1 00:00:00,050 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,810 under a Creative Commons license. 3 00:00:03,810 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,120 to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:18,050 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,050 --> 00:00:18,995 at ocw.mit.edu. 8 00:00:21,650 --> 00:00:24,200 PROFESSOR: It's time to get started. 9 00:00:24,200 --> 00:00:27,040 Today, we'll complete our discussion 10 00:00:27,040 --> 00:00:30,020 of nuclear magnetic resonance. 11 00:00:30,020 --> 00:00:32,290 And we started that last time. 12 00:00:32,290 --> 00:00:38,020 And after that, we will talk about multi-particle states, 13 00:00:38,020 --> 00:00:40,860 the kind of quantum mechanical states 14 00:00:40,860 --> 00:00:45,840 that you need if your system has more than one particle. 15 00:00:45,840 --> 00:00:48,460 That will bring us naturally to the idea 16 00:00:48,460 --> 00:00:52,260 of denser products of vector spaces, which 17 00:00:52,260 --> 00:00:57,910 is the key mathematical idea behind operations 18 00:00:57,910 --> 00:01:01,870 such as addition of angular momentum. 19 00:01:01,870 --> 00:01:05,980 By introducing a little before, we really 20 00:01:05,980 --> 00:01:08,460 talk about addition of angular momentum, 21 00:01:08,460 --> 00:01:13,350 the idea of denser products of vector spaces. 22 00:01:13,350 --> 00:01:16,860 I really hope that this subject will 23 00:01:16,860 --> 00:01:20,130 look a lot less mysterious in a couple of weeks 24 00:01:20,130 --> 00:01:24,020 when we will really be adding angular momentum. 25 00:01:24,020 --> 00:01:31,000 So you will have seen the main mathematical structures. 26 00:01:31,000 --> 00:01:32,790 And you will know what you're supposed 27 00:01:32,790 --> 00:01:34,350 to do when you have a formula. 28 00:01:34,350 --> 00:01:37,110 So that should be of some help. 29 00:01:37,110 --> 00:01:40,140 So last time, we were talking about 30 00:01:40,140 --> 00:01:41,770 nuclear magnetic resonance. 31 00:01:41,770 --> 00:01:43,200 And I described the problem. 32 00:01:43,200 --> 00:01:48,110 Here is a summary of where more or less we got last time. 33 00:01:48,110 --> 00:01:51,970 We said in the set up for nuclear magnetic resonance, 34 00:01:51,970 --> 00:01:57,250 we have a large magnetic field, the longitudinal magnetic field 35 00:01:57,250 --> 00:02:01,070 it's sometimes called, along the z direction. 36 00:02:01,070 --> 00:02:04,860 In addition to that longitudinal magnetic field, 37 00:02:04,860 --> 00:02:09,289 there's a rotating magnetic field in the xy plane. 38 00:02:09,289 --> 00:02:12,835 And it's rotating in the direction that you see here. 39 00:02:12,835 --> 00:02:20,350 As seen on the xy plane, it's clockwise on the plane. 40 00:02:20,350 --> 00:02:23,470 In terms of a rotation around the z direction, 41 00:02:23,470 --> 00:02:26,810 it's negative in the sense that the positive rotation 42 00:02:26,810 --> 00:02:30,484 around the z plane would go like this. 43 00:02:30,484 --> 00:02:32,787 And this magnetic field is rotating 44 00:02:32,787 --> 00:02:34,020 in the other direction. 45 00:02:34,020 --> 00:02:35,390 So I wrote it here. 46 00:02:35,390 --> 00:02:42,820 The magnitude of this radio frequency component is B1. 47 00:02:42,820 --> 00:02:47,080 And typically, it's much smaller than B0 48 00:02:47,080 --> 00:02:49,880 in the problems of interest. 49 00:02:49,880 --> 00:02:54,310 Then, as you recall, the spin Hamiltonian 50 00:02:54,310 --> 00:02:58,490 is always given by this constant gamma that 51 00:02:58,490 --> 00:03:02,960 relates to the dipole moment of a particle to its spin angular 52 00:03:02,960 --> 00:03:09,260 momentum B dot S, the S operator. 53 00:03:09,260 --> 00:03:11,440 And the S operator has components 54 00:03:11,440 --> 00:03:17,270 Sx, Sy, Sz multiplied by B. Well, the z component 55 00:03:17,270 --> 00:03:23,172 picks up a z, the x component Sx, the y component Sy. 56 00:03:23,172 --> 00:03:24,380 And this is your Hamiltonian. 57 00:03:24,380 --> 00:03:28,510 And as we emphasized last time, this Hamiltonian 58 00:03:28,510 --> 00:03:30,880 is time dependent. 59 00:03:30,880 --> 00:03:35,090 Moreover, at different times, the Hamiltonians don't commute. 60 00:03:35,090 --> 00:03:41,370 So the HS at time T1, HS at time T2, they don't commute. 61 00:03:41,370 --> 00:03:45,770 Therefore, any of the formulas we have for solving problems 62 00:03:45,770 --> 00:03:47,325 like this don't apply. 63 00:03:49,850 --> 00:03:53,860 The only formula that applies is this time ordered exponential 64 00:03:53,860 --> 00:03:56,610 that we spoke about that in general 65 00:03:56,610 --> 00:03:58,500 is very difficult to use. 66 00:03:58,500 --> 00:04:01,770 The time ordered exponential just never quits. 67 00:04:01,770 --> 00:04:05,520 And we're writing terms, term after term. 68 00:04:05,520 --> 00:04:08,100 And then, you have to add it all up. 69 00:04:08,100 --> 00:04:10,110 And adding it all up is difficult. 70 00:04:10,110 --> 00:04:14,960 So most of the time, it's not a great help. 71 00:04:14,960 --> 00:04:18,680 So what did we say we would do? 72 00:04:18,680 --> 00:04:22,870 We wondered if we could somehow do quantum mechanics, 73 00:04:22,870 --> 00:04:26,760 or think of quantum mechanics, in the rotating frame, 74 00:04:26,760 --> 00:04:31,600 in a frame that is rotating with a magnetic field in such a way 75 00:04:31,600 --> 00:04:36,460 that perhaps in that frame the physics would simplify. 76 00:04:36,460 --> 00:04:39,830 And physics simplified would be that the Hamiltonian becomes 77 00:04:39,830 --> 00:04:41,030 time independent. 78 00:04:41,030 --> 00:04:44,740 So we're trying to figure out, how would the Hamiltonian 79 00:04:44,740 --> 00:04:47,720 look in another frame? 80 00:04:47,720 --> 00:04:53,380 Now, we've not discussed in general very explicitly 81 00:04:53,380 --> 00:04:56,620 transformations of Hamiltonians within different frames. 82 00:04:56,620 --> 00:04:59,750 But we can sort of roughly try to guess 83 00:04:59,750 --> 00:05:02,320 what it would have to do. 84 00:05:02,320 --> 00:05:04,490 So what we said last time is that we 85 00:05:04,490 --> 00:05:08,380 would imagine that you have an original system 86 00:05:08,380 --> 00:05:11,650 and no magnetic field, nothing whatsoever. 87 00:05:11,650 --> 00:05:15,000 So forget about all that-- no magnetic field. 88 00:05:15,000 --> 00:05:16,630 You put the spin state. 89 00:05:16,630 --> 00:05:20,980 It stays there forever, doesn't precess. 90 00:05:20,980 --> 00:05:24,480 It doesn't precess because there's no magnetic field. 91 00:05:24,480 --> 00:05:27,720 Then, you jump into a rotating frame. 92 00:05:27,720 --> 00:05:30,600 So this was a static frame, no magnetic field. 93 00:05:30,600 --> 00:05:32,480 And then, you start rotating. 94 00:05:32,480 --> 00:05:34,220 And then, you look at the spins. 95 00:05:34,220 --> 00:05:37,680 And the spins are not moving. 96 00:05:37,680 --> 00:05:40,020 But you say, oh, I'm rotating. 97 00:05:40,020 --> 00:05:45,425 So for me, they seem to be precessing, in fact precessing 98 00:05:45,425 --> 00:05:49,300 with the angular frequency that I am rotating. 99 00:05:49,300 --> 00:05:52,800 And therefore, in my rotating frame, 100 00:05:52,800 --> 00:05:54,680 I would have a Hamiltonian that I 101 00:05:54,680 --> 00:05:59,100 would call the rotating Hamiltonian, HR. 102 00:05:59,100 --> 00:06:02,160 And how would it look? 103 00:06:02,160 --> 00:06:09,960 We claim that in fact spin states in my rotating frame 104 00:06:09,960 --> 00:06:13,700 would have to be rotating really this way. 105 00:06:13,700 --> 00:06:16,050 Because I'm rotating with the magnetic field. 106 00:06:16,050 --> 00:06:19,880 So for me, they're rotating in the positive z direction 107 00:06:19,880 --> 00:06:22,430 with angular frequency omega. 108 00:06:22,430 --> 00:06:24,730 And this would be the operator that 109 00:06:24,730 --> 00:06:27,970 makes them rotate exactly that way. 110 00:06:27,970 --> 00:06:31,580 Because if you remember, an operator here minus 111 00:06:31,580 --> 00:06:36,990 i angle Sz over h bar, this is the angle 112 00:06:36,990 --> 00:06:38,530 as a function of time. 113 00:06:38,530 --> 00:06:43,790 Omega t, that's how these operators rotate states. 114 00:06:43,790 --> 00:06:48,720 So this U of t is the U of t that rotates 115 00:06:48,720 --> 00:06:52,150 states the way I want them in my rotating frame. 116 00:06:52,150 --> 00:06:56,720 Now, you remember for time independent systems, 117 00:06:56,720 --> 00:07:04,560 the unitary operator is e to the minus iht over h bar. 118 00:07:04,560 --> 00:07:08,340 So you identify the Hamiltonian as omega Sz. 119 00:07:13,130 --> 00:07:18,840 In general, given a unitary operator, 120 00:07:18,840 --> 00:07:21,120 the Hamiltonian is obtained this way. 121 00:07:21,120 --> 00:07:24,620 You take the time derivative of it multiplied by U dagger. 122 00:07:24,620 --> 00:07:28,080 And that's the formula for the Hamiltonian given the time 123 00:07:28,080 --> 00:07:32,220 evolution operator, something we did about a couple of weeks 124 00:07:32,220 --> 00:07:34,640 ago. 125 00:07:34,640 --> 00:07:39,900 So that's where we stood-- some idea about the rotating frame. 126 00:07:39,900 --> 00:07:44,930 So the way we now make this complete 127 00:07:44,930 --> 00:07:48,290 is to say, well, I'm going to attempt to do the following. 128 00:07:48,290 --> 00:07:59,282 I'm going to act with U, this U of t, on this complicated state 129 00:07:59,282 --> 00:08:02,590 that I don't know how it behaves. 130 00:08:02,590 --> 00:08:09,100 And this I'm going to call the rotated wave function. 131 00:08:09,100 --> 00:08:11,740 It's a definition. 132 00:08:11,740 --> 00:08:13,800 We're going to use this operator that 133 00:08:13,800 --> 00:08:18,200 seems to induce a rotation of the frames 134 00:08:18,200 --> 00:08:22,630 and act it on the state to find this. 135 00:08:22,630 --> 00:08:25,290 And we're going to hope that the Schrodinger 136 00:08:25,290 --> 00:08:29,970 equation for this one is simpler. 137 00:08:29,970 --> 00:08:34,239 So actually, last time, I actually did a computation 138 00:08:34,239 --> 00:08:36,980 and found what is the Schrodinger Hamiltonian 139 00:08:36,980 --> 00:08:43,390 for this, psi R. I want to do it in a slightly different way. 140 00:08:43,390 --> 00:08:48,190 I think it's a little easier perhaps, or maybe less 141 00:08:48,190 --> 00:08:50,670 familiar but more interesting. 142 00:08:50,670 --> 00:08:53,300 I write this like that-- U of t. 143 00:08:53,300 --> 00:08:56,625 And I say, look, this state evolves 144 00:08:56,625 --> 00:08:59,470 due to its own Hamiltonian. 145 00:08:59,470 --> 00:09:01,840 U of t is an extra thing I've put here. 146 00:09:01,840 --> 00:09:04,710 So here is U of t. 147 00:09:04,710 --> 00:09:06,240 But how do I distinguish it? 148 00:09:06,240 --> 00:09:15,390 I put an S for the spin system, for HS, psi 0. 149 00:09:15,390 --> 00:09:22,430 So psi of t is being evolved by the Schrodinger Hamiltonian, 150 00:09:22,430 --> 00:09:26,120 the unitary operator associated to the Hamiltonian 151 00:09:26,120 --> 00:09:29,030 that we want times time of 0. 