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 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,305 at ocw.mit.edu. 8 00:00:21,670 --> 00:00:27,620 PROFESSOR: OK, so let me go back to what we were doing. 9 00:00:27,620 --> 00:00:30,090 The plan for today is as follows. 10 00:00:30,090 --> 00:00:34,240 We're going to look at this unitary time evolution 11 00:00:34,240 --> 00:00:39,800 and calculate this operator u, given the Hamiltonian. 12 00:00:39,800 --> 00:00:44,170 That will be the first order of business today. 13 00:00:44,170 --> 00:00:46,660 Then we will look at the Heisenberg 14 00:00:46,660 --> 00:00:50,090 picture of quantum mechanics. 15 00:00:50,090 --> 00:00:53,270 And the Heisenberg picture of quantum mechanics 16 00:00:53,270 --> 00:00:57,140 is one where the operators, the Schrodinger operators, 17 00:00:57,140 --> 00:00:59,510 acquire time dependence. 18 00:00:59,510 --> 00:01:07,480 And it's a pretty useful way of seeing things, a pretty useful 19 00:01:07,480 --> 00:01:11,620 way of calculating things as well, 20 00:01:11,620 --> 00:01:16,360 and makes the relation between classical mechanics and quantum 21 00:01:16,360 --> 00:01:19,270 mechanics more obvious. 22 00:01:19,270 --> 00:01:22,530 So it's a very important tool. 23 00:01:22,530 --> 00:01:23,890 So we'll discuss that. 24 00:01:23,890 --> 00:01:27,020 We'll find the Heisenberg equations of motion 25 00:01:27,020 --> 00:01:30,270 and solve them for a particular case today. 26 00:01:30,270 --> 00:01:36,290 All this material is not so to be covered in the test. 27 00:01:36,290 --> 00:01:38,740 The only part-- of course, the first few things 28 00:01:38,740 --> 00:01:42,750 I will say today about solving for the unitary operator 29 00:01:42,750 --> 00:01:46,770 you've done in other ways, and I will do it again this time. 30 00:01:49,790 --> 00:01:53,510 So going back to what we were saying last time, 31 00:01:53,510 --> 00:01:56,570 we postulated unitary time evolution. 32 00:01:56,570 --> 00:02:06,300 We said that psi at t was given by some operator U of t t0 psi 33 00:02:06,300 --> 00:02:08,800 t0. 34 00:02:08,800 --> 00:02:12,170 And then we found that this equation 35 00:02:12,170 --> 00:02:16,870 implied the Schrodinger equation with a Hamiltonian 36 00:02:16,870 --> 00:02:21,480 given by the following expression. 37 00:02:21,480 --> 00:02:32,830 ih dU dt of t t0 u dagger of t t0. 38 00:02:32,830 --> 00:02:36,740 So that was our derivation of the Schrodinger equation. 39 00:02:36,740 --> 00:02:38,820 We start with the time evolution. 40 00:02:38,820 --> 00:02:43,070 We found that, whenever we declare that states evolve 41 00:02:43,070 --> 00:02:46,870 in time in that way, they satisfy a first order time 42 00:02:46,870 --> 00:02:51,230 differential equation of the Schrodinger form 43 00:02:51,230 --> 00:02:56,491 in which the Hamiltonian is given in terms of U 44 00:02:56,491 --> 00:02:58,240 by this equation. 45 00:02:58,240 --> 00:03:01,160 And we talked about this operator. 46 00:03:01,160 --> 00:03:04,890 First we showed that it doesn't depend really on t0. 47 00:03:04,890 --> 00:03:07,250 Then we showed that it's Hermitian. 48 00:03:07,250 --> 00:03:09,430 It has units of energy. 49 00:03:09,430 --> 00:03:15,020 And as you may have seen already in the notes, 50 00:03:15,020 --> 00:03:18,610 there is a very clear correspondence 51 00:03:18,610 --> 00:03:22,330 between this operator and the way 52 00:03:22,330 --> 00:03:29,870 the dynamics follows with the ideas of Poisson brackets 53 00:03:29,870 --> 00:03:33,740 that are the precursors of commutators 54 00:03:33,740 --> 00:03:35,480 from classical mechanics. 55 00:03:35,480 --> 00:03:37,040 So that's in the notes. 56 00:03:37,040 --> 00:03:40,050 I will not go in detail in this. 57 00:03:40,050 --> 00:03:43,130 Many of you may have not heard of Poisson brackets. 58 00:03:43,130 --> 00:03:46,880 It's an interesting thing, and really that 59 00:03:46,880 --> 00:03:48,680 will be good enough. 60 00:03:48,680 --> 00:03:53,180 So our goal today is to find U given 61 00:03:53,180 --> 00:03:57,740 H, because as we mentioned last time, for physics 62 00:03:57,740 --> 00:04:01,380 it is typically more easy to invent a quantum 63 00:04:01,380 --> 00:04:04,340 system by postulating a Hamiltonian 64 00:04:04,340 --> 00:04:09,290 and then solving it than postulating a time evolution 65 00:04:09,290 --> 00:04:10,180 operator. 66 00:04:10,180 --> 00:04:24,270 So our goal in general is to find U of t t0 given H of t. 67 00:04:26,820 --> 00:04:29,060 That's what we're supposed to do. 68 00:04:29,060 --> 00:04:31,320 So the first thing I'm going to do 69 00:04:31,320 --> 00:04:34,510 is multiply this equation by u. 70 00:04:37,340 --> 00:04:41,500 By multiplying this equation by a u from the right, 71 00:04:41,500 --> 00:04:50,110 I will write first this term. ih dU dt of t t0 72 00:04:50,110 --> 00:04:55,012 is equal to H of t U of t t0. 73 00:04:57,610 --> 00:05:01,990 So I multiplied this equation by u from the right. 74 00:05:01,990 --> 00:05:06,090 This operator is unitary, so u dagger u is one. 75 00:05:06,090 --> 00:05:09,430 That's why this equation cleaned up to this. 76 00:05:09,430 --> 00:05:13,420 Now there's no confusion really here with derivatives, 77 00:05:13,420 --> 00:05:17,930 so I might this well write them with normal derivatives. 78 00:05:17,930 --> 00:05:27,886 So I'll write this equation as d dt of U t t0 79 00:05:27,886 --> 00:05:31,780 is equal to H of t U of t t0. 80 00:05:43,140 --> 00:05:45,960 You should be able to look at that equation 81 00:05:45,960 --> 00:05:50,230 and say I see the Schrodinger equation there. 