:38,724 --> 00:08:45,024 So, another thing I want to mention here is that any wave-function, so what,what 107 00:08:45,024 --> 00:08:51,234 basis really means is that any vector in a linear vector space can be represented as 108 00:08:51,234 --> 00:08:57,496 a linear combination of the basis vectors. Just like here, I represented my vector a 109 00:08:57,496 --> 00:09:02,612 as a linear combination of ex and ey. So, the fact that I have a basis here 110 00:09:02,612 --> 00:09:08,420 implies that I can represent my wave-function or my state vector as a 111 00:09:08,420 --> 00:09:13,754 linear combination of these q's. And the matrix elements between q and psi 112 00:09:13,754 --> 00:09:18,542 is what we actually call the wave-function in the q representation. 113 00:09:18,542 --> 00:09:25,862 So, this is much like these coordinates. So, these coordinates, for instance here, 114 00:09:25,862 --> 00:09:34,914 ax is can be written as ex dagger times a, and here psi of q can be written as a 115 00:09:34,914 --> 00:09:40,198 bracket product of q and psi. So, and again, in principle, we can have 116 00:09:40,198 --> 00:09:44,752 two situations, either we have a discrete spectrum or we have a continuum spectrum 117 00:09:44,752 --> 00:09:49,108 in the former case, we have a sum over q, in the latter case we have an integral 118 00:09:49,108 --> 00:09:53,510 over q. But now, a question remains of how to 119 00:09:53,510 --> 00:10:00,075 actually choose a basis or representation to describe our state vector or 120 00:10:00,075 --> 00:10:04,210 wave-function. Just like in the usual Linear Algebra 121 00:10:04,210 --> 00:10:09,030 there is no absolute or correct answer. The choice of a basis is a question of 122 00:10:09,030 --> 00:10:14,212 convenience. But there is some standard or natural 123 00:10:14,212 --> 00:10:20,928 choices that can be constructed, and so I'm going to discuss now how it's how it's 124 00:10:20,928 --> 00:10:23,704 done. So, we already know that physical 125 00:10:23,704 --> 00:10:28,852 observables in Quantum Mechanics are associated with linear Hermitian or 126 00:10:28,852 --> 00:10:33,272 self-adjoined operators. For a generic such operator, one can 127 00:10:33,272 --> 00:10:39,110 define the eigenvalue problem as here, which basically is the problem of finding 128 00:10:39,110 --> 00:10:45,116 a vector or vectors, set of vectors, which under the action of the operator don't 129 00:10:45,116 --> 00:10:50,050 really change but just are stretched with a certain eigenvalue a. 130 00:10:50,050 --> 00:10:55,554 And for the Hermitian and self-adjoined operator, these values of a are real 131 00:10:55,554 --> 00:10:59,590 numbers. So what's essential is that, in many 132 00:10:59,590 --> 00:11:07,180 cases, this set of eigenvalues of such operators form a basis in the Hilbert 133 00:11:07,180 --> 00:11:10,610 space. This means that we can expand our 134 00:11:10,610 --> 00:11:14,855 wave-function in terms of this basis vectors. 135 00:11:14,855 --> 00:11:21,623 And the corresponding matrix elements, so let's say for this vector a, whatever it 136 00:11:21,623 --> 00:11:26,230 is, is called the wave-function in the a representation. 137 00:11:26,230 --> 00:11:30,863 So, one can construct an infinite number of such representations, just like there 138 00:11:30,863 --> 00:11:35,201 exist an infinite number of observables and corresponding operators. 139 00:11:35,201 --> 00:11:40,407 But sort of a natural way to choose a basis is to use operators that are 140 00:11:40,408 --> 00:11:45,682 irrelevant to our problem, such as for instance, the coordinate operator, it's 141 00:11:45,682 --> 00:11:51,379 just here in one dimensional quantum mechanics, so the momentum operator. 142 00:11:51,380 --> 00:11:56,201 And the corresponding matrix elements, psi of x or psi of p, are called the 143 00:11:56,201 --> 00:12:01,077 coordinate representation and the momentum representation correspondingly. 144 00:12:01,078 --> 00:12:07,395 So for example, psi of x or psi of r, we have already seen many times in our 145 00:12:07,395 --> 00:12:11,152 lectures. Now, the meaning of this of this basis 146 00:12:11,152 --> 00:12:15,717 vector is, is essentially this tightly localized wave packets. 147 00:12:15,718 --> 00:12:19,991 We shall locate it in a precise point and space. 148 00:12:19,992 --> 00:12:27,347 So in contrast, the vectors p correspond to states with the well-defined momentum, 149 00:12:27,347 --> 00:12:31,140 but which are completely delocalized in space. 150 00:12:31,140 --> 00:12:35,628 So, these are completely different vectors, completely different basis, and 151 00:12:35,628 --> 00:12:39,143 which one to choose is entirely the question of convenience.