152 00:09:29,030 --> 00:09:34,320 So this is the total unitary operator 153 00:09:34,320 --> 00:09:38,660 that evolves this state. 154 00:09:38,660 --> 00:09:41,490 Therefore, I'm going to say, I'm going 155 00:09:41,490 --> 00:09:45,190 to recover the Hamiltonian using this formula. 156 00:09:45,190 --> 00:09:51,190 So I will simply say that the rotating Hamiltonian 157 00:09:51,190 --> 00:10:04,660 is going to be HR is ih bar dt of this whole thing, U 158 00:10:04,660 --> 00:10:11,420 of t US of t, times its dagger. 159 00:10:11,420 --> 00:10:13,045 Because this is the general formula 160 00:10:13,045 --> 00:10:15,180 to obtain the Hamiltonian when you 161 00:10:15,180 --> 00:10:17,560 know the time evolution operator. 162 00:10:17,560 --> 00:10:24,290 So the dagger of this would be US dagger U of t dagger. 163 00:10:27,900 --> 00:10:31,370 Now we have to evaluate this derivative. 164 00:10:31,370 --> 00:10:34,410 It's is not all that bad. 165 00:10:34,410 --> 00:10:37,850 If the time derivative hits the U, 166 00:10:37,850 --> 00:10:43,480 this one, then U and U dagger annihilate each other. 167 00:10:43,480 --> 00:10:46,170 So you're left with U dagger of t here. 168 00:10:46,170 --> 00:10:56,430 So you get ih bar dtU U dagger. 169 00:10:56,430 --> 00:11:00,640 Now, when it acts on this second one, what do we get? 170 00:11:00,640 --> 00:11:02,590 Well, I'll write it this way. 171 00:11:02,590 --> 00:11:06,860 Plus U of t, because this will be first. 172 00:11:06,860 --> 00:11:08,787 Then, ih bar dtUS. 173 00:11:16,250 --> 00:11:24,260 And then, you have US dagger, and then U dagger of t. 174 00:11:30,150 --> 00:11:34,840 Now, this thing is how much? 175 00:11:34,840 --> 00:11:39,530 Well, in this thing is in fact this Hamiltonian 176 00:11:39,530 --> 00:11:45,070 associated to what we called U, which is just omega Sz. 177 00:11:48,060 --> 00:11:50,610 So actually, I don't know if I have-- yeah, 178 00:11:50,610 --> 00:11:52,370 I have this notation here. 179 00:11:52,370 --> 00:11:59,570 I think it's useful-- H of U, the Hamiltonian associated 180 00:11:59,570 --> 00:12:01,550 to whatever U it is. 181 00:12:01,550 --> 00:12:07,740 So you get a nice formula-- plus U of t. 182 00:12:07,740 --> 00:12:14,941 This is the Hamiltonian HS U dagger of t. 183 00:12:17,770 --> 00:12:21,290 So this formula we derived also last time 184 00:12:21,290 --> 00:12:29,850 by looking at the differential equations satisfied by psi R. 185 00:12:29,850 --> 00:12:32,690 So it's a nice formula. 186 00:12:32,690 --> 00:12:35,700 If somebody gives you something like this 187 00:12:35,700 --> 00:12:37,920 and says, here's a unitary operator, 188 00:12:37,920 --> 00:12:40,200 what is the new Hamiltonian? 189 00:12:40,200 --> 00:12:47,550 The new Hamiltonian is equal to the transformed old Hamiltonian 190 00:12:47,550 --> 00:12:51,220 plus the Hamiltonian associated to this time evolution 191 00:12:51,220 --> 00:12:52,115 operator. 192 00:12:52,115 --> 00:12:53,530 It's a nice formula. 193 00:12:56,190 --> 00:12:59,300 Now, more concretely now, we can see 194 00:12:59,300 --> 00:13:01,840 what this is supposed to be. 195 00:13:01,840 --> 00:13:08,625 So here is the important crucial step. 196 00:13:08,625 --> 00:13:13,170 Let's see, HR is supposed to be what? 197 00:13:13,170 --> 00:13:18,290 We now know we're going to choose U to be this thing. 198 00:13:18,290 --> 00:13:25,050 Therefore, HU is omega Sz hat. 199 00:13:25,050 --> 00:13:30,540 But now we have U of t, U dagger of t. 200 00:13:30,540 --> 00:13:35,720 So I have plus e to the minus i omega 201 00:13:35,720 --> 00:13:42,880 tSz over h bar times the Hamiltonian 202 00:13:42,880 --> 00:13:47,900 that we have over there, this full Hamiltonian HS. 203 00:13:47,900 --> 00:13:49,614 So what is it? 204 00:13:49,614 --> 00:14:01,050 Minus gamma big parentheses B0 Sz hat plus B1 cosine omega 205 00:14:01,050 --> 00:14:11,150 tSx minus sine omega tSy-- like that-- 206 00:14:11,150 --> 00:14:16,510 e to the i omega tSz over h bar. 207 00:14:16,510 --> 00:14:20,260 OK, that's supposed to be our new Hamiltonian. 208 00:14:20,260 --> 00:14:25,170 And unless it simplifies, we have not gained all that much. 209 00:14:29,550 --> 00:14:34,410 So before simplifying, are there questions, 210 00:14:34,410 --> 00:14:36,225 anything we've gotten so far? 211 00:14:39,005 --> 00:14:41,440 Yes. 212 00:14:41,440 --> 00:14:47,284 AUDIENCE: So U of t is the unitary operator that 213 00:14:47,284 --> 00:14:50,693 enacts the rotation, and it's not 214 00:14:50,693 --> 00:14:55,780 corresponding to any Hamiltonian of the spin? 215 00:14:55,780 --> 00:14:59,740 PROFESSOR: Right, U of t is the thing that essentially moves us 216 00:14:59,740 --> 00:15:01,510 into the rotating frame. 217 00:15:01,510 --> 00:15:05,380 This is the hope that we have. 218 00:15:05,380 --> 00:15:08,965 Any unitary operator, if it's what evolves your state, 219 00:15:08,965 --> 00:15:10,880 it corresponds to Hamiltonian. 220 00:15:10,880 --> 00:15:14,210 But that's not anything a priori to do 221 00:15:14,210 --> 00:15:15,890 with our original Hamiltonian. 222 00:15:15,890 --> 00:15:18,540 It's just another Hamiltonian. 223 00:15:18,540 --> 00:15:24,150 So you choose your U of t behind some Hamiltonian. 224 00:15:24,150 --> 00:15:26,930 But what you learn is that the Hamiltonian 225 00:15:26,930 --> 00:15:30,200 for your new system, or for the new wave function, 226 00:15:30,200 --> 00:15:33,136 will involve that other Hamiltonian. 227 00:15:33,136 --> 00:15:34,582 Yes? 228 00:15:34,582 --> 00:15:36,992 AUDIENCE: HR equals [INAUDIBLE]? 229 00:15:39,900 --> 00:15:41,250 PROFESSOR: This is-- yeah. 230 00:15:41,250 --> 00:15:42,500 It's HU. 231 00:15:42,500 --> 00:15:44,130 I think you are probably right. 232 00:15:44,130 --> 00:15:46,920 I should call it just HU for-- yeah, 233 00:15:46,920 --> 00:15:50,590 it's a bad idea to call it HR. 234 00:15:50,590 --> 00:15:54,410 U, HU, much better-- thank you. 235 00:15:54,410 --> 00:16:01,220 Here, HR is the Hamiltonian associated to the Schrodinger 236 00:16:01,220 --> 00:16:02,410 equation here. 237 00:16:02,410 --> 00:16:05,420 So the claim behind this calculation 238 00:16:05,420 --> 00:16:12,320 is that if you calculated ih bar d dt of psi R of t, 239 00:16:12,320 --> 00:16:19,030 you would find HR psi R of t. 240 00:16:19,030 --> 00:16:21,015 So that's the calculation I think 241 00:16:21,015 --> 00:16:23,760 we did last time in which we just 242 00:16:23,760 --> 00:16:25,480 calculated these derivatives. 243 00:16:25,480 --> 00:16:29,000 And you find this answer. 244 00:16:29,000 --> 00:16:33,590 OK, so let's continue here and see what happens. 245 00:16:33,590 --> 00:16:39,040 Well, I have an Sz and a Sz similarity transformation here. 246 00:16:39,040 --> 00:16:40,390 But here is Sz. 247 00:16:40,390 --> 00:16:43,835 So that Sz doesn't care. 248 00:16:43,835 --> 00:16:45,900 And it can just go out. 249 00:16:45,900 --> 00:16:47,080 So what do we have? 250 00:16:47,080 --> 00:17:01,130 We have omega minus gamma B0 Sz hat like that. 251 00:17:04,300 --> 00:17:06,500 Then, the rest would be what? 252 00:17:06,500 --> 00:17:11,400 I can take out the numbers gamma, the B1. 253 00:17:11,400 --> 00:17:14,790 And then, I have this exponential-- e to the minus i 254 00:17:14,790 --> 00:17:19,040 omega t Sz hat over h bar. 255 00:17:19,040 --> 00:17:27,760 I'm just having here cosine omega tSx hat minus sine omega 256 00:17:27,760 --> 00:17:36,210 tSy hat, e to the i omega tSz hat over h bar. 257 00:17:41,860 --> 00:17:49,810 OK, now there's two ways to do this. 258 00:17:49,810 --> 00:17:53,640 You have an exponential here and another exponential. 259 00:17:53,640 --> 00:17:57,690 This is the kind of thing you've done quite a few times 260 00:17:57,690 --> 00:17:58,870 in different ways. 261 00:17:58,870 --> 00:18:03,190 You could expand exponentials and just multiply. 262 00:18:03,190 --> 00:18:09,150 Or you can sort of do this usual formula of e to the AB e 263 00:18:09,150 --> 00:18:13,850 to the minus A for this term and for this term, 264 00:18:13,850 --> 00:18:17,670 and simplify, and do many things. 265 00:18:17,670 --> 00:18:19,270 So let's do something different. 266 00:18:22,630 --> 00:18:24,830 When one looks at this, one could say, 267 00:18:24,830 --> 00:18:26,550 well, here's my problem. 268 00:18:26,550 --> 00:18:30,940 HR-- I know U of t. 269 00:18:30,940 --> 00:18:32,380 I fixed it. 270 00:18:32,380 --> 00:18:37,150 So knowing psi is knowing psi R. But to know psi R, 271 00:18:37,150 --> 00:18:39,580 I need to know HR. 272 00:18:39,580 --> 00:18:42,620 And HR still looks very complicated 273 00:18:42,620 --> 00:18:46,460 unless somehow this whole idea has worked out very well 274 00:18:46,460 --> 00:18:49,630 and this simplifies. 275 00:18:49,630 --> 00:18:51,610 And the hope is that it simplifies. 276 00:18:51,610 --> 00:18:55,230 Because in the rotating frame, the field 277 00:18:55,230 --> 00:18:57,380 is not rotating anymore. 278 00:18:57,380 --> 00:19:05,800 So I'll call this M of t. 279 00:19:05,800 --> 00:19:09,050 And I think one way of doing this, 280 00:19:09,050 --> 00:19:11,810 doing something, if you're in a rush 281 00:19:11,810 --> 00:19:15,940 and you don't have a formula sheet or any of those things, 282 00:19:15,940 --> 00:19:20,080 is to just take its derivative, its time derivative. 283 00:19:20,080 --> 00:19:22,270 Now, if you have a function of time, 284 00:19:22,270 --> 00:19:24,990 and you don't know what it is, take its time derivative. 285 00:19:24,990 --> 00:19:28,500 And maybe you'll see a differential equation 286 00:19:28,500 --> 00:19:29,990 or see something. 287 00:19:29,990 --> 00:19:38,220 So let's take the d dt of M of t. 288 00:19:38,220 --> 00:19:41,040 OK, so what do we get? 289 00:19:41,040 --> 00:19:43,860 Well, we get the following-- e to the minus 290 00:19:43,860 --> 00:19:49,210 i omega tSz over h bar. 291 00:19:49,210 --> 00:19:53,860 And I'll put the big parentheses here. 292 00:19:53,860 --> 00:19:57,280 OK, here I have an object. 293 00:19:57,280 --> 00:20:00,160 This is something I think you are familiar. 294 00:20:00,160 --> 00:20:02,830 You've been computing these things for coherence state, 295 00:20:02,830 --> 00:20:05,010 for squeeze state, and for all that. 296 00:20:05,010 --> 00:20:08,150 So this will certainly sound familiar. 297 00:20:08,150 --> 00:20:12,200 If you take a time derivative of this exponential, 298 00:20:12,200 --> 00:20:15,320 it will bring an operator down. 299 00:20:15,320 --> 00:20:17,610 When you take the time derivative of this operator, 300 00:20:17,610 --> 00:20:22,420 it will bring the operator down but with a different sign. 301 00:20:22,420 --> 00:20:24,660 And therefore, it will form the commutator 302 00:20:24,660 --> 00:20:27,220 with the thing like this in the middle. 