82 00:05:50,230 --> 00:05:51,930 How? 83 00:05:51,930 --> 00:05:59,720 Imagine that you have a psi of t0 here, and you put it in. 84 00:05:59,720 --> 00:06:06,730 Then the right hand side becomes h and t acting on psi of t. 85 00:06:06,730 --> 00:06:14,110 And on the left hand side, this psi of t0 86 00:06:14,110 --> 00:06:18,200 can be put inside the derivative because it doesn't depend on t. 87 00:06:18,200 --> 00:06:23,640 Therefore this becomes ih bar d dt of psi of t. 88 00:06:23,640 --> 00:06:29,530 So the Schrodinger equation is there. 89 00:06:29,530 --> 00:06:31,850 OK so now let's solve this. 90 00:06:31,850 --> 00:06:33,680 We'll go through three cases. 91 00:06:33,680 --> 00:06:37,400 Case one, h is time independent. 92 00:06:43,560 --> 00:06:48,480 So we're doing this sort of quickly. 93 00:06:48,480 --> 00:06:54,240 So H of t is really H like that. 94 00:06:54,240 --> 00:06:58,700 No explicit time dependence there. 95 00:06:58,700 --> 00:07:00,510 So what do we have? 96 00:07:00,510 --> 00:07:02,730 ih bar. 97 00:07:02,730 --> 00:07:13,210 Let's write dU dt is equal H times U. 98 00:07:13,210 --> 00:07:18,020 And we tried to write a solution of the form U 99 00:07:18,020 --> 00:07:24,630 use equal to e to the minus iHt over h bar times U0. 100 00:07:29,500 --> 00:07:32,600 Does that work? 101 00:07:32,600 --> 00:07:40,030 Well, we can think du dt and ih. 102 00:07:40,030 --> 00:07:42,090 So we get ih. 103 00:07:42,090 --> 00:07:48,420 When I take dU dt, I have to differentiate this exponential. 104 00:07:48,420 --> 00:07:53,530 And now in this exponential, this full operator H is there. 105 00:07:53,530 --> 00:07:57,850 But we are differentiating with respect to time. 106 00:07:57,850 --> 00:08:00,970 And H doesn't depend on time, so this is not 107 00:08:00,970 --> 00:08:03,330 a very difficult situation. 108 00:08:03,330 --> 00:08:06,640 You could imagine the power series expansion. 109 00:08:06,640 --> 00:08:10,340 And H, as far as this derivative goes, 110 00:08:10,340 --> 00:08:12,340 is like if it would be even a number. 111 00:08:12,340 --> 00:08:15,200 It wouldn't make any difference if it's an operator. 112 00:08:15,200 --> 00:08:20,040 So the derivative with respect to time of this thing 113 00:08:20,040 --> 00:08:25,600 is minus iH over h times the same exponential. 114 00:08:32,000 --> 00:08:38,159 Moreover, the position of this h could be here, 115 00:08:38,159 --> 00:08:41,419 or it could be to the right. 116 00:08:41,419 --> 00:08:45,030 It cannot be to the right of U0 though, 117 00:08:45,030 --> 00:08:49,630 because this is a matrix, a constant matrix that we've put 118 00:08:49,630 --> 00:08:54,590 in here as a possible thing for boundary condition. 119 00:08:54,590 --> 00:08:58,320 So so far we've taken this derivative, 120 00:08:58,320 --> 00:09:03,706 and then i's cancel, the h bar cancels, and you get H. 121 00:09:03,706 --> 00:09:07,340 But this whole thing is, again, U. 122 00:09:07,340 --> 00:09:10,940 So the equation has been solved. 123 00:09:10,940 --> 00:09:14,440 So try this. 124 00:09:14,440 --> 00:09:15,630 And it works. 125 00:09:15,630 --> 00:09:25,120 So having this solution we can write, for example, 126 00:09:25,120 --> 00:09:31,410 that U of t t0 is going to be e to the minus 127 00:09:31,410 --> 00:09:38,110 iHt over h bar, some constant matrix. 128 00:09:38,110 --> 00:09:44,940 When t is equal to t0, this matrix becomes the unit matrix. 129 00:09:44,940 --> 00:09:51,180 So this is e to the minus iHt0 over h bar times U0. 130 00:09:54,500 --> 00:09:59,140 And therefore from here, U0 is the inverse 131 00:09:59,140 --> 00:10:06,520 of this matrix, which is nothing else but e to the iHt0 over h 132 00:10:06,520 --> 00:10:08,140 bar. 133 00:10:08,140 --> 00:10:18,910 So I can substitute back here what U0 is and finally 134 00:10:18,910 --> 00:10:31,170 obtain U of t t0 is e to the minus iH over h bar t minus t0. 135 00:10:33,890 --> 00:10:41,180 And this is for h time independent. 136 00:10:41,180 --> 00:10:43,900 And that's our solution. 137 00:10:43,900 --> 00:10:46,435 There's very little to add to this. 138 00:10:50,790 --> 00:10:53,710 We discussed that in recitation on Thursday. 139 00:10:53,710 --> 00:10:56,560 This unitary operator you've been seeing that 140 00:10:56,560 --> 00:10:59,420 from the beginning of the course in some sense, 141 00:10:59,420 --> 00:11:02,800 that you evolve energy eigenstate. 142 00:11:02,800 --> 00:11:07,110 If this acts on any energy eigenstate, 143 00:11:07,110 --> 00:11:13,380 h is an energy-- if you act here on an energy eigenstate, 144 00:11:13,380 --> 00:11:17,030 the energy eigenstate is an eigenstate precisely for H, 145 00:11:17,030 --> 00:11:19,670 you can put just the number here. 146 00:11:19,670 --> 00:11:26,700 That is e to the, say, alpha h on a state psi 147 00:11:26,700 --> 00:11:38,290 n is equal to e to the alpha en psi n if h on psi n 148 00:11:38,290 --> 00:11:43,110 is equal to en on psi n. 149 00:11:43,110 --> 00:11:48,910 So the function of an operator acting on an eigenstate 150 00:11:48,910 --> 00:11:54,560 is just the function evaluated at the eigenvalue. 151 00:11:54,560 --> 00:12:01,600 So this is a rule that you've been using a really long time. 152 00:12:01,600 --> 00:12:05,140 OK, so when h is time independent, that's what it is. 153 00:12:05,140 --> 00:12:10,110 How about when h has a little time dependence? 154 00:12:10,110 --> 00:12:13,880 What do I call a little time dependence? 155 00:12:13,880 --> 00:12:19,590 A little time dependence is an idea, the sign 156 00:12:19,590 --> 00:12:24,050 to make it possible for you to solve the equation, 157 00:12:24,050 --> 00:12:26,800 even though it has some time dependence. 158 00:12:26,800 --> 00:12:31,180 So you could have Hamiltonians that are time dependent, 159 00:12:31,180 --> 00:12:33,870 but still have a simplifying virtue. 160 00:12:33,870 --> 00:12:40,290 So H of t is time dependent. 161 00:12:40,290 --> 00:12:54,655 But assume that H at t1 and h at t2 commute for all t1 and t2. 162 00:12:58,080 --> 00:13:00,140 So what could that be? 163 00:13:00,140 --> 00:13:03,890 For example, you know that the particle in a magnetic field, 164 00:13:03,890 --> 00:13:15,435 the spin in a magnetic field is minus gamma B dot the spin. 165 00:13:18,210 --> 00:13:24,970 And you could have a time dependent magnetic field, B 166 00:13:24,970 --> 00:13:28,380 of t times the spin. 167 00:13:28,380 --> 00:13:30,680 I'm not sure this is the constant gamma that they 168 00:13:30,680 --> 00:13:35,570 usually call gamma, but it may be. 169 00:13:35,570 --> 00:13:41,500 Now then if the magnetic field is time dependent, but imagine 170 00:13:41,500 --> 00:13:46,350 its direction is not time dependent. 171 00:13:46,350 --> 00:13:51,690 So if its direction is not time dependent, then, for example, 172 00:13:51,690 --> 00:13:59,450 you would have here minus gamma Bz of t times Sz. 173 00:13:59,450 --> 00:14:02,250 And the Hamiltonian at different times commute 174 00:14:02,250 --> 00:14:06,260 because Sz commutes with itself, and the fact that it's time 175 00:14:06,260 --> 00:14:12,770 independent doesn't make it fail to commute. 176 00:14:12,770 --> 00:14:17,900 So if you have a magnetic field that is fixed in one direction 177 00:14:17,900 --> 00:14:21,330 but change in time, you can have a situation where 178 00:14:21,330 --> 00:14:24,370 your Hamiltonian is time dependent, 179 00:14:24,370 --> 00:14:26,970 but still at different times it commutes. 180 00:14:26,970 --> 00:14:30,640 And you will discuss such case because it's interesting. 181 00:14:30,640 --> 00:14:34,290 But later on as we do nuclear magnetic resonance, 182 00:14:34,290 --> 00:14:36,550 we will have the more interesting case 183 00:14:36,550 --> 00:14:41,400 in which a magnetic field rotates and therefore it's 184 00:14:41,400 --> 00:14:43,900 not that simple. 185 00:14:43,900 --> 00:14:49,540 So what happens if you have a time dependent Hamiltonian that 186 00:14:49,540 --> 00:14:51,560 actually commutes? 187 00:14:51,560 --> 00:14:56,780 Well, the claim is that U of t t0 188 00:14:56,780 --> 00:15:02,660 is given by a natural extension of what we had before. 189 00:15:02,660 --> 00:15:08,000 You would want to put exponential of minus iHt, 190 00:15:08,000 --> 00:15:11,880 but the reason this worked was because the derivative 191 00:15:11,880 --> 00:15:15,880 with respect to time brought down an iH over h bar. 192 00:15:15,880 --> 00:15:24,140 So one way to fix this is to put t t0 H of t prime dt prime. 193 00:15:28,980 --> 00:15:34,635 So this is an answer to try this. 194 00:15:39,760 --> 00:15:40,630 Look at this. 195 00:15:40,630 --> 00:15:44,810 If the Hamiltonian were to be time independent, 196 00:15:44,810 --> 00:15:46,520 you could take it out. 197 00:15:46,520 --> 00:15:48,480 And then you would get t minus t0. 198 00:15:48,480 --> 00:15:53,390 That brings you back to this case, so this looks reasonable. 199 00:15:53,390 --> 00:15:58,280 So let me call this quantity R of t. 200 00:16:01,030 --> 00:16:05,660 And then you notice that R dot of t, 201 00:16:05,660 --> 00:16:10,050 the derivative of this quantity with respect to time. 202 00:16:10,050 --> 00:16:14,090 Well, when you differentiate an integral the upper argument, 203 00:16:14,090 --> 00:16:16,460 you get just the integrand evaluated 204 00:16:16,460 --> 00:16:21,350 at the time represented by the upper argument 205 00:16:21,350 --> 00:16:24,140 of the upper limit of integration. 206 00:16:24,140 --> 00:16:26,645 So this is H of t. 207 00:16:29,610 --> 00:16:36,370 And now here comes a crucial point. 208 00:16:36,370 --> 00:16:38,900 You're trying to differentiate. 209 00:16:38,900 --> 00:16:43,580 This U is really e to the R. And you're 210 00:16:43,580 --> 00:16:49,910 trying to differentiate to see if the equation holds dU dt. 211 00:16:49,910 --> 00:16:51,480 So what is the dU dt? 212 00:16:55,960 --> 00:17:05,897 Would be d dt of 1 plus R plus RR plus 1 3 factor RRR. 213 00:17:12,140 --> 00:17:15,520 And now what happens? 214 00:17:15,520 --> 00:17:23,280 You differentiate here, and the first term is R dot. 215 00:17:23,280 --> 00:17:29,520 Here You, would have one half R dot R plus R R dot. 216 00:17:34,220 --> 00:17:39,480 And then 1 over 3 factorial, but three factors. 217 00:17:39,480 --> 00:17:46,910 R dot RR plus R R dot R plus RR R dot. 218 00:17:53,900 --> 00:17:58,720 But here is the claim R dot commutes 219 00:17:58,720 --> 00:18:09,050 with R. Claim R dot and R commute. 220 00:18:09,050 --> 00:18:10,470 Why is that? 221 00:18:10,470 --> 00:18:16,320 Well, R dot depends on H. And R is an integral 222 00:18:16,320 --> 00:18:19,850 of H as well, but the H at different times commute 223 00:18:19,850 --> 00:18:22,610 anyway, so this must be true. 224 00:18:22,610 --> 00:18:25,250 There's no place where you can get a contribution, 225 00:18:25,250 --> 00:18:30,100 because R dot is like an H, and here's an integral of H. 226 00:18:30,100 --> 00:18:34,270 So since the Hamiltonians are assumed to commute, 227 00:18:34,270 --> 00:18:38,470 R dot commutes with R. And this becomes 228 00:18:38,470 --> 00:18:41,820 like a normal derivative of an exponential in which you 229 00:18:41,820 --> 00:18:45,840 can move the R dot to the left everywhere. 