303 00:20:27,220 --> 00:20:32,540 So the first thing in this thing is the commutator 304 00:20:32,540 --> 00:20:39,240 of-- you take the time derivative of this-- minus i 305 00:20:39,240 --> 00:20:49,650 omega Sz hat over h bar with the rest of this thing, 306 00:20:49,650 --> 00:20:57,320 with cosine omega tSx minus sine omega tSy. 307 00:21:03,335 --> 00:21:07,330 Because the first derivative brings down the minus i omega 308 00:21:07,330 --> 00:21:08,410 this over that. 309 00:21:08,410 --> 00:21:12,205 So I'll actually take the minus i omega over h bar out. 310 00:21:20,070 --> 00:21:22,300 OK, that's the first term that you 311 00:21:22,300 --> 00:21:25,330 get from differentiating this and that. 312 00:21:25,330 --> 00:21:27,520 And then, you have to differentiate 313 00:21:27,520 --> 00:21:29,510 the term in the middle. 314 00:21:29,510 --> 00:21:39,060 So then, you get minus omega sine omega tSx 315 00:21:39,060 --> 00:21:48,500 hat minus omega cosine omega tSy hat. 316 00:21:48,500 --> 00:21:51,340 And now, we can close the big parentheses 317 00:21:51,340 --> 00:21:56,010 and put the other exponential like that. 318 00:21:59,550 --> 00:22:14,880 OK, so let's see this. 319 00:22:14,880 --> 00:22:22,540 Sz with Sx is ih bar Sy. 320 00:22:22,540 --> 00:22:31,020 So you get here minus i omega over h bar, ih bar Sy 321 00:22:31,020 --> 00:22:32,900 cosine omega t. 322 00:22:36,540 --> 00:22:40,840 Because Sz with Sx-- you should remember that I think. 323 00:22:40,840 --> 00:22:44,470 Otherwise, you'll waste time. x, y, and z, 324 00:22:44,470 --> 00:22:48,660 when you have Sz with Sx, you get Sy. 325 00:22:48,660 --> 00:22:50,850 Sx with Sy, you get a z. 326 00:22:50,850 --> 00:22:54,310 And that order helps. 327 00:22:54,310 --> 00:22:58,520 So ih bar this, and then the other term 328 00:22:58,520 --> 00:23:03,040 would be minus sine omega t. 329 00:23:03,040 --> 00:23:13,394 But now, Sz with Sy is minus ih bar, so plus ih bar Sx. 330 00:23:16,572 --> 00:23:18,720 OK, I won't copy the other term. 331 00:23:18,720 --> 00:23:21,150 And let's see if it worked out. 332 00:23:21,150 --> 00:23:24,645 Minus i and i is a plus. 333 00:23:24,645 --> 00:23:30,970 The h bar cancels, so plus omega Sy cosine omega t 334 00:23:30,970 --> 00:23:34,730 minus omega cosine omega tSy. 335 00:23:34,730 --> 00:23:38,110 So this term cancels with this. 336 00:23:38,110 --> 00:23:42,330 And this-- there's an ih bar. 337 00:23:42,330 --> 00:23:46,780 The sign is the same, so it's a plus omega sine 338 00:23:46,780 --> 00:23:49,540 Sx minus omega sine. 339 00:23:49,540 --> 00:23:51,510 So these two cancel. 340 00:23:51,510 --> 00:23:54,140 And happily, this whole thing is 0. 341 00:23:56,790 --> 00:24:02,260 So that is the signal your strategy worked. 342 00:24:02,260 --> 00:24:02,760 Why? 343 00:24:02,760 --> 00:24:06,840 Because M of t looks like it has lots of time dependence. 344 00:24:06,840 --> 00:24:10,300 But in fact, it has none. 345 00:24:10,300 --> 00:24:13,010 Its derivative is 0. 346 00:24:13,010 --> 00:24:17,920 So if its derivative is 0, you can evaluate it 347 00:24:17,920 --> 00:24:20,420 at any time you want. 348 00:24:20,420 --> 00:24:28,270 Well, the best time to evaluate it is time equals 0 now. 349 00:24:28,270 --> 00:24:32,500 And you put here 0-- disappears, disappears. 350 00:24:32,500 --> 00:24:33,890 This term disappears. 351 00:24:33,890 --> 00:24:42,300 The whole thing is just Sx hat, nothing else. 352 00:24:42,300 --> 00:24:48,150 So you may want to do it the other way by doing 353 00:24:48,150 --> 00:24:51,430 this similarity on this and on that, 354 00:24:51,430 --> 00:24:53,200 or do it whichever way you want. 355 00:24:53,200 --> 00:24:55,930 But this is good enough. 356 00:24:55,930 --> 00:24:58,960 So we have that this whole Hamiltonian 357 00:24:58,960 --> 00:25:00,440 has become very simple. 358 00:25:00,440 --> 00:25:06,730 So HR has become-- and I'll copy it here, 359 00:25:06,730 --> 00:25:17,780 I want to make sure-- minus gamma B0 plus omega Sz 360 00:25:17,780 --> 00:25:25,140 hat minus gamma B1Sx hat. 361 00:25:25,140 --> 00:25:30,920 Very nice Hamiltonian-- just two pieces. 362 00:25:30,920 --> 00:25:35,620 The Sz part of the Hamiltonian that used to be this 363 00:25:35,620 --> 00:25:38,310 got this extra piece from the rotation. 364 00:25:38,310 --> 00:25:42,340 That's our omega Sz. 365 00:25:42,340 --> 00:25:45,280 And this piece over here is all that 366 00:25:45,280 --> 00:25:47,190 is left of the rotating one. 367 00:25:47,190 --> 00:25:51,090 Because in the rotating frame, the magnetic field 368 00:25:51,090 --> 00:25:57,120 is always along the x-axis as it was at time equals 0. 369 00:25:57,120 --> 00:25:59,490 So it's a perfectly nice Hamiltonian. 370 00:25:59,490 --> 00:26:01,820 Let's factor a few things. 371 00:26:01,820 --> 00:26:06,410 Let's remember the notation that omega 0 372 00:26:06,410 --> 00:26:10,180 is [INAUDIBLE] more frequency associated to B0. 373 00:26:10,180 --> 00:26:22,060 So I'll write here, this is equal to minus gamma B0 374 00:26:22,060 --> 00:26:32,410 minus omega over gamma Sz hat plus B1Sx hat, 375 00:26:32,410 --> 00:26:43,160 which is minus gamma B0 1 minus omega over omega 0 Sz 376 00:26:43,160 --> 00:26:48,780 hat plus B1Sx hat. 377 00:26:48,780 --> 00:26:51,780 So let me look at what we've done. 378 00:26:51,780 --> 00:26:56,650 First step, I just factor a gamma here, so nothing else. 379 00:26:56,650 --> 00:26:59,370 And the same gamma went out. 380 00:26:59,370 --> 00:27:02,330 Next step was to use this equation, 381 00:27:02,330 --> 00:27:08,020 omega 0 equals gamma. [INAUDIBLE] to eliminate gamma. 382 00:27:08,020 --> 00:27:10,310 So you eliminated gamma by that. 383 00:27:10,310 --> 00:27:14,240 So it's equal to omega 0 over B0. 384 00:27:14,240 --> 00:27:15,780 So the B0 went out. 385 00:27:15,780 --> 00:27:19,100 And you have 1 minus this thing. 386 00:27:19,100 --> 00:27:23,810 And then, you can think of this HR 387 00:27:23,810 --> 00:27:33,220 as minus gamma BR times S. In the usual notation for spin 388 00:27:33,220 --> 00:27:38,420 Hamiltonians, minus gamma BS, this 389 00:27:38,420 --> 00:27:43,780 is the rotating B, the so-called rotating B. 390 00:27:43,780 --> 00:27:47,550 But of course, it's not rotating anymore, happily. 391 00:27:47,550 --> 00:27:56,320 This is just B0 times 1 minus omega over omega 0 Sz 392 00:27:56,320 --> 00:27:57,100 plus B1Sx. 393 00:28:05,710 --> 00:28:10,020 And then, how about our answer? 394 00:28:10,020 --> 00:28:12,260 What is the answer to this problem? 395 00:28:12,260 --> 00:28:15,940 Well, the answer is the psi and t. 396 00:28:15,940 --> 00:28:33,700 So psi t is U dagger times psi R of t. 397 00:28:33,700 --> 00:28:46,000 And U dagger was e to the i omega tSz over h bar-- omega t, 398 00:28:46,000 --> 00:28:46,730 yeah. 399 00:28:46,730 --> 00:28:48,480 And what is the other U? 400 00:28:48,480 --> 00:28:53,430 Well, the other U associated to a time evolution 401 00:28:53,430 --> 00:28:59,540 is just e to the minus iHRt over h bar. 402 00:28:59,540 --> 00:29:02,840 Because HR is time independent. 403 00:29:02,840 --> 00:29:09,310 So this equation that tells us how psi R evolves is with HR. 404 00:29:09,310 --> 00:29:11,970 HR is time independent. 405 00:29:11,970 --> 00:29:15,420 And therefore, you'll have here e 406 00:29:15,420 --> 00:29:23,500 to the minus iHR, which is this. 407 00:29:23,500 --> 00:29:37,300 So it becomes plus i gamma BR dot S over h bar t acting 408 00:29:37,300 --> 00:29:42,680 on the state psi at time equals 0. 409 00:29:50,020 --> 00:29:53,290 You see, the state psi R at time equals 410 00:29:53,290 --> 00:29:56,860 0 is the same as the state psi at time 411 00:29:56,860 --> 00:30:00,820 equals 0 because the unitary operator is the same. 412 00:30:00,820 --> 00:30:09,690 So let me box this as well. 413 00:30:09,690 --> 00:30:13,740 This is the complete solution for the problem of the rotating 414 00:30:13,740 --> 00:30:14,800 spins. 415 00:30:14,800 --> 00:30:17,860 So the quantum mechanical problem 416 00:30:17,860 --> 00:30:21,250 is you've gotten this time dependent magnetic field. 417 00:30:21,250 --> 00:30:23,270 You want to know the spin evolution. 418 00:30:23,270 --> 00:30:26,280 Well, the spin evolution is a little complicated. 419 00:30:26,280 --> 00:30:29,210 It's this whole thing. 420 00:30:29,210 --> 00:30:33,460 And let me just make sure everything is good here. 421 00:30:36,540 --> 00:30:37,790 Perfect. 422 00:30:37,790 --> 00:30:38,380 Fine. 423 00:30:38,380 --> 00:30:39,360 All right. 424 00:30:39,360 --> 00:30:40,080 Questions. 425 00:30:40,080 --> 00:30:41,482 Yes? 426 00:30:41,482 --> 00:30:46,150 AUDIENCE: So you have x hat, so xc hat and xs hat-- 427 00:30:46,150 --> 00:30:47,280 AUDIENCE: BR. 428 00:30:47,280 --> 00:30:48,695 BARTON ZWIEBACH: Oh. 429 00:30:48,695 --> 00:30:49,920 I'm sorry. 430 00:30:49,920 --> 00:30:52,253 It should be BR. 431 00:30:52,253 --> 00:30:53,120 I'm sorry. 432 00:30:53,120 --> 00:30:54,930 What is the mistake here? 433 00:30:54,930 --> 00:30:55,470 Sorry. 434 00:30:55,470 --> 00:30:57,460 This is terrible. 435 00:30:57,460 --> 00:31:00,320 z hat, x hat. 436 00:31:00,320 --> 00:31:01,748 Thank you. 437 00:31:01,748 --> 00:31:04,522 That's a magnetic field. 438 00:31:04,522 --> 00:31:05,230 Another question. 439 00:31:05,230 --> 00:31:07,452 Yes? 440 00:31:07,452 --> 00:31:09,160 AUDIENCE: This is a question [INAUDIBLE]. 441 00:31:09,160 --> 00:31:12,800 But when we take d dt of f of t, how 442 00:31:12,800 --> 00:31:17,990 can we take the computation relation with hu 443 00:31:17,990 --> 00:31:19,918 in that computation? 444 00:31:19,918 --> 00:31:24,582 Why is that the Hamiltonian that we take in the computation? 445 00:31:24,582 --> 00:31:26,082 BARTON ZWIEBACH: I don't understand. 446 00:31:26,082 --> 00:31:29,120 When we take the tan derivative of m of t, 447 00:31:29,120 --> 00:31:31,720 here was m of t from here to here. 448 00:31:31,720 --> 00:31:32,636 AUDIENCE: [INAUDIBLE]. 449 00:31:35,835 --> 00:31:37,501 BARTON ZWIEBACH: Why is this thing here? 450 00:31:37,501 --> 00:31:39,324 AUDIENCE: Yeah, omega sc. 451 00:31:39,324 --> 00:31:40,740 BARTON ZWIEBACH: It's because when 452 00:31:40,740 --> 00:31:43,300 you take the time derivative of this-- 453 00:31:43,300 --> 00:31:47,960 let me give you a formula that you should check. 454 00:31:47,960 --> 00:31:51,590 tA minus tA. 455 00:31:51,590 --> 00:31:59,560 d dt of this is equal to e to the tA, A, B, 456 00:31:59,560 --> 00:32:03,150 e to the minus tA. 457 00:32:07,400 --> 00:32:09,370 That's the formula we used. 458 00:32:09,370 --> 00:32:14,130 It should be all right. 459 00:32:14,130 --> 00:32:16,910 So we have our time dependent states, 460 00:32:16,910 --> 00:32:23,580 and let's analyze it and see what it's doing. 461 00:32:23,580 --> 00:32:24,240 Yes? 462 00:32:24,240 --> 00:32:27,820 One more question. 463 00:32:27,820 --> 00:32:32,040 AUDIENCE: So [INAUDIBLE] that we have these magnetic fields, 464 00:32:32,040 --> 00:32:34,546 but somehow, in our Hamiltonian, it all 465 00:32:34,546 --> 00:32:38,830 boils down to just the spin operating in the x direction. 466 00:32:38,830 --> 00:32:41,210 BARTON ZWIEBACH: Well, they're still in the z direction. 467 00:32:41,210 --> 00:32:42,876 AUDIENCE: There's still the z component, 468 00:32:42,876 --> 00:32:45,509 but the time variant components in the y and x direction-- 469 00:32:45,509 --> 00:32:47,050 BARTON ZWIEBACH: Has become something 470 00:32:47,050 --> 00:32:48,518 like in the x direction. 471 00:32:48,518 --> 00:32:49,017 Right. 472 00:32:52,290 --> 00:32:54,056 AUDIENCE: It's hard to say physically, 473 00:32:54,056 --> 00:32:55,986 but what does the system actually look like? 474 00:32:55,986 --> 00:32:58,360 BARTON ZWIEBACH: That's what we're going to do right now. 475 00:32:58,360 --> 00:33:00,370 AUDIENCE: All right. 476 00:33:00,370 --> 00:33:04,090 BARTON ZWIEBACH: This formula is very pleasant to look at. 477 00:33:04,090 --> 00:33:06,390 You say, oh, I'm very accomplished. 478 00:33:06,390 --> 00:33:08,090 I solved it. 479 00:33:08,090 --> 00:33:09,760 But what does that do? 480 00:33:09,760 --> 00:33:11,600 What does that describe? 481 00:33:11,600 --> 00:33:13,260 That's what we have to do now. 482 00:33:16,948 --> 00:33:20,430 And that's the interesting part, of course. 483 00:33:20,430 --> 00:33:26,580 So applications always have B1 much smaller than B0. 484 00:33:26,580 --> 00:33:32,650 This is the longitudinal component, and that's the case. 485 00:33:32,650 --> 00:33:34,830 Let me describe for you what this 486 00:33:34,830 --> 00:33:43,250 is doing in case one, omega much smaller than omega 0. 487 00:33:43,250 --> 00:33:47,808 You see, given B0, there is an omega 488 00:33:47,808 --> 00:33:52,900 0, the Larmor one over there. 489 00:33:52,900 --> 00:34:00,710 And B0 is very large, so omega 0 is very large as well. 490 00:34:00,710 --> 00:34:03,010 So omega very slow is reasonable. 491 00:34:03,010 --> 00:34:06,880 The magnetic field is rotating much smaller 492 00:34:06,880 --> 00:34:10,310 than whatever rotation this other magnetic field would 493 00:34:10,310 --> 00:34:11,414 create on the things. 494 00:34:14,860 --> 00:34:19,580 Moreover, if omega is much smaller than omega 0, 495 00:34:19,580 --> 00:34:36,199 BR is sort of like B0 z plus B1 x, roughly that. 496 00:34:36,199 --> 00:34:37,760 So what does this do? 497 00:34:37,760 --> 00:34:43,980 Well, this is a magnetic field mostly along the z-axis. 498 00:34:43,980 --> 00:34:46,980 So here is x, y, z. 499 00:34:46,980 --> 00:34:51,949 This is a magnetic field that's like this, BR. 500 00:35:04,050 --> 00:35:12,260 Now, this BR, let's assume also for an assumption that 501 00:35:12,260 --> 00:35:17,530 at t equals 0, the spin of whatever spin state you have 502 00:35:17,530 --> 00:35:23,690 is up plus z. 503 00:35:23,690 --> 00:35:30,370 So what does this magnetic field do to the spin? 504 00:35:30,370 --> 00:35:31,740 Well, you have it here. 505 00:35:35,530 --> 00:35:39,800 So this is going to rotate the spin 506 00:35:39,800 --> 00:35:44,240 states around the axis of BR. 507 00:35:54,380 --> 00:35:59,330 The rotation operator here, this is minus the angular velocity, 508 00:35:59,330 --> 00:36:02,100 so there's a minus sign here that you always 509 00:36:02,100 --> 00:36:03,350 must think about. 510 00:36:03,350 --> 00:36:06,470 But forget about it for a second. 511 00:36:06,470 --> 00:36:10,140 This is rotating around BR, so this, 512 00:36:10,140 --> 00:36:13,640 supposed to have a spin state here in the z direction, 513 00:36:13,640 --> 00:36:17,670 is going to start to rotating in a little cone like that. 514 00:36:17,670 --> 00:36:21,660 It's very close to the BR, so it's just 515 00:36:21,660 --> 00:36:25,820 going to rotate around it like that. 516 00:36:25,820 --> 00:36:28,350 And that's what this part is going to do. 517 00:36:28,350 --> 00:36:31,290 But you say, but that's not the whole thing. 518 00:36:31,290 --> 00:36:34,170 The whole thing is this as well. 519 00:36:34,170 --> 00:36:41,920 Well, intuitively, this rotation that this produces 520 00:36:41,920 --> 00:36:45,680 a rotation around the z-axis, but it's 521 00:36:45,680 --> 00:36:51,910 much slower than this rotation because this BR field, 522 00:36:51,910 --> 00:36:59,050 its magnitude is like B0 roughly. 523 00:36:59,050 --> 00:37:03,620 Therefore, it produces a very fast rotation. 524 00:37:03,620 --> 00:37:07,960 So what you must imagine is this little spin state 525 00:37:07,960 --> 00:37:11,560 generating this cone here and rotating a million times 526 00:37:11,560 --> 00:37:15,390 a second, but this whole thing is still 527 00:37:15,390 --> 00:37:20,530 being rotated by this exponential around the z-axis. 528 00:37:20,530 --> 00:37:23,300 So this whole cone, in some sense, 529 00:37:23,300 --> 00:37:26,500 is precessing around the z-axis. 530 00:37:29,460 --> 00:37:31,870 Now, if you want to know where the spin is 531 00:37:31,870 --> 00:37:35,840 at any instant of time, you do the following thing. 532 00:37:35,840 --> 00:37:38,350 You say, time equals one second. 533 00:37:38,350 --> 00:37:42,340 So you come here and say, one second, 534 00:37:42,340 --> 00:37:44,400 a billion turns and a quarter. 535 00:37:44,400 --> 00:37:46,720 It ends up here. 536 00:37:46,720 --> 00:37:52,000 But in one second, this rotates it by five degrees, 537 00:37:52,000 --> 00:37:55,240 so then it rotates a little more. 538 00:37:55,240 --> 00:37:56,730 You could do it that way. 539 00:37:56,730 --> 00:37:58,680 That's a good way to think about it. 540 00:37:58,680 --> 00:38:02,290 But in the approximation, which is so fast, 541 00:38:02,290 --> 00:38:06,050 basically, this cone has now, by the second exponential, 542 00:38:06,050 --> 00:38:07,418 been rotated like that. 543 00:38:11,410 --> 00:38:14,130 Now let's do the case that is really 544 00:38:14,130 --> 00:38:19,810 the important one, the case, as you could imagine in which 545 00:38:19,810 --> 00:38:26,230 we know some resonance and omega is set equal to omega 0. 546 00:38:30,630 --> 00:38:37,880 So the physicists know what omega 0 is for the spins, 547 00:38:37,880 --> 00:38:41,140 and then they make the radio frequency coincide 548 00:38:41,140 --> 00:38:43,810 with that omega 0. 549 00:38:43,810 --> 00:38:49,760 In that case, you lose completely the z component here 550 00:38:49,760 --> 00:38:53,750 of this BR. 551 00:38:53,750 --> 00:39:02,410 So BR is going to be B1 x. 552 00:39:02,410 --> 00:39:05,030 And then, here is what's going to happen. 553 00:39:07,890 --> 00:39:14,130 You have a B1 x here, a B1 over here in the x direction. 554 00:39:14,130 --> 00:39:18,260 The spin state is sitting at time equals 0 here. 555 00:39:21,670 --> 00:39:25,570 And since normal exponentials that 556 00:39:25,570 --> 00:39:28,890 do time evolutions have a minus sign here, 557 00:39:28,890 --> 00:39:32,910 I claim that instead of rotating the spin around B1, 558 00:39:32,910 --> 00:39:35,830 it's really rotating it around minus B1. 559 00:39:35,830 --> 00:39:41,650 So the spin will just go down here. 560 00:39:41,650 --> 00:39:43,838 Rotate, rotate, rotate, rotate. 561 00:39:48,050 --> 00:39:57,810 Now, it's rotating with B1, which is much smaller than B0. 562 00:39:57,810 --> 00:40:00,610 So it's rotating with an angular frequency 563 00:40:00,610 --> 00:40:05,990 that is much smaller than omega 0, but it's rotating down here. 564 00:40:09,130 --> 00:40:14,935 And now, what's really happening is the following. 565 00:40:14,935 --> 00:40:17,590 Let me draw this if I can. 566 00:40:26,290 --> 00:40:27,800 Here it is. 567 00:40:27,800 --> 00:40:36,450 The spin is up, and it begins to go into the y-axis. 568 00:40:36,450 --> 00:40:37,125 Here is z. 569 00:40:37,125 --> 00:40:41,940 B1 is turning into the y-axis, so it rotates a little. 570 00:40:41,940 --> 00:40:46,880 So let me ask you, what is the second exponential going to do? 571 00:40:46,880 --> 00:40:49,400 It's going to rotate it further, but this time 572 00:40:49,400 --> 00:40:52,600 around the z-axis. 573 00:40:52,600 --> 00:40:58,230 So this one goes down a little bit in some time, 574 00:40:58,230 --> 00:41:01,250 but then this z exponential, the other exponential, 575 00:41:01,250 --> 00:41:03,580 is going to rotate it around the z-axis. 576 00:41:03,580 --> 00:41:07,910 So this is going to go down a little but rotate. 577 00:41:07,910 --> 00:41:09,630 So actually, what's going to happen 578 00:41:09,630 --> 00:41:18,240 is that it's going to do a spiral. 579 00:41:18,240 --> 00:41:20,615 As it begins to go down, the other one rotates. 580 00:41:20,615 --> 00:41:23,290 It goes down a little, the other one rotates it. 581 00:41:23,290 --> 00:41:33,830 In fact, this, since B1 is much smaller than B0 or omega 0, 582 00:41:33,830 --> 00:41:36,890 here now, omega is equal to omega 0, 583 00:41:36,890 --> 00:41:40,930 so this is rotating around the z direction with omega 0. 584 00:41:40,930 --> 00:41:46,910 But B1 is much smaller than B0, so this rotation is very slow. 585 00:41:46,910 --> 00:41:50,730 As it goes down a little, it's rotating very fast 586 00:41:50,730 --> 00:41:56,990 and fills out the spiral until it gets down here. 587 00:41:56,990 --> 00:42:03,880 So that's what the poor spin is doing in this thing. 588 00:42:03,880 --> 00:42:09,840 It's actually of interest to time 589 00:42:09,840 --> 00:42:15,830 the radio frequency signals, to time the systems of B1 590 00:42:15,830 --> 00:42:19,310 so that you get it to go into the plane 591 00:42:19,310 --> 00:42:23,350 so that the spin is maximally perpendicular 592 00:42:23,350 --> 00:42:25,430 to the original direction. 593 00:42:25,430 --> 00:42:29,365 For that, you choose omega 1 times 594 00:42:29,365 --> 00:42:35,450 time, which is the Larmor frequency associated 595 00:42:35,450 --> 00:42:40,390 to B1 times time, to be equal to pi over 2. 596 00:42:40,390 --> 00:42:50,110 And omega 1, just like any omega, is gamma B1 pi over 2. 597 00:42:50,110 --> 00:42:56,160 So t is pi over 2 gamma B1. 598 00:42:56,160 --> 00:42:59,890 It's called the half pulse, something like that. 599 00:43:05,410 --> 00:43:08,380 No, a 90 degree pulse. 600 00:43:08,380 --> 00:43:10,330 That's a normal name for it. 601 00:43:10,330 --> 00:43:13,810 So if you keep the radio frequency 602 00:43:13,810 --> 00:43:19,880 on for this much time, then the spin, at the end of the day, 603 00:43:19,880 --> 00:43:26,250 it got to the equator of the sphere, and B1 has disappeared. 604 00:43:26,250 --> 00:43:29,710 So B1 is gone, BR is gone, and then 605 00:43:29,710 --> 00:43:33,475 it keeps rotating because there is the other magnetic field 606 00:43:33,475 --> 00:43:38,010 still, the longitudinal magnetic field. 607 00:43:38,010 --> 00:43:40,750 One exercise that you still have in the homework 608 00:43:40,750 --> 00:43:45,770 is to figure out the equation of the spiral that comes out 609 00:43:45,770 --> 00:43:46,430 of here. 610 00:43:46,430 --> 00:43:50,160 You will find it sounds complicated, 611 00:43:50,160 --> 00:43:51,690 but it's a couple of lines. 612 00:43:51,690 --> 00:43:54,010 It's very simple, actually, to do it, 613 00:43:54,010 --> 00:43:56,840 after you think about it for a second. 