230 00:18:45,840 --> 00:18:48,270 And you're differentiating the usual thing. 231 00:18:48,270 --> 00:18:59,240 So this is R dot and times the exponential of R. 232 00:18:59,240 --> 00:19:01,800 So actually that means that we've 233 00:19:01,800 --> 00:19:07,010 got pretty much our answer, because R dot is minus 234 00:19:07,010 --> 00:19:11,300 i over h bar H of t. 235 00:19:11,300 --> 00:19:17,810 And e to the R is U, so we got dU dt equals 236 00:19:17,810 --> 00:19:21,330 this, which is the same as this equation. 237 00:19:34,430 --> 00:19:39,140 The only reason a derivative with respect to time will not 238 00:19:39,140 --> 00:19:45,880 give the usual thing is if R and R dot fail to commute, 239 00:19:45,880 --> 00:19:46,790 and they don't. 240 00:19:46,790 --> 00:19:48,810 So you could put the R dot here. 241 00:19:48,810 --> 00:19:50,840 You can put R dot on the other side, 242 00:19:50,840 --> 00:19:54,660 because it commutes with R, but it's better here. 243 00:19:54,660 --> 00:20:01,800 And therefore you've got this very nice solution. 244 00:20:01,800 --> 00:20:04,650 So the solution is not that bad. 245 00:20:04,650 --> 00:20:09,790 Now finally, I want to discuss for a second the general case. 246 00:20:12,710 --> 00:20:20,806 So that's case-- there was a 1, a 2, a 3 H of t general. 247 00:20:23,740 --> 00:20:25,270 What can you do? 248 00:20:25,270 --> 00:20:28,570 Well, if H of t is general, there's 249 00:20:28,570 --> 00:20:31,560 not too much you can do. 250 00:20:31,560 --> 00:20:34,350 You can write something that will 251 00:20:34,350 --> 00:20:38,570 get you started doing things, but it's not 252 00:20:38,570 --> 00:20:42,310 obviously terribly useful. 253 00:20:42,310 --> 00:20:45,060 But it's interesting anyway that there's 254 00:20:45,060 --> 00:20:47,530 a way to write something that makes sense. 255 00:20:47,530 --> 00:20:50,830 So here it is. 256 00:20:50,830 --> 00:20:52,450 U of t and t0. 257 00:20:52,450 --> 00:20:58,230 I'll write the answer and explain how it looks, 258 00:20:58,230 --> 00:21:04,090 and then you will see that it's OK. 259 00:21:04,090 --> 00:21:05,200 It's interesting. 260 00:21:05,200 --> 00:21:08,710 But it probably is not the most practical way 261 00:21:08,710 --> 00:21:09,895 you can solve this problem. 262 00:21:15,310 --> 00:21:16,790 So here it is. 263 00:21:16,790 --> 00:21:19,680 There's an acronym for this thing. 264 00:21:19,680 --> 00:21:23,900 T it's called the time ordered exponential. 265 00:21:23,900 --> 00:21:27,750 This operator does something to the exponential function. 266 00:21:27,750 --> 00:21:29,470 So it's a definition. 267 00:21:29,470 --> 00:21:31,350 So I have to say what this time ordered 268 00:21:31,350 --> 00:21:37,830 exponential is, and it's the following. 269 00:21:37,830 --> 00:21:40,970 You take the exponential and just begin to expand. 270 00:21:40,970 --> 00:21:45,217 So 1 minus i over h bar-- or I'll 271 00:21:45,217 --> 00:21:51,790 put like this, plus minus i over h bar integral 272 00:21:51,790 --> 00:22:01,360 from t0 to t of dt1 H of t1. 273 00:22:01,360 --> 00:22:03,130 So far, so good. 274 00:22:03,130 --> 00:22:07,270 I've just expanded this. 275 00:22:07,270 --> 00:22:10,960 Now if I would continue expanding, 276 00:22:10,960 --> 00:22:15,460 I would get something that doesn't provide the solution. 277 00:22:15,460 --> 00:22:20,110 You see, this thing is the solution 278 00:22:20,110 --> 00:22:23,580 when the Hamiltonian at different times commute. 279 00:22:23,580 --> 00:22:26,910 So it's unlikely to be the solution when 280 00:22:26,910 --> 00:22:28,000 they don't commute. 281 00:22:28,000 --> 00:22:29,610 In fact, it's not the solution. 282 00:22:29,610 --> 00:22:32,730 So what is the next term here? 283 00:22:32,730 --> 00:22:36,390 The next term is you think of the exponential 284 00:22:36,390 --> 00:22:38,180 as you would expand as usual. 285 00:22:38,180 --> 00:22:43,010 So you will have here plus one half of this thing squared. 286 00:22:43,010 --> 00:22:48,320 So I will put something and then erase it, so maybe don't copy. 287 00:22:48,320 --> 00:22:53,370 One half minus i over h bar squared. 288 00:22:53,370 --> 00:23:02,700 And you would say, well, t0 to t dt prime H of t prime. 289 00:23:02,700 --> 00:23:10,680 t0 to t dt double prime H of double prime. 290 00:23:10,680 --> 00:23:13,010 Well, that would be just an exponential. 291 00:23:13,010 --> 00:23:16,620 So what is a time ordered exponential? 292 00:23:16,620 --> 00:23:18,480 You erase the one half. 293 00:23:21,040 --> 00:23:26,885 And then for notation call this t1 and t1. 294 00:23:30,010 --> 00:23:37,210 And then the next integral do it only up to time t1, 295 00:23:37,210 --> 00:23:38,375 and call this t2. 296 00:23:44,370 --> 00:23:49,670 So t1 will always be greater than t2, 297 00:23:49,670 --> 00:23:53,760 because t2 is integrated from t0 to t1. 298 00:23:53,760 --> 00:24:00,410 And as you integrate here over the various t1's, you just 299 00:24:00,410 --> 00:24:03,740 integrate up to that value. 300 00:24:03,740 --> 00:24:06,580 So you're doing less of the full integral 301 00:24:06,580 --> 00:24:08,210 then you should be doing, and that's 302 00:24:08,210 --> 00:24:12,080 why the factor of one half has disappeared. 303 00:24:12,080 --> 00:24:14,350 This can be continued. 304 00:24:14,350 --> 00:24:19,330 I can write the next one would be minus i over h bar cubed 305 00:24:19,330 --> 00:24:33,080 integral t0 to t H of t1 integral t0 to t1 dt2 H of t2. 