614 00:43:56,840 --> 00:43:59,790 That's one thing you'll have to do. 615 00:43:59,790 --> 00:44:04,740 So this, in fact, is basically the solution of the problem, 616 00:44:04,740 --> 00:44:07,260 and let's just talk for a few minutes 617 00:44:07,260 --> 00:44:10,140 about what it's used for. 618 00:44:10,140 --> 00:44:13,500 So this is basically the technique 619 00:44:13,500 --> 00:44:20,766 that is exactly used for Magnetic Resonance Imaging, 620 00:44:20,766 --> 00:44:21,265 MRIs. 621 00:44:24,590 --> 00:44:28,340 If you have to say one of the interesting applications 622 00:44:28,340 --> 00:44:31,490 of quantum mechanics to technology, 623 00:44:31,490 --> 00:44:35,910 MRIs, Magnetic Resonance Imaging, 624 00:44:35,910 --> 00:44:39,650 is one of the great ones. 625 00:44:39,650 --> 00:44:42,930 So how does it work? 626 00:44:42,930 --> 00:44:46,830 Well, it's a very useful device. 627 00:44:46,830 --> 00:44:50,540 In fact, it revolutionized medicine 628 00:44:50,540 --> 00:44:54,130 because it goes much beyond what you can do with x-rays. 629 00:44:54,130 --> 00:44:55,660 It's very popular. 630 00:44:55,660 --> 00:44:59,790 I imagine a good fraction of you have had an MRI. 631 00:44:59,790 --> 00:45:00,360 Let's see. 632 00:45:00,360 --> 00:45:04,160 How many people have had an MRI in their lives? 633 00:45:04,160 --> 00:45:09,800 We're pretty much, I think, 60% of people here. 634 00:45:09,800 --> 00:45:13,130 So you remember getting into this cavity? 635 00:45:13,130 --> 00:45:16,140 Well, you've experienced there a large-- 636 00:45:16,140 --> 00:45:19,830 if you worry about cell phone radiation, 637 00:45:19,830 --> 00:45:22,790 well, there, you were with two Tesla, 638 00:45:22,790 --> 00:45:27,350 20,000 Gauss, a big, big magnet. 639 00:45:27,350 --> 00:45:33,210 It's big enough that it has to be cooled with liquid helium 640 00:45:33,210 --> 00:45:37,310 at 3 degrees Kelvin so you can get 641 00:45:37,310 --> 00:45:41,810 the currents big enough and the magnetic field big enough. 642 00:45:41,810 --> 00:45:51,980 So they put you into this cylinder, a solenoid, 643 00:45:51,980 --> 00:45:52,855 and there you go. 644 00:45:56,360 --> 00:46:00,840 And it's not dangerous unless you 645 00:46:00,840 --> 00:46:06,200 forget some metal or some device like that. 646 00:46:06,200 --> 00:46:10,710 In fact, it says in WebMD that if you 647 00:46:10,710 --> 00:46:15,030 have some sort of tattoos, they used iron ink, 648 00:46:15,030 --> 00:46:20,550 it can burn your skin or you could have some problems. 649 00:46:20,550 --> 00:46:22,520 Anyway, if you're claustrophobic, 650 00:46:22,520 --> 00:46:26,240 may have MRIs that are open air, but they're 651 00:46:26,240 --> 00:46:29,500 less powerful, less strong magnetic field. 652 00:46:29,500 --> 00:46:33,380 So this is two Tesla, roughly. 653 00:46:33,380 --> 00:46:39,350 And so what happens is that basically, 654 00:46:39,350 --> 00:46:41,910 this thing just is trying to figure out 655 00:46:41,910 --> 00:46:44,480 the local concentration of water. 656 00:46:44,480 --> 00:46:46,800 It begins like that. 657 00:46:46,800 --> 00:46:51,530 The magnetic fields react with the hydrogen atoms, in fact, 658 00:46:51,530 --> 00:46:54,590 with the protons inside there. 659 00:46:54,590 --> 00:46:58,310 Each proton has a magnetic dipole moment. 660 00:46:58,310 --> 00:47:04,310 It gets roughly aligned to the magnetic field, to this B0. 661 00:47:08,760 --> 00:47:13,730 So the protons get aligned to the B0 662 00:47:13,730 --> 00:47:17,890 that is going in here, B0. 663 00:47:17,890 --> 00:47:21,630 So the proton spin is up there. 664 00:47:21,630 --> 00:47:24,500 In fact, because of temperature, not all 665 00:47:24,500 --> 00:47:27,040 of your protons in your body get aligned. 666 00:47:27,040 --> 00:47:32,015 Maybe one in a million does, but that's high enough. 667 00:47:32,015 --> 00:47:36,400 And then they send this 90 degree pulse. 668 00:47:36,400 --> 00:47:39,300 So this thing starts spiraling, and then 669 00:47:39,300 --> 00:47:41,840 it finally goes into this direction 670 00:47:41,840 --> 00:47:45,440 and it rotates, rotates like crazy. 671 00:47:45,440 --> 00:47:51,410 Then, as it rotates like crazy, a rotating dipole moment 672 00:47:51,410 --> 00:47:54,420 is like a generator of electromagnetic waves. 673 00:47:54,420 --> 00:47:57,400 So it generates an electromagnetic wave, 674 00:47:57,400 --> 00:48:03,740 and there are detectors all over that pick up this signal. 675 00:48:03,740 --> 00:48:07,230 The strength of that signal is proportional 676 00:48:07,230 --> 00:48:10,510 to the concentration of water. 677 00:48:10,510 --> 00:48:14,480 So it gives you an idea, not to distinguish 678 00:48:14,480 --> 00:48:20,935 solid matter versus soft matter, but all kinds of liquids. 679 00:48:23,750 --> 00:48:26,860 So it's very, very useful. 680 00:48:26,860 --> 00:48:30,480 You pick a signal over here and you get it from a receiver, 681 00:48:30,480 --> 00:48:34,070 and it tells you about the local concentration of water. 682 00:48:34,070 --> 00:48:38,560 And therefore, it allows you to distinguish different tissues. 683 00:48:38,560 --> 00:48:42,160 Some tissues have lots of water, some tissues have less water, 684 00:48:42,160 --> 00:48:44,535 so it begins to distinguish different tissues. 685 00:48:47,090 --> 00:48:50,480 Now, there's two more things that people do. 686 00:48:50,480 --> 00:48:58,270 This pin is rotating very fast, but there's a relaxation time. 687 00:48:58,270 --> 00:49:02,910 After it rotates for a while, there's 688 00:49:02,910 --> 00:49:12,170 a time, T2, is relaxation of this rotation. 689 00:49:12,170 --> 00:49:13,990 Relaxation time. 690 00:49:13,990 --> 00:49:16,600 And then there's another time, T1, 691 00:49:16,600 --> 00:49:20,610 which is the time after this is turned off 692 00:49:20,610 --> 00:49:25,180 that it takes the spin to go back 693 00:49:25,180 --> 00:49:26,810 again to the original position. 694 00:49:29,830 --> 00:49:33,480 So two times, the relaxation time 695 00:49:33,480 --> 00:49:37,340 in which you lose your rotation here 696 00:49:37,340 --> 00:49:40,380 because it's interacting with other spins. 697 00:49:40,380 --> 00:49:42,850 It's called spin, spin, relaxation. 698 00:49:42,850 --> 00:49:48,180 And then an interaction with a whole set of neighboring atoms 699 00:49:48,180 --> 00:49:50,740 that brings it eventually back up 700 00:49:50,740 --> 00:49:53,440 and aligns it to the magnetic field. 701 00:49:53,440 --> 00:49:57,840 So you measure two things, T1 and T2, 702 00:49:57,840 --> 00:50:01,260 and those are very good clues because you 703 00:50:01,260 --> 00:50:05,000 can put any liquid in the machine 704 00:50:05,000 --> 00:50:07,870 and measure its T1 and T2. 705 00:50:07,870 --> 00:50:11,430 And then you place a table of T1's and T2's, and then 706 00:50:11,430 --> 00:50:13,720 if you want to figure out what kind of thing 707 00:50:13,720 --> 00:50:16,550 you have in your body, they look it up, 708 00:50:16,550 --> 00:50:18,730 and immediately, they may know what 709 00:50:18,730 --> 00:50:20,240 are the possible candidates. 710 00:50:20,240 --> 00:50:27,490 In fact, T2 is actually enough to distinguish 711 00:50:27,490 --> 00:50:31,550 white matter, gray matter, and fluids in your brain. 712 00:50:31,550 --> 00:50:34,940 Totally different T2's. 713 00:50:34,940 --> 00:50:41,640 T1 is helpful to discuss all kinds of things as well. 714 00:50:41,640 --> 00:50:44,760 So basically, people have figured out how to do that. 715 00:50:44,760 --> 00:50:46,570 Finally, one last thing. 716 00:50:46,570 --> 00:50:51,450 When you go into the machine, it sometimes makes noises. 717 00:50:51,450 --> 00:50:53,040 Big noises. 718 00:50:53,040 --> 00:50:57,530 And those are gradient magnets that are being moved. 719 00:50:57,530 --> 00:51:01,320 You see, basically, if this thing would be like that, 720 00:51:01,320 --> 00:51:02,940 you would pick up a signal and you 721 00:51:02,940 --> 00:51:05,420 wouldn't know where it comes from. 722 00:51:05,420 --> 00:51:07,690 So what these people do now-- this 723 00:51:07,690 --> 00:51:09,740 has gotten extremely sophisticated. 724 00:51:09,740 --> 00:51:15,890 They put a gradient magnet which changes the value of B0 locally 725 00:51:15,890 --> 00:51:17,550 by increasing it. 726 00:51:17,550 --> 00:51:20,540 For example, the B0 is not really constant in z, 727 00:51:20,540 --> 00:51:22,270 but it changes. 728 00:51:22,270 --> 00:51:28,390 So the omega 0 of rotation of the spins changes as well. 729 00:51:28,390 --> 00:51:32,650 So with a sufficiently high gradient of magnetic field, 730 00:51:32,650 --> 00:51:34,660 they can have spatial resolution. 731 00:51:34,660 --> 00:51:37,370 If they pick a signal a little bit higher frequency, 732 00:51:37,370 --> 00:51:40,550 they know it's a little bit higher up in your body. 733 00:51:40,550 --> 00:51:43,470 So they get resolutions with these magnets 734 00:51:43,470 --> 00:51:46,390 of about how much? 735 00:51:46,390 --> 00:51:48,350 Very high resolutions. 736 00:51:48,350 --> 00:51:51,290 Half a millimeter. 737 00:51:51,290 --> 00:51:56,780 So at the end of the day, it's a great machine, 738 00:51:56,780 --> 00:52:00,980 and lots of mathematics in reconstructing images, 739 00:52:00,980 --> 00:52:05,480 lots of computation, lots of experimental analysis 740 00:52:05,480 --> 00:52:08,530 of constants, much of it phenomenological. 741 00:52:08,530 --> 00:52:11,930 It would be hard to predict those constants, 742 00:52:11,930 --> 00:52:14,260 but you can measure them. 743 00:52:14,260 --> 00:52:19,090 So it's very, very practical and very nice. 744 00:52:19,090 --> 00:52:26,040 So it's a nice application, and we'll leave it at that. 745 00:52:26,040 --> 00:52:28,370 Are there any questions? 746 00:52:28,370 --> 00:52:32,840 Not that I know much more about that. 747 00:52:32,840 --> 00:52:36,745 But it's a junior lab experiment as well. 748 00:52:36,745 --> 00:52:40,520 So I believe the calculation is fairly straightforward 749 00:52:40,520 --> 00:52:42,980 and the technology is just amazing. 750 00:52:42,980 --> 00:52:44,010 It's great. 751 00:52:44,010 --> 00:52:45,110 Yes? 752 00:52:45,110 --> 00:52:47,200 AUDIENCE: So about getting your spin back up 753 00:52:47,200 --> 00:52:49,550 along the z direction, is that mechanism 754 00:52:49,550 --> 00:52:52,079 the same as when you relax [INAUDIBLE]? 755 00:52:52,079 --> 00:52:53,620 BARTON ZWIEBACH: When you relax what? 756 00:52:53,620 --> 00:52:57,370 AUDIENCE: So is the mechanism for getting your spin back 757 00:52:57,370 --> 00:52:59,620 along the z direction to find T1, 758 00:52:59,620 --> 00:53:02,062 is that the same mechanism as when 759 00:53:02,062 --> 00:53:04,970 you do all of this [INAUDIBLE]? 760 00:53:04,970 --> 00:53:06,130 BARTON ZWIEBACH: No. 761 00:53:06,130 --> 00:53:08,670 It's sort of a different mechanism. 762 00:53:08,670 --> 00:53:09,930 We don't analyze it. 763 00:53:09,930 --> 00:53:17,040 It's not being driven by Schrodinger time evolution. 764 00:53:17,040 --> 00:53:20,510 All these relaxations are more complicated phenomena. 765 00:53:20,510 --> 00:53:23,380 In fact, if you maintain the magnetic field, 766 00:53:23,380 --> 00:53:26,270 this is supposed to go on forever and never stop 767 00:53:26,270 --> 00:53:27,180 rotating. 