306 00:24:33,080 --> 00:24:36,370 And then they next integral goes up to t2. 307 00:24:36,370 --> 00:24:41,315 So t0 to t2 dt3 H of t3. 308 00:24:46,620 --> 00:24:50,190 Anyway, that's a time ordered exponential. 309 00:24:50,190 --> 00:24:58,970 And I leave it to you to take the time derivative, 310 00:24:58,970 --> 00:25:02,660 at least to see that the first few terms are working exactly 311 00:25:02,660 --> 00:25:04,550 the way they should. 312 00:25:04,550 --> 00:25:07,940 That is, if you take a time derivative of this, 313 00:25:07,940 --> 00:25:12,710 you will get H times that thing. 314 00:25:12,710 --> 00:25:14,670 So since it's a power series, you 315 00:25:14,670 --> 00:25:16,950 will differentiate the first term, 316 00:25:16,950 --> 00:25:18,680 and you will get the right thing. 317 00:25:18,680 --> 00:25:22,710 Then the second term and you will start getting everything 318 00:25:22,710 --> 00:25:25,310 that you need. 319 00:25:25,310 --> 00:25:27,770 So it's a funny object. 320 00:25:27,770 --> 00:25:30,950 It's reassuring that something like this success, 321 00:25:30,950 --> 00:25:33,150 but in general, you would want to be 322 00:25:33,150 --> 00:25:37,560 able to do all these integrals and to sum them up. 323 00:25:37,560 --> 00:25:41,420 And in general, it's not that easy. 324 00:25:41,420 --> 00:25:45,370 So it's of limited usefulness. 325 00:25:45,370 --> 00:25:47,670 It's a nice thing that you can write it, 326 00:25:47,670 --> 00:25:52,570 and you can prove things about it and manipulate it. 327 00:25:52,570 --> 00:25:54,750 But when you have a practical problem, 328 00:25:54,750 --> 00:25:57,820 generally that's not the way you solve it. 329 00:25:57,820 --> 00:26:01,920 In fact, when we will discuss the rotating magnetic fields 330 00:26:01,920 --> 00:26:06,680 for magnetic resonance, we will not solve it in this way. 331 00:26:06,680 --> 00:26:10,870 We will try to figure out the solution some other way. 332 00:26:10,870 --> 00:26:13,290 But in terms of completeness, it's 333 00:26:13,290 --> 00:26:16,150 kind of pretty in that you go from 334 00:26:16,150 --> 00:26:19,230 the exponential to the time ordered exponential. 335 00:26:19,230 --> 00:26:24,080 And I think you'll see more of this in 806. 336 00:26:24,080 --> 00:26:29,100 So that's basically our solution for H 337 00:26:29,100 --> 00:26:35,200 and for the unitary operator U in terms of H. 338 00:26:35,200 --> 00:26:36,810 And what we're going to do now is 339 00:26:36,810 --> 00:26:39,170 turn to the Heisenberg picture of quantum mechanics. 340 00:26:39,170 --> 00:26:40,300 Yes, questions? 341 00:26:40,300 --> 00:26:42,176 AUDIENCE: Why does R dot [INAUDIBLE]? 342 00:26:47,210 --> 00:26:53,090 PROFESSOR: Because that's really a property of integrals. 343 00:26:53,090 --> 00:26:59,180 d dx integral up to x from x0 g of x 344 00:26:59,180 --> 00:27:07,360 prime dx prime is just equal to g of x. 345 00:27:07,360 --> 00:27:09,570 This is a constant here, so you're not 346 00:27:09,570 --> 00:27:12,310 varying the integral over in this limit. 347 00:27:12,310 --> 00:27:16,170 So if this limit would also be x dependent, 348 00:27:16,170 --> 00:27:17,830 you would get another contribution, 349 00:27:17,830 --> 00:27:20,290 but we only get the contribution from here. 350 00:27:20,290 --> 00:27:22,640 What's really happening is you're integrating up 351 00:27:22,640 --> 00:27:25,740 to x, then up to x plus epsilon subtracting, 352 00:27:25,740 --> 00:27:30,810 so you pick up the value of the function of the upper limit. 353 00:27:30,810 --> 00:27:31,800 Yes? 354 00:27:31,800 --> 00:27:35,740 AUDIENCE: So what happens to the T that was pre factor? 355 00:27:35,740 --> 00:27:37,452 PROFESSOR: What happens to this T? 356 00:27:37,452 --> 00:27:38,660 AUDIENCE: Yeah, what happens? 357 00:27:38,660 --> 00:27:40,830 PROFESSOR: That's just a symbol. 358 00:27:40,830 --> 00:27:45,280 It says time order the following exponential. 359 00:27:45,280 --> 00:27:49,370 So at this stage, this is a definition 360 00:27:49,370 --> 00:27:52,670 of what t on an exponential means. 361 00:27:52,670 --> 00:27:54,140 AUDIENCE: OK. 362 00:27:54,140 --> 00:27:57,260 PROFESSOR: It's not-- let me say T is not 363 00:27:57,260 --> 00:28:00,290 an operator in the usual sense of quantum mechanics 364 00:28:00,290 --> 00:28:01,490 or anything like that. 365 00:28:01,490 --> 00:28:03,680 It's an instruction. 366 00:28:03,680 --> 00:28:07,560 Whenever you have an exponential of this form, 367 00:28:07,560 --> 00:28:10,380 the time ordered exponential is this series 368 00:28:10,380 --> 00:28:13,050 that we've written down. 369 00:28:13,050 --> 00:28:14,250 It's just a definition. 370 00:28:14,250 --> 00:28:14,926 Yes? 371 00:28:14,926 --> 00:28:17,050 AUDIENCE: So when we have operators in differential 372 00:28:17,050 --> 00:28:20,850 equations, do we still get [INAUDIBLE]? 373 00:28:20,850 --> 00:28:22,600 PROFESSOR: If we have what? 374 00:28:22,600 --> 00:28:24,933 AUDIENCE: If we have operators in differential equations 375 00:28:24,933 --> 00:28:27,500 do we still get unique [INAUDIBLE] solutions? 376 00:28:27,500 --> 00:28:29,680 PROFESSOR: Yes, pretty much. 377 00:28:29,680 --> 00:28:32,930 Because at the end of the day, this 378 00:28:32,930 --> 00:28:37,940 is a first order matrix differential equation. 379 00:28:37,940 --> 00:28:41,730 So it's a collection of first order differential equations 380 00:28:41,730 --> 00:28:43,445 for every element of a matrix. 381 00:28:47,600 --> 00:28:51,760 It's pretty much the same as you have before. 