768 00:53:27,180 --> 00:53:30,700 But due to interactions with other things, 769 00:53:30,700 --> 00:53:34,400 it's sort of stops rotating at some stage, 770 00:53:34,400 --> 00:53:38,070 and then eventually, many of them also go back up. 771 00:53:38,070 --> 00:53:41,170 So these are not easily calculable things 772 00:53:41,170 --> 00:53:42,700 within our analysis. 773 00:53:42,700 --> 00:53:45,970 These are phenomenological constants 774 00:53:45,970 --> 00:53:52,310 that need to be measured or done by experiment. 775 00:53:52,310 --> 00:53:56,890 So last part of today's lecture is multi-particle states 776 00:53:56,890 --> 00:53:59,520 and tensor products. 777 00:53:59,520 --> 00:54:03,980 Let's get on with that. 778 00:54:03,980 --> 00:54:07,600 Let's see we have there. 779 00:54:07,600 --> 00:54:17,230 So multi-particle states and tensor products. 780 00:54:30,030 --> 00:54:31,440 The idea is the following. 781 00:54:31,440 --> 00:54:33,980 You have a system with more than one particle. 782 00:54:33,980 --> 00:54:36,660 Let's talk two particles. 783 00:54:36,660 --> 00:54:40,030 It won't matter at this moment whether they're 784 00:54:40,030 --> 00:54:41,980 distinguishable, not distinguishable. 785 00:54:41,980 --> 00:54:43,810 Those are things that come later, 786 00:54:43,810 --> 00:54:48,320 and we will probably not discuss much of that in this semester. 787 00:54:48,320 --> 00:54:50,880 This will be more in 806. 788 00:54:50,880 --> 00:54:56,110 But let's consider if we have particle one, 789 00:54:56,110 --> 00:54:58,100 and we'll keep the possibility that this 790 00:54:58,100 --> 00:55:00,230 is completely distinguishable, in fact. 791 00:55:00,230 --> 00:55:03,010 Particle one. 792 00:55:03,010 --> 00:55:08,810 Its quantum mechanics is described by a complex vector 793 00:55:08,810 --> 00:55:22,165 space, v. And you have some operators, T1, T2. 794 00:55:25,120 --> 00:55:31,850 Particle two, complex vector space, 795 00:55:31,850 --> 00:55:42,300 w, and the operators, S1, S2, all that. 796 00:55:42,300 --> 00:55:45,340 You see, the list of operators is something 797 00:55:45,340 --> 00:55:48,050 that you are already familiar with. 798 00:55:48,050 --> 00:55:52,930 Quantum mechanics operators can include momentum, position, 799 00:55:52,930 --> 00:55:57,380 if it's three dimensional, three positions, three momenta, 800 00:55:57,380 --> 00:56:04,560 angular momentum, spin, Hamiltonians, all those things, 801 00:56:04,560 --> 00:56:08,860 and those exist for both particles. 802 00:56:08,860 --> 00:56:13,130 So the question is, how do we describe the composite system, 803 00:56:13,130 --> 00:56:17,200 the system of the two particles that 804 00:56:17,200 --> 00:56:21,720 exist at the same time, that possibly could interact even 805 00:56:21,720 --> 00:56:23,780 with each other? 806 00:56:23,780 --> 00:56:27,570 So we need a description of these two things. 807 00:56:27,570 --> 00:56:32,710 Well, a description of particle one 808 00:56:32,710 --> 00:56:37,145 is described by some vector, v, in the vector space. 809 00:56:39,950 --> 00:56:45,350 Description of particle two is some state, some vector omega, 810 00:56:45,350 --> 00:56:49,330 in the vector space w. 811 00:56:49,330 --> 00:56:55,820 So we imagine that it's a reasonable thing 812 00:56:55,820 --> 00:57:00,200 to give you those two vectors and say, look, 813 00:57:00,200 --> 00:57:02,360 that's what particle one is doing, 814 00:57:02,360 --> 00:57:06,370 that's what particle two is doing, and that is correct. 815 00:57:06,370 --> 00:57:09,330 It's a bit more subtle than that, as we'll see now, 816 00:57:09,330 --> 00:57:17,360 but we could list v and list w, and this 817 00:57:17,360 --> 00:57:25,345 is the information about each particle. 818 00:57:31,970 --> 00:57:38,690 And this is, in some ways, too naive, as we will see. 819 00:57:38,690 --> 00:57:44,400 It has the right idea, but the amazing possibilities 820 00:57:44,400 --> 00:57:47,280 that can take place when you have these two particles 821 00:57:47,280 --> 00:57:53,340 are not quite reflected on that thing yet. 822 00:57:53,340 --> 00:58:04,620 So to make this idea clearer, we'll use a notation. 823 00:58:04,620 --> 00:58:10,650 So I will say that I will encode these things 824 00:58:10,650 --> 00:58:21,940 and I will write them as v tensor product w. 825 00:58:27,190 --> 00:58:30,640 It's reflecting that we're going to do something more than just 826 00:58:30,640 --> 00:58:32,920 view this as the possibility. 827 00:58:32,920 --> 00:58:33,670 You have a system. 828 00:58:33,670 --> 00:58:35,510 You know what the particle one is doing, 829 00:58:35,510 --> 00:58:38,830 you know what the second particle is doing. 830 00:58:38,830 --> 00:58:41,890 List those two, that's all that can happen. 831 00:58:41,890 --> 00:58:43,840 Not quite true. 832 00:58:43,840 --> 00:58:48,630 So let's put it like that, and this will be the information. 833 00:58:48,630 --> 00:58:50,680 BARTON ZWIEBACH: I'm not multiplying 834 00:58:50,680 --> 00:58:52,870 this in any obvious way. 835 00:58:52,870 --> 00:58:55,270 It's not that I'm supposed to multiply and do 836 00:58:55,270 --> 00:58:56,910 a calculation here. 837 00:58:56,910 --> 00:58:59,030 I'm just putting the two pieces of data 838 00:58:59,030 --> 00:59:02,630 but putting this curly ball here as 839 00:59:02,630 --> 00:59:05,570 to say that it's the two together, 840 00:59:05,570 --> 00:59:09,810 and we'll have some ways of dealing with this object. 841 00:59:09,810 --> 00:59:16,760 And this will be said to be an element 842 00:59:16,760 --> 00:59:28,526 of a new, complex vector space, v tensor w. 843 00:59:36,550 --> 00:59:48,550 So what we are saying here, with v belonging to v and w 844 00:59:48,550 --> 00:59:54,560 belonging to w, this thing belongs to v plus w. 845 00:59:58,200 --> 01:00:00,440 Tensor product of the vector spaces. 846 01:00:05,610 --> 01:00:08,270 At this moment, this is a little funny, 847 01:00:08,270 --> 01:00:16,130 but let me ask you some things about it. 848 01:00:16,130 --> 01:00:21,734 We have v w. 849 01:00:21,734 --> 01:00:24,850 So this is an element of that space. 850 01:00:24,850 --> 01:00:27,300 I put a vector of the first vector space here 851 01:00:27,300 --> 01:00:29,880 and I put a vector of the second vector space, 852 01:00:29,880 --> 01:00:32,440 and this will be a vector here. 853 01:00:32,440 --> 01:00:35,090 I'm not saying it's the most general vector there, 854 01:00:35,090 --> 01:00:39,690 or how we operate with them, but it's a vector there. 855 01:00:39,690 --> 01:00:43,000 Now, you say, look, when I had states, 856 01:00:43,000 --> 01:00:48,350 you had states plus minus, you put constants in front of them. 857 01:00:48,350 --> 01:00:54,825 So I'll put a constant in front of this v, alpha. 858 01:01:02,730 --> 01:01:08,710 Well, is this something related? 859 01:01:08,710 --> 01:01:13,370 We had this vector that was a vector in the tensor product. 860 01:01:13,370 --> 01:01:16,050 How is this vector related to this vector? 861 01:01:24,680 --> 01:01:37,060 Well, unless you declare that these things are somewhat 862 01:01:37,060 --> 01:01:41,520 related, this object is going to be very, very large 863 01:01:41,520 --> 01:01:44,940 because if this vector has nothing to do with this, 864 01:01:44,940 --> 01:01:46,800 this is going to be yet another vector 865 01:01:46,800 --> 01:01:48,610 linearly independent with this. 866 01:01:48,610 --> 01:01:52,820 So it's, again, your choice what you're going to declare. 867 01:01:52,820 --> 01:01:56,480 If you will be constructing what is called a direct product, 868 01:01:56,480 --> 01:02:00,280 not a tensor product, and that's apologies 869 01:02:00,280 --> 01:02:04,400 to [? Shanker. ?] He uses the word "direct products" wrong 870 01:02:04,400 --> 01:02:05,380 mathematically. 871 01:02:05,380 --> 01:02:08,690 Mathematicians don't call this a direct product. 872 01:02:08,690 --> 01:02:10,470 Direct product is something in which 873 01:02:10,470 --> 01:02:12,930 this has nothing to do with this. 874 01:02:12,930 --> 01:02:16,340 But in physics, you're thinking of the amplitude 875 01:02:16,340 --> 01:02:20,470 to have a first particle doing that, and suddenly, 876 01:02:20,470 --> 01:02:24,280 that amplitude becomes twice as big and this other one 877 01:02:24,280 --> 01:02:25,010 doesn't change. 878 01:02:25,010 --> 01:02:26,870 Well, the amplitude defining this one here 879 01:02:26,870 --> 01:02:29,850 and that one there has become twice as big 880 01:02:29,850 --> 01:02:32,930 just because this one is twice more likely to be here 881 01:02:32,930 --> 01:02:34,250 and that's the same. 882 01:02:34,250 --> 01:02:38,230 So we're going to say that this is the same as that. 883 01:02:38,230 --> 01:02:42,460 The alpha can go out and it's the same thing, 884 01:02:42,460 --> 01:02:47,740 and it's also the same thing as v tensor alpha w. 885 01:02:50,400 --> 01:02:55,390 So numbers go out, and they don't go out 886 01:02:55,390 --> 01:02:57,270 with a complex conjugate. 887 01:02:57,270 --> 01:02:59,345 They go out just like that. 888 01:03:03,180 --> 01:03:05,760 So that's one thing we declare that 889 01:03:05,760 --> 01:03:07,770 will happen with these things, and that's 890 01:03:07,770 --> 01:03:11,010 a property that we impose. 891 01:03:11,010 --> 01:03:13,990 It's what is usually understood to take place 892 01:03:13,990 --> 01:03:15,050 in tensor products. 893 01:03:20,280 --> 01:03:27,580 Now, if this is a vector in this space, 894 01:03:27,580 --> 01:03:37,260 then if, for example, v1 is a vector 895 01:03:37,260 --> 01:03:43,650 and v2 cross w2 is a vector in v cross w, 896 01:03:43,650 --> 01:03:47,250 then this is supposed to be a linear vector space. 897 01:03:47,250 --> 01:03:59,610 So it should be true that alpha v1 omega 1 plus beta v2 omega 2 898 01:03:59,610 --> 01:04:03,520 also belongs to this vector space. 899 01:04:03,520 --> 01:04:08,430 And suddenly, you see why this was a little too naive. 900 01:04:08,430 --> 01:04:11,810 If you tell me, this is what particle one is doing 901 01:04:11,810 --> 01:04:14,770 and this is what particle is doing, end of story, 902 01:04:14,770 --> 01:04:18,010 this list, well, quantum mechanics 903 01:04:18,010 --> 01:04:21,430 seems to say that there's more interesting possibilities 904 01:04:21,430 --> 01:04:25,280 in which the state is a superposition now 905 01:04:25,280 --> 01:04:29,020 in which first particle is doing this, which 906 01:04:29,020 --> 01:04:31,450 may be a superposition itself, and this one 907 01:04:31,450 --> 01:04:36,090 is doing this, plus first particle is doing that 908 01:04:36,090 --> 01:04:39,510 and second particle is doing that. 909 01:04:39,510 --> 01:04:42,940 So it's not enough to say to the state of two particles, 910 01:04:42,940 --> 01:04:46,280 say, you just need to know the state of one, 911 01:04:46,280 --> 01:04:49,190 state of the other, list the two. 912 01:04:49,190 --> 01:04:51,895 That is a possible state of the system, 913 01:04:51,895 --> 01:04:54,180 but it's not the most general. 914 01:04:54,180 --> 01:04:56,520 The most general is a superposition 915 01:04:56,520 --> 01:04:58,750 in which particle one does something, 916 01:04:58,750 --> 01:05:02,340 particle two does something, plus another possibility, 917 01:05:02,340 --> 01:05:04,500 particle one is doing something else, 918 01:05:04,500 --> 01:05:06,240 particle two is doing another thing. 919 01:05:08,770 --> 01:05:14,180 This is the origin of this famous idea of entanglement, 920 01:05:14,180 --> 01:05:16,490 as we will see you next time. 921 01:05:16,490 --> 01:05:19,120 But let's just continue here. 922 01:05:19,120 --> 01:05:23,020 These particles, roughly speaking, 923 01:05:23,020 --> 01:05:26,110 seem to be entangled, in which you 924 01:05:26,110 --> 01:05:29,660 can't say what particle one is doing without knowing what 925 01:05:29,660 --> 01:05:32,155 particle two is doing and things like that. 926 01:05:34,720 --> 01:05:41,330 So something new has happened by taking this idea slowly 927 01:05:41,330 --> 01:05:43,360 and figuring out what's going on, 928 01:05:43,360 --> 01:05:47,600 but we need yet one more thing to cut down 929 01:05:47,600 --> 01:05:48,610 this tensor product. 930 01:05:48,610 --> 01:05:51,650 This tensor product is still a little bigger 931 01:05:51,650 --> 01:05:57,920 than what you may want it to be in the following sense. 932 01:05:57,920 --> 01:06:01,310 Suppose you have v1 plus v2, which 933 01:06:01,310 --> 01:06:06,875 is another vector, tensor w, another vector. 934 01:06:09,590 --> 01:06:12,330 Unless you tell me what this is, I 935 01:06:12,330 --> 01:06:22,420 don't know that this is actually v1 plus v2 unless you 936 01:06:22,420 --> 01:06:23,520 declare it. 937 01:06:23,520 --> 01:06:27,150 This is just another vector here and that. 938 01:06:27,150 --> 01:06:30,810 Why should that be equal to this? 939 01:06:30,810 --> 01:06:34,520 Well, it's the way you feel about quantum mechanics. 940 01:06:34,520 --> 01:06:37,660 That's how you should think of the composite system. 941 01:06:37,660 --> 01:06:40,970 If the first state can be either of two possibilities 942 01:06:40,970 --> 01:06:43,690 while the other one is not, well it's 943 01:06:43,690 --> 01:06:52,500 a superposition in which, yes, the first state is doing this 944 01:06:52,500 --> 01:06:54,410 and the first state is doing something 945 01:06:54,410 --> 01:06:57,340 different with the second doing the same. 946 01:06:57,340 --> 01:07:02,680 So this is part of also an axiom of the tensor product. 947 01:07:02,680 --> 01:07:04,510 Whenever you have tensor products, 948 01:07:04,510 --> 01:07:07,170 you will simplify in that way. 949 01:07:07,170 --> 01:07:10,430 It looks almost trivial, but it's certainly 950 01:07:10,430 --> 01:07:12,410 something that has to be stated. 951 01:07:12,410 --> 01:07:16,190 If you had a direct product, you form a vector space 952 01:07:16,190 --> 01:07:19,540 by just putting first vector, second vector, 953 01:07:19,540 --> 01:07:21,260 and you don't explain anything. 954 01:07:21,260 --> 01:07:23,210 This would not be true. 955 01:07:23,210 --> 01:07:36,740 So similarly, we'll also have v1 w1 plus w2 equals v1 w1 plus v1 956 01:07:36,740 --> 01:07:39,320 w2. 957 01:07:39,320 --> 01:07:44,910 So these are our last operations. 958 01:07:44,910 --> 01:07:49,190 So with these two operations, we've 959 01:07:49,190 --> 01:07:55,390 defined completely how to calculate with this two 960 01:07:55,390 --> 01:08:01,920 particle Hilbert space, and how to define the states. 961 01:08:01,920 --> 01:08:09,170 Let me add some intuition to this by stating the following. 962 01:08:09,170 --> 01:08:16,390 So what is this space, v tensor w, 963 01:08:16,390 --> 01:08:19,944 is the space spanned by things like this. 964 01:08:23,779 --> 01:08:29,560 So with these rules, here comes the thing 965 01:08:29,560 --> 01:08:33,319 that puts it all, in a sense, together. 966 01:08:33,319 --> 01:08:55,819 If e1 up to en is a basis for v, and f1 up to fm 967 01:08:55,819 --> 01:09:09,399 is a basis for w, this set of things, ei tensor fj, 968 01:09:09,399 --> 01:09:11,649 and how many of them are there? 969 01:09:11,649 --> 01:09:15,500 There are n of this and m of those. 970 01:09:15,500 --> 01:09:23,270 This for i from 1 up to n, j from 1 up to m, 971 01:09:23,270 --> 01:09:31,594 form a basis for v tensor w. 972 01:09:34,229 --> 01:09:39,880 So basically, if you have a four dimensional vector space 973 01:09:39,880 --> 01:09:43,646 v and a 10 dimensional vector space w, 974 01:09:43,646 --> 01:09:47,490 v cross w is 40 dimensional. 975 01:09:47,490 --> 01:09:50,240 You multiply the dimensions. 976 01:09:50,240 --> 01:09:52,310 You don't sum them. 977 01:09:52,310 --> 01:09:56,810 In the tensor product, the dimensions multiply. 978 01:09:56,810 --> 01:09:59,290 You see, the elements of this tensor product 979 01:09:59,290 --> 01:10:01,760 was one vector here, one vector there. 980 01:10:01,760 --> 01:10:05,740 So it behooves you that you would expect that, well, you 981 01:10:05,740 --> 01:10:09,480 can get all vectors that can sit here with all the ei's, all 982 01:10:09,480 --> 01:10:11,510 the vectors here, but only if you 983 01:10:11,510 --> 01:10:15,520 have these rules of superposition and linearity. 984 01:10:15,520 --> 01:10:20,350 So this is the whole basis of the vector space. 985 01:10:20,350 --> 01:10:27,240 And dimension of v cross w would be 986 01:10:27,240 --> 01:10:33,210 equal to the dimension of v times the dimension of w. 987 01:10:43,800 --> 01:10:44,390 Questions? 988 01:10:44,390 --> 01:10:45,194 Yes? 989 01:10:45,194 --> 01:10:48,644 AUDIENCE: So for the first box thing up there, 990 01:10:48,644 --> 01:10:53,711 so v and w are two different vector spaces. 991 01:10:53,711 --> 01:10:56,169 Let's say that they're vector spaces over different fields. 992 01:10:56,169 --> 01:10:57,111 How do you know that-- 993 01:10:57,111 --> 01:10:57,590 BARTON ZWIEBACH: Oh no. 994 01:10:57,590 --> 01:10:58,465 Different fields, no. 995 01:10:58,465 --> 01:11:00,390 Both are over complex numbers. 996 01:11:00,390 --> 01:11:01,376 AUDIENCE: Oh, OK. 997 01:11:01,376 --> 01:11:03,629 They're both complex. 998 01:11:03,629 --> 01:11:05,920 BARTON ZWIEBACH: Both spaces are complex vector spaces. 999 01:11:05,920 --> 01:11:07,110 Did I say it? 1000 01:11:07,110 --> 01:11:09,830 Complex vector space, complex vector space. 1001 01:11:09,830 --> 01:11:20,460 Then v cross w is an element of a new complex vector space. 1002 01:11:20,460 --> 01:11:22,380 So yes, everybody's complex here. 1003 01:11:28,550 --> 01:11:31,740 Let me continue a little more because we 1004 01:11:31,740 --> 01:11:34,400 need to get a little further. 1005 01:11:34,400 --> 01:11:36,930 I would like to get a little further here today. 1006 01:11:36,930 --> 01:11:41,020 We still have about 10 minutes. 1007 01:11:41,020 --> 01:11:44,630 These are really, in some ways simple 1008 01:11:44,630 --> 01:11:47,870 but in some ways very subtle ideas. 1009 01:11:47,870 --> 01:11:49,080 I hope you appreciate it. 1010 01:11:49,080 --> 01:11:51,040 At some moments, you think, this is obvious. 1011 01:11:51,040 --> 01:11:52,030 It's ridiculous. 1012 01:11:52,030 --> 01:11:53,820 It's taking too long to explain it. 1013 01:11:53,820 --> 01:11:57,790 At some point, you step back and say, I'm now confused. 1014 01:11:57,790 --> 01:12:00,630 What does that mean? 1015 01:12:00,630 --> 01:12:03,800 It is subtle. 1016 01:12:03,800 --> 01:12:06,170 So let's see now what happens. 1017 01:12:06,170 --> 01:12:13,430 We need to do operators on v cross w. 1018 01:12:13,430 --> 01:12:22,450 So suppose T is an operator in v, and s is an operator in w. 1019 01:12:25,880 --> 01:12:29,990 So we call them like that, T operators and the S operator 1020 01:12:29,990 --> 01:12:30,500 on w. 1021 01:12:30,500 --> 01:12:31,730 So here it is. 1022 01:12:31,730 --> 01:12:37,500 We define an s tensor T that will 1023 01:12:37,500 --> 01:12:43,970 be an operator on the tensor product, v cross w. 1024 01:12:43,970 --> 01:12:45,720 Let's see how it is. 1025 01:12:45,720 --> 01:12:50,120 So if this is an operator on the tensor product, 1026 01:12:50,120 --> 01:12:55,800 it should know how to act on v any element. 1027 01:12:55,800 --> 01:12:59,750 In any basic element or element of this form, 1028 01:12:59,750 --> 01:13:02,970 since all the elements can be written as superposition 1029 01:13:02,970 --> 01:13:05,660 of these things, if this is a linear operator 1030 01:13:05,660 --> 01:13:09,110 and I define it acting on this, we've got it. 1031 01:13:09,110 --> 01:13:13,120 Now, what I'm going to write in the right hand side 1032 01:13:13,120 --> 01:13:15,440 is a matter of common sense. 1033 01:13:15,440 --> 01:13:20,030 It's a definition, but it's hardly possible 1034 01:13:20,030 --> 01:13:22,890 to have any other definition than what I'm going to do. 1035 01:13:29,860 --> 01:13:31,480 Lights. 1036 01:13:31,480 --> 01:13:32,975 Main screen, window shades. 1037 01:13:38,836 --> 01:13:40,195 Oh, lights here. 1038 01:13:40,195 --> 01:13:42,010 There we go. 1039 01:13:42,010 --> 01:13:56,280 Well, you know, I got the order wrong here, T cross S. 1040 01:13:56,280 --> 01:13:57,970 And the only thing you could do is 1041 01:13:57,970 --> 01:14:04,458 if T, you know how it acts on v. Well, let it act on v. 1042 01:14:04,458 --> 01:14:10,085 And it will be a vector in v, and you know how S acts 1043 01:14:10,085 --> 01:14:13,560 on omega, so let it act on omega, 1044 01:14:13,560 --> 01:14:16,630 and that's the definition. 1045 01:14:16,630 --> 01:14:21,140 Not that complicated, really, in a sense. 1046 01:14:21,140 --> 01:14:26,840 Each operator acts on its thing, so you just let it act. 1047 01:14:26,840 --> 01:14:32,400 Whoever has to act, acts wherever it can do it. 1048 01:14:32,400 --> 01:14:38,640 And it's linear operator, and if you act on more things, 1049 01:14:38,640 --> 01:14:39,860 it will be linear. 1050 01:14:39,860 --> 01:14:42,620 It will all be fine here. 1051 01:14:42,620 --> 01:14:45,310 Now, this is a fine operator. 1052 01:14:45,310 --> 01:14:50,050 Now, the operators that are a little more surprising, 1053 01:14:50,050 --> 01:14:53,090 perhaps, at first sight are the following. 1054 01:14:57,960 --> 01:15:09,105 Suppose you have an operator, T1, that is an operator on v, 1055 01:15:09,105 --> 01:15:13,580 and you want it to act on the tensor product. 1056 01:15:13,580 --> 01:15:16,350 You say, well, I need another operator. 1057 01:15:16,350 --> 01:15:20,070 This acts on v. I need another operator attached on the tensor 1058 01:15:20,070 --> 01:15:26,430 product, but you don't give me one, so what am I to do? 1059 01:15:26,430 --> 01:15:29,850 So this is what is called upgrading an operator that 1060 01:15:29,850 --> 01:15:32,800 acts on one vector space to an operator that 1061 01:15:32,800 --> 01:15:35,940 acts on the whole thing because you need everything 1062 01:15:35,940 --> 01:15:37,840 to act now on the tensor product. 1063 01:15:37,840 --> 01:15:42,060 So what you do is the following. 1064 01:15:42,060 --> 01:15:48,700 You let the operator become T1 that acts on v, 1065 01:15:48,700 --> 01:15:52,740 and you put tensor product with the identity operator. 1066 01:15:55,280 --> 01:15:57,360 This is more or less what you would imagine. 1067 01:15:57,360 --> 01:16:03,570 You have now an operator that belongs to the linear operators 1068 01:16:03,570 --> 01:16:05,680 on the tensor product, but it's the operator 1069 01:16:05,680 --> 01:16:07,680 you got times that. 1070 01:16:07,680 --> 01:16:14,740 If you had an operator, S1, belonging to L of w, 1071 01:16:14,740 --> 01:16:18,560 it would go into 1 times S1. 1072 01:16:25,600 --> 01:16:29,494 Now I ask you the question, do these operators commute? 1073 01:16:32,672 --> 01:16:35,450 Strange question, but it's a fundamental question. 1074 01:16:35,450 --> 01:16:39,470 It's really the basis of most of your intuition about two 1075 01:16:39,470 --> 01:16:42,820 particle systems. 1076 01:16:42,820 --> 01:16:44,750 Well, let's see if they commute. 1077 01:16:44,750 --> 01:16:54,230 You have T1 tensor 1 multiplied by 1 tensor S1. 1078 01:16:54,230 --> 01:16:59,500 It's going to act on v tensor w. 1079 01:16:59,500 --> 01:17:08,750 Or you have it in the other order, 1 S1 T1 1. 1080 01:17:14,560 --> 01:17:22,020 Well, when it acts, the first part, it gives me T1 tensor 1, 1081 01:17:22,020 --> 01:17:26,747 but now acting on this, this is v tensor Sw. 1082 01:17:29,870 --> 01:17:31,710 And then when I act on this one, I 1083 01:17:31,710 --> 01:17:39,285 get now T1 on v tensor S1 of w. 1084 01:17:42,430 --> 01:17:48,670 And when I act here, the first step, it gives me a T1 on v, 1085 01:17:48,670 --> 01:17:51,310 and the second gives me the S1 on w, 1086 01:17:51,310 --> 01:17:53,090 so it gives me the same thing. 1087 01:17:53,090 --> 01:17:57,830 T1 on v, S1 on w. 1088 01:17:57,830 --> 01:18:02,680 So these two operators, because they originated, 1089 01:18:02,680 --> 01:18:05,750 they've operators of the first particle, operator 1090 01:18:05,750 --> 01:18:11,000 of the second particle, they can act on the whole system, 1091 01:18:11,000 --> 01:18:12,850 but they still commute. 1092 01:18:12,850 --> 01:18:15,520 They don't know about each other, 1093 01:18:15,520 --> 01:18:18,390 and the communication now is the calculation 1094 01:18:18,390 --> 01:18:21,660 which you just have seen. 1095 01:18:21,660 --> 01:18:24,350 What this is a good example of this thing 1096 01:18:24,350 --> 01:18:32,120 is that when you try to write the Hamiltonian 1097 01:18:32,120 --> 01:18:36,810 of the whole system, H total, you would say, 1098 01:18:36,810 --> 01:18:40,090 oh, it's the Hamiltonian of the first system, 1099 01:18:40,090 --> 01:18:46,340 tensor 1, plus 1 tensor the Hamiltonian 1100 01:18:46,340 --> 01:18:47,590 of the second system. 1101 01:18:53,350 --> 01:18:55,880 Time for an example. 1102 01:18:55,880 --> 01:19:03,490 So the example is a famous one, and this is a great example 1103 01:19:03,490 --> 01:19:08,250 because it's at the basis of combining angular momentum. 1104 01:19:08,250 --> 01:19:13,100 So it's an example you're going to see now and see many times. 1105 01:19:13,100 --> 01:19:14,715 Two spin one 1/2 particles. 1106 01:19:25,530 --> 01:19:31,130 The first partoc;e has a state plus and a state minus 1107 01:19:31,130 --> 01:19:33,910 for the first article. 1108 01:19:33,910 --> 01:19:38,805 The second particle has a state plus and a state minus. 1109 01:19:41,930 --> 01:19:46,050 So how do we form the tensor product? 1110 01:19:46,050 --> 01:19:48,300 Well, we say, these are our basis 1111 01:19:48,300 --> 01:19:50,340 states for the first Hilbert space, 1112 01:19:50,340 --> 01:19:52,800 the basis states for the second Hilbert space. 1113 01:19:52,800 --> 01:19:55,920 We're supposed to take the product state. 1114 01:19:55,920 --> 01:20:02,320 So our tensor product is going to be spanned, so two spins. 1115 01:20:02,320 --> 01:20:10,260 The tensor vector space is spanned 1116 01:20:10,260 --> 01:20:18,570 by a vector in the first times a vector in the second, 1117 01:20:18,570 --> 01:20:20,890 the basis vectors. 1118 01:20:20,890 --> 01:20:30,770 You could have plus in 1, minus in 2, minus in 1, 1119 01:20:30,770 --> 01:20:38,320 plus in 2, and minus in 1, minus in 2. 1120 01:20:38,320 --> 01:20:45,220 So two spin states form a four-dimensional complex vector 1121 01:20:45,220 --> 01:20:47,940 space. 1122 01:20:47,940 --> 01:20:52,120 That's how you describe them, with those little products. 1123 01:20:52,120 --> 01:20:55,310 And the most general state is the following. 1124 01:21:01,390 --> 01:21:10,900 Most general state is a psi, which 1125 01:21:10,900 --> 01:21:21,340 is alpha plus plus-- you could put 1 and 2-- plus here's alpha 1126 01:21:21,340 --> 01:21:33,172 1, alpha 2, plus, minus, plus alpha 3 minus, plus, 1127 01:21:33,172 --> 01:21:37,920 plus alpha 4 minus, minus. 1128 01:21:45,850 --> 01:21:51,340 Let's do just one simple computation, be done for today. 1129 01:21:51,340 --> 01:21:57,130 I want you to try to figure out what 1130 01:21:57,130 --> 01:22:04,130 is the result of acting with a total z component of angular 1131 01:22:04,130 --> 01:22:10,190 momentum on this state, whatever that means. 1132 01:22:10,190 --> 01:22:13,910 Total z component of angular momentum. 1133 01:22:13,910 --> 01:22:18,810 So naively speaking, the total z component of angular momentum 1134 01:22:18,810 --> 01:22:21,350 would be the z component of angular momentum 1135 01:22:21,350 --> 01:22:25,910 of the first particle plus the z component of the angular 1136 01:22:25,910 --> 01:22:27,850 momentum of the second particle, but you 1137 01:22:27,850 --> 01:22:30,400 know that's not the way we should write it. 1138 01:22:30,400 --> 01:22:31,915 So how is it really? 1139 01:22:31,915 --> 01:22:40,270 It should be Sz of the first particle tensor product with 1 1140 01:22:40,270 --> 01:22:46,280 plus 1 tensor product with Sz of the second particle. 1141 01:22:46,280 --> 01:22:48,536 You can say 2 or 1 here. 1142 01:22:51,660 --> 01:22:53,230 This is really what it is. 1143 01:22:53,230 --> 01:22:57,740 You see, how do we upgrade an operator in the first Hilbert 1144 01:22:57,740 --> 01:23:00,380 space to an operator on the tensor product? 1145 01:23:00,380 --> 01:23:02,810 You just let it act on the first state 1146 01:23:02,810 --> 01:23:05,070 but do nothing on the second. 1147 01:23:05,070 --> 01:23:09,590 So summing the two angular momenta really 1148 01:23:09,590 --> 01:23:16,120 means constructing this new operator in the new, larger, 1149 01:23:16,120 --> 01:23:22,650 more extended space in which it acts in this way. 1150 01:23:22,650 --> 01:23:27,720 So let's just calculate what that is and we'll stop there. 1151 01:23:36,930 --> 01:23:39,150 Let's see. 1152 01:23:39,150 --> 01:23:45,350 For example, we'll have the first term, Sz 1 1153 01:23:45,350 --> 01:23:52,630 tensor 1 acting on psi on the whole state. 1154 01:23:52,630 --> 01:23:57,400 Well, it acts on the first term, alpha 1. 1155 01:23:57,400 --> 01:24:02,680 Now, this operator is supposed to act on this thing, 1156 01:24:02,680 --> 01:24:06,710 so this acts on that and that acts on that. 1157 01:24:06,710 --> 01:24:09,450 So you get-- I'll write this whole thing-- Sz 1158 01:24:09,450 --> 01:24:16,280 hat on plus tensor plus for the first term. 1159 01:24:16,280 --> 01:24:20,100 This term just acted on that. 1160 01:24:20,100 --> 01:24:22,470 We can act with the second one now. 1161 01:24:22,470 --> 01:24:27,010 Well, I was putting the first, so let me leave the first. 1162 01:24:27,010 --> 01:24:30,370 Let's do the second term, plus alpha 2. 1163 01:24:30,370 --> 01:24:35,520 It still acts on the first only, Sz hat plus tensor 1164 01:24:35,520 --> 01:24:46,230 minus plus alpha 2 Sz hat on minus tensor plus, plus alpha 1165 01:24:46,230 --> 01:24:53,030 4 Sz hat on minus tensor minus. 1166 01:24:53,030 --> 01:24:54,650 Now, what is this? 1167 01:24:54,650 --> 01:25:00,530 Well, this thing is h bar over 2 times plus, 1168 01:25:00,530 --> 01:25:03,200 and the number, of course, goes out of the tensor product. 1169 01:25:03,200 --> 01:25:04,630 You don't worry about that. 1170 01:25:04,630 --> 01:25:07,340 There's h bar over 2 everywhere. 1171 01:25:07,340 --> 01:25:10,250 Alpha 1 plus, plus. 1172 01:25:12,830 --> 01:25:20,750 Here the same, another plus, so plus alpha 2 plus, minus. 1173 01:25:20,750 --> 01:25:26,350 And here is minus, so minus alpha 3 minus, 1174 01:25:26,350 --> 01:25:32,075 plus, minus alpha 4, also minus, minus, minus. 1175 01:25:36,390 --> 01:25:37,530 So that's what it is. 1176 01:25:40,040 --> 01:25:44,440 If I do the other one, you could do it with me now quickly. 1177 01:25:44,440 --> 01:25:49,650 1 tensor Sz 2 on psi. 1178 01:25:49,650 --> 01:25:53,050 Once you get accustomed, these are pretty direct. 1179 01:25:53,050 --> 01:25:56,620 So I have to act with Sz hat on these ones, 1180 01:25:56,620 --> 01:26:00,780 so I just act on the second one, on the second state. 1181 01:26:00,780 --> 01:26:03,860 So I get h bar over 2. 1182 01:26:03,860 --> 01:26:05,970 For the first one, you get a plus, 1183 01:26:05,970 --> 01:26:07,790 because it's acting on this thing, 1184 01:26:07,790 --> 01:26:13,110 so you get alpha 1 plus, plus. 1185 01:26:13,110 --> 01:26:16,030 For the second one, however, you get a minus 1186 01:26:16,030 --> 01:26:19,200 because it's the second operator, so minus alpha 1187 01:26:19,200 --> 01:26:22,710 2 plus, minus. 1188 01:26:22,710 --> 01:26:26,110 For the third one, it's a plus, so you 1189 01:26:26,110 --> 01:26:32,160 get a plus alpha 3 minus, plus. 1190 01:26:32,160 --> 01:26:35,230 And for the last one, it's a minus, 1191 01:26:35,230 --> 01:26:42,540 so you get minus alpha 4 minus, minus. 1192 01:26:42,540 --> 01:26:45,710 So when you add them together, these two pieces, 1193 01:26:45,710 --> 01:26:52,290 it's the total, Sz total, acting on the state, 1194 01:26:52,290 --> 01:26:53,480 and what did you get? 1195 01:26:53,480 --> 01:27:03,360 You get h bar, this one's at, alpha plus, plus alpha 1. 1196 01:27:03,360 --> 01:27:06,010 These two cancel, these two cancel. 1197 01:27:06,010 --> 01:27:13,680 Minus alpha 4 minus, minus. 1198 01:27:13,680 --> 01:27:20,120 That's the whole action of Sz on this thing. 1199 01:27:20,120 --> 01:27:32,100 And if you wanted to have a state with total Sz equals 0, 1200 01:27:32,100 --> 01:27:36,330 then you would have to put alpha 1 and alpha 4 to 0. 1201 01:27:41,620 --> 01:27:44,050 You will, if you want, at some stage, 1202 01:27:44,050 --> 01:27:49,510 calculate, maybe in recitation, how much is Sy at this state 1203 01:27:49,510 --> 01:27:53,080 and how much is Sx at this state, 1204 01:27:53,080 --> 01:27:55,370 and try to figure out if there is 1205 01:27:55,370 --> 01:27:59,510 a state whose total spin angular momentum is 0. 1206 01:27:59,510 --> 01:28:02,530 It's just a calculation like this. 1207 01:28:02,530 --> 01:28:06,080 So next week, we'll continue with this 1208 01:28:06,080 --> 01:28:11,490 and with teleportation and Bell inequalities.