382 00:28:51,760 --> 00:28:57,440 If you know the operator at any time, initial time, 383 00:28:57,440 --> 00:28:59,980 with the differential equation you know the operator 384 00:28:59,980 --> 00:29:01,510 at a little bit time later. 385 00:29:01,510 --> 00:29:04,460 So the operator is completely determined 386 00:29:04,460 --> 00:29:07,280 if you know it initially and the differential equation. 387 00:29:07,280 --> 00:29:09,930 So I think it's completely analogous. 388 00:29:09,930 --> 00:29:13,890 It's just that it's harder to solve. 389 00:29:13,890 --> 00:29:15,150 Nothing else. 390 00:29:15,150 --> 00:29:16,556 One last question. 391 00:29:16,556 --> 00:29:18,306 AUDIENCE: So let's say that we can somehow 392 00:29:18,306 --> 00:29:21,620 fly in this unitary operator, and then we have a differential 393 00:29:21,620 --> 00:29:23,859 equation, and we somehow, let's say, get 394 00:29:23,859 --> 00:29:27,400 a wave function out of it. 395 00:29:27,400 --> 00:29:30,140 What is the interpretation of that wave function? 396 00:29:30,140 --> 00:29:31,515 PROFESSOR: Well, it's not that we 397 00:29:31,515 --> 00:29:34,040 get the wave function out of this. 398 00:29:34,040 --> 00:29:38,300 What really is happening is that you 399 00:29:38,300 --> 00:29:45,020 have learned how to calculate this operator given H. 400 00:29:45,020 --> 00:29:49,330 And therefore now you're able to evolve any wave function. 401 00:29:49,330 --> 00:29:51,940 So you have solved the dynamical system. 402 00:29:51,940 --> 00:29:55,690 If somebody tells you a time equals 0, your system is here, 403 00:29:55,690 --> 00:29:59,270 you can now calculate where it's going to be at the later time. 404 00:29:59,270 --> 00:30:01,200 So that's really all you have achieved. 405 00:30:01,200 --> 00:30:05,537 You now know the solution. 406 00:30:05,537 --> 00:30:07,370 When you're doing mechanics and they ask you 407 00:30:07,370 --> 00:30:10,060 for an orbit problem, they say at this time 408 00:30:10,060 --> 00:30:11,130 the planet is here. 409 00:30:11,130 --> 00:30:12,980 What are you supposed to find? 410 00:30:12,980 --> 00:30:14,830 x is a function of time. 411 00:30:14,830 --> 00:30:16,910 You now know how it's going to develop. 412 00:30:16,910 --> 00:30:19,040 You've solved equations of motion. 413 00:30:19,040 --> 00:30:19,860 Here it's the same. 414 00:30:19,860 --> 00:30:22,020 You know the wave function of time equals. 415 00:30:22,020 --> 00:30:26,490 If you know it at any time, you've solved problem. 416 00:30:26,490 --> 00:30:29,650 OK, so Heisenberg picture of quantum mechanics. 417 00:30:34,920 --> 00:30:38,230 Heisenberg picture. 418 00:30:43,050 --> 00:30:48,540 So basically the Heisenberg picture 419 00:30:48,540 --> 00:30:54,210 exists thanks to the existence of the Schrodinger picture. 420 00:30:54,210 --> 00:30:56,240 Heisenberg picture of quantum mechanics 421 00:30:56,240 --> 00:30:58,690 is not something that you necessarily 422 00:30:58,690 --> 00:31:00,380 invent from the beginning. 423 00:31:00,380 --> 00:31:05,190 The way we think of it is we assume there is a Schrodinger 424 00:31:05,190 --> 00:31:10,020 picture that we've developed in which we have operators like x, 425 00:31:10,020 --> 00:31:14,790 p, spin, Hamiltonians, and wave functions. 426 00:31:14,790 --> 00:31:19,020 And then we are going to define a new way of thinking 427 00:31:19,020 --> 00:31:21,840 about this, which is called the Heisenberg 428 00:31:21,840 --> 00:31:24,040 picture of the quantum mechanics. 429 00:31:24,040 --> 00:31:29,690 So it all begins by considering a Schrodinger operator 430 00:31:29,690 --> 00:31:35,070 As hat, which is s is for Schrodinger. 431 00:31:42,940 --> 00:31:48,700 And the motivation comes from expectation values. 432 00:31:48,700 --> 00:31:51,190 Suppose you have time dependent states, 433 00:31:51,190 --> 00:31:53,440 in fact, matrix elements. 434 00:31:53,440 --> 00:31:58,780 One time dependent state alpha of t, one time dependent state 435 00:31:58,780 --> 00:32:00,640 beta of t. 436 00:32:00,640 --> 00:32:06,000 Two independent time dependent states. 437 00:32:06,000 --> 00:32:17,090 So you could ask what is the matrix element of A 438 00:32:17,090 --> 00:32:20,520 between these two time dependent states, a matrix element. 439 00:32:26,180 --> 00:32:31,800 But then, armed with our unitary operator, 440 00:32:31,800 --> 00:32:37,980 we know that As is here, and this state beta at time t 441 00:32:37,980 --> 00:32:47,190 is equal to U of t comma 0 beta at time 0. 442 00:32:47,190 --> 00:32:53,390 And alpha t is equal to alpha at 0 U dagger of t0. 443 00:32:56,160 --> 00:33:00,590 So the states have time dependence. 444 00:33:00,590 --> 00:33:02,280 But the time dependence has already 445 00:33:02,280 --> 00:33:06,850 been found, say, in principle, if you know U dagger. 446 00:33:06,850 --> 00:33:13,200 And then you can speak about the time dependent matrix elements 447 00:33:13,200 --> 00:33:19,510 of the operator As or the matrix element 448 00:33:19,510 --> 00:33:24,960 of this time dependent operator between the time 449 00:33:24,960 --> 00:33:28,480 equals 0 states. 450 00:33:28,480 --> 00:33:31,820 And this operator is sufficiently important 451 00:33:31,820 --> 00:33:35,520 that this operator is called the Heisenberg 452 00:33:35,520 --> 00:33:38,760 version of the operator s. 453 00:33:38,760 --> 00:33:44,685 Has time dependence, and it's defined by this equation. 454 00:33:52,820 --> 00:33:57,560 So whenever you have Schrodinger operator, 455 00:33:57,560 --> 00:34:02,190 whether it be time dependent or time independent, 456 00:34:02,190 --> 00:34:04,790 whatever the Schrodinger operator is, 457 00:34:04,790 --> 00:34:07,150 I have now a definition of what I 458 00:34:07,150 --> 00:34:11,409 will call the Heisenberg operator. 459 00:34:11,409 --> 00:34:17,719 And it is obtained by acting with a unitary operator, U. 460 00:34:17,719 --> 00:34:23,900 And operators always act on operators from the left 461 00:34:23,900 --> 00:34:25,360 and from the right. 462 00:34:25,360 --> 00:34:30,000 That's something that operators act on states from the left. 463 00:34:30,000 --> 00:34:31,850 They act on the state. 464 00:34:31,850 --> 00:34:35,730 But operators act on operator from the left 465 00:34:35,730 --> 00:34:38,360 and from the right, as you see them here, 466 00:34:38,360 --> 00:34:41,739 is the natural, ideal thing to happen. 467 00:34:41,739 --> 00:34:45,510 If you have an operator that's on another from the right only 468 00:34:45,510 --> 00:34:47,969 or from the left only, I think you 469 00:34:47,969 --> 00:34:50,270 have grounds to be suspicious that maybe you're 470 00:34:50,270 --> 00:34:53,000 not doing things right. 471 00:34:53,000 --> 00:34:55,440 So this is the Heisenberg operator. 472 00:34:55,440 --> 00:34:57,360 And as you can imagine, there's a lot 473 00:34:57,360 --> 00:35:01,090 of things to be said about this operator. 474 00:35:01,090 --> 00:35:03,060 So let's begin with a remark. 475 00:35:03,060 --> 00:35:07,970 Are there questions about this Heisenberg operator. 476 00:35:07,970 --> 00:35:08,700 Yes? 477 00:35:08,700 --> 00:35:12,180 AUDIENCE: Do we know anything about the Schrodinger operator? 478 00:35:12,180 --> 00:35:13,680 PROFESSOR: You have to speak louder. 479 00:35:13,680 --> 00:35:15,221 AUDIENCE: Is the Schrodinger operator 480 00:35:15,221 --> 00:35:17,850 related to the Hamiltonian [INAUDIBLE]? 481 00:35:17,850 --> 00:35:19,830 PROFESSOR: Any Schrodinger operator, 482 00:35:19,830 --> 00:35:22,260 this could be the Hamiltonian, this 483 00:35:22,260 --> 00:35:27,550 could be x hat, it could be Sz, could 484 00:35:27,550 --> 00:35:29,090 be any of the operators you know. 485 00:35:29,090 --> 00:35:34,420 All the operators you know are Schrodinger operators. 486 00:35:34,420 --> 00:35:38,330 So remarks, comments. 487 00:35:38,330 --> 00:35:39,306 OK, comments. 488 00:35:43,130 --> 00:35:57,730 One, at t equals 0 A Heisenberg becomes 489 00:35:57,730 --> 00:36:05,630 identical to A Schrodinger at t equals 0. 490 00:36:14,190 --> 00:36:16,300 So look why. 491 00:36:16,300 --> 00:36:22,260 Because when t is equal to 0, U of t-- of 0 0 is the operator 492 00:36:22,260 --> 00:36:26,020 propagates no state, so it's equal to the identity. 493 00:36:26,020 --> 00:36:28,980 So this is a wonderful relation that 494 00:36:28,980 --> 00:36:34,220 tell us you that time equals 0 the two operators are really 495 00:36:34,220 --> 00:36:36,190 the same. 496 00:36:36,190 --> 00:36:38,880 And another simple remark. 497 00:36:38,880 --> 00:36:44,440 If you have the unit operator in the Schrodinger picture, 498 00:36:44,440 --> 00:36:49,210 what is the unit operator in the Heisenberg picture? 499 00:36:49,210 --> 00:37:00,790 Well, it would be U t 0 dagger 1 U t 0. 500 00:37:00,790 --> 00:37:03,300 But 1 doesn't matter. 501 00:37:03,300 --> 00:37:05,890 U dagger with U is 1. 502 00:37:05,890 --> 00:37:09,460 This is a 1 Schrodinger, and therefore it's 503 00:37:09,460 --> 00:37:11,300 the same operator. 504 00:37:11,300 --> 00:37:16,170 So the unit operator is the same. 505 00:37:16,170 --> 00:37:20,300 It just doesn't change whatsoever. 506 00:37:20,300 --> 00:37:23,080 OK, so that's good. 507 00:37:23,080 --> 00:37:26,470 But now this is something interesting also happens. 508 00:37:26,470 --> 00:37:31,380 Suppose you have Schrodinger operator 509 00:37:31,380 --> 00:37:37,526 C that is equal to the product of A with B, two Schrodingers. 510 00:37:41,200 --> 00:37:47,460 If I try to figure out what is CH, 511 00:37:47,460 --> 00:37:53,640 I would put U dagger-- avoid all the letters, the t 0. 512 00:37:53,640 --> 00:37:55,750 It's supposed to be t 0. 513 00:37:55,750 --> 00:38:07,520 Cs U. But that's equal U dagger As Bs U. 514 00:38:07,520 --> 00:38:10,270 But now, in between the two operators, 515 00:38:10,270 --> 00:38:14,980 you can put a U U dagger, which is equal to 1. 516 00:38:14,980 --> 00:38:23,670 So As U U dagger Bs U. And then you 517 00:38:23,670 --> 00:38:26,780 see why this is really nice. 518 00:38:26,780 --> 00:38:31,890 Because what do you get is that C Heisenberg is just 519 00:38:31,890 --> 00:38:34,650 A Heisenberg times B Heisenberg. 520 00:38:34,650 --> 00:38:38,882 So if you have C Schrodinger equals A Schrodinger, B 521 00:38:38,882 --> 00:38:45,330 Schrodinger, C Heisenberg is A Heisenberg B Heisenberg. 522 00:38:45,330 --> 00:38:49,260 So there's a nice correspondence between those operators. 523 00:38:52,360 --> 00:38:58,670 Also you can do is for commutators. 524 00:38:58,670 --> 00:39:02,350 So you don't have to worry about this thing. 525 00:39:02,350 --> 00:39:13,500 So for example, if A Schrodinger with B Schrodinger 526 00:39:13,500 --> 00:39:19,380 is equal to C Schrodinger, then by doing exactly 527 00:39:19,380 --> 00:39:27,760 the same things, you see that A Heisenberg with B Heisenberg 528 00:39:27,760 --> 00:39:31,400 would be the commutator equal to C Heisenberg. 529 00:39:34,140 --> 00:39:35,670 Yes? 530 00:39:35,670 --> 00:39:38,828 AUDIENCE: That argument for the identity operators 531 00:39:38,828 --> 00:39:41,120 being the same in both pictures. 532 00:39:41,120 --> 00:39:43,970 If the Hamiltonian is time independent, 533 00:39:43,970 --> 00:39:46,470 does that work for any operator that commutes 534 00:39:46,470 --> 00:39:48,727 with the Hamiltonian? 535 00:39:48,727 --> 00:39:50,310 PROFESSOR: Hamiltonian is [INAUDIBLE]. 536 00:39:56,070 --> 00:39:58,280 AUDIENCE: Because then you can push the operator just 537 00:39:58,280 --> 00:40:00,850 through the exponential of the Hamiltonian. 538 00:40:00,850 --> 00:40:05,010 PROFESSOR: Yeah, we'll see things like that. 539 00:40:05,010 --> 00:40:08,430 We could discuss that maybe a little later. 540 00:40:08,430 --> 00:40:11,950 But there are some cases, as we will see immediately, 541 00:40:11,950 --> 00:40:16,650 in which some operators are the same in the two pictures. 542 00:40:16,650 --> 00:40:20,700 So basically operators that commute with the Hamiltonian 543 00:40:20,700 --> 00:40:26,320 as you say, since U involves the Hamiltonian, 544 00:40:26,320 --> 00:40:28,790 and this is the Hamiltonian, if the operator 545 00:40:28,790 --> 00:40:31,310 commutes with the Hamiltonian and you can move them across, 546 00:40:31,310 --> 00:40:32,268 then they are the same. 547 00:40:32,268 --> 00:40:35,880 So I think it's definitely true. 548 00:40:35,880 --> 00:40:40,640 So we will have an interesting question, 549 00:40:40,640 --> 00:40:43,450 in fact, whether the Heisenberg Hamiltonian is 550 00:40:43,450 --> 00:40:45,070 equal to the Schrodinger Hamiltonian, 551 00:40:45,070 --> 00:40:48,290 and we'll answer that very soon. 552 00:40:48,290 --> 00:40:52,720 So the one example that here I think you should keep in mind 553 00:40:52,720 --> 00:40:54,860 is this one. 554 00:40:54,860 --> 00:40:56,900 You know this is true. 555 00:40:56,900 --> 00:41:00,730 So what do you knowing the Heisenberg picture? 556 00:41:00,730 --> 00:41:08,150 That X Heisenberg of t times P Heisenberg of t commutator 557 00:41:08,150 --> 00:41:12,210 is equal to the Heisenberg version of this. 558 00:41:12,210 --> 00:41:15,000 But here was the unit operator. 559 00:41:15,000 --> 00:41:20,510 And therefore this is just ih bar times the unit operator 560 00:41:20,510 --> 00:41:22,790 again, because the units operator 561 00:41:22,790 --> 00:41:25,900 is the same in all pictures. 562 00:41:25,900 --> 00:41:31,840 So these commutation relation is true for any Heisenberg 563 00:41:31,840 --> 00:41:33,950 operator. 564 00:41:33,950 --> 00:41:37,100 Whatever commutation relation you have of Schrodinger, 565 00:41:37,100 --> 00:41:38,990 it's true for Heisenberg as well. 566 00:41:42,000 --> 00:41:46,670 OK, so then let's talk about Hamiltonians. 567 00:41:46,670 --> 00:41:53,180 Three, Hamiltonians. 568 00:41:53,180 --> 00:42:00,160 So Heisenberg Hamiltonian by definition 569 00:42:00,160 --> 00:42:07,890 would be equal to U dagger t 0 Schrodinger Hamiltonian 570 00:42:07,890 --> 00:42:09,980 times U of t 0. 571 00:42:14,500 --> 00:42:18,100 So if the Schrodinger Hamiltonian-- actually, 572 00:42:18,100 --> 00:42:25,490 if Hs at t1 commutes units with Hs at t2, 573 00:42:25,490 --> 00:42:31,270 the Schrodinger Hamiltonian is such that for all t1 and t2 574 00:42:31,270 --> 00:42:34,890 they commute with each other. 575 00:42:34,890 --> 00:42:38,560 Remember, if that is the case, the unitary operator 576 00:42:38,560 --> 00:42:42,790 is any way built by an exponential. 577 00:42:42,790 --> 00:42:43,880 It's this one. 578 00:42:47,590 --> 00:42:51,010 And the Schrodinger Hamiltonians commute. 579 00:42:51,010 --> 00:42:53,770 So as was asked in the question before, 580 00:42:53,770 --> 00:42:57,700 this thing commutes with that, and you 581 00:42:57,700 --> 00:43:00,180 get that they are the same. 582 00:43:00,180 --> 00:43:05,730 So if this is happening, the two Hamiltonians are identical. 583 00:43:12,030 --> 00:43:17,110 And we'll have the chance to check 584 00:43:17,110 --> 00:43:21,220 this today in a nice example. 585 00:43:21,220 --> 00:43:29,230 So I will write in this as saying the Heisenberg 586 00:43:29,230 --> 00:43:31,550 Hamiltonian as a function of time 587 00:43:31,550 --> 00:43:34,580 then is equal to the Schrodinger Hamiltonian 588 00:43:34,580 --> 00:43:37,160 as a function of time. 589 00:43:37,160 --> 00:43:45,160 And this goes Hs of t1 and Hs of t2 commute. 590 00:43:48,130 --> 00:43:55,990 OK, now I want you to notice this thing. 591 00:43:59,700 --> 00:44:15,260 Suppose the Hs of t is some Hs of x,p, and t, for example. 592 00:44:15,260 --> 00:44:21,150 OK, now you come and turn it into Heisenberg 593 00:44:21,150 --> 00:44:26,930 by putting a U dagger from the left and a U from the right. 594 00:44:26,930 --> 00:44:29,560 What will that do? 595 00:44:29,560 --> 00:44:32,360 It will put U dagger from the left, U dagger on the right. 596 00:44:32,360 --> 00:44:36,750 And then it will start working it's way inside, 597 00:44:36,750 --> 00:44:41,990 and any x that it will find will turn into a Heisenberg x. 598 00:44:41,990 --> 00:44:44,690 Any p will turn into Heisenberg p. 599 00:44:44,690 --> 00:44:47,160 Imagine, for example, any Hamiltonian 600 00:44:47,160 --> 00:44:49,630 is some function of x. 601 00:44:49,630 --> 00:44:51,090 It has an x squared. 602 00:44:51,090 --> 00:44:53,970 Well the U dagger and U come and turn 603 00:44:53,970 --> 00:44:57,390 this into x Heisenberg squared. 604 00:44:57,390 --> 00:45:05,720 So what I claim here happens is that H Heisenberg of t 605 00:45:05,720 --> 00:45:15,540 is equal to U dagger H Schrodinger of x, p, t, U. 606 00:45:15,540 --> 00:45:22,770 And therefore this becomes H Schrodinger of x 607 00:45:22,770 --> 00:45:29,260 Heisenberg of t, P Heisenberg of t, and t. 608 00:45:34,040 --> 00:45:41,550 So here is what the Heisenberg Hamiltonian is. 609 00:45:41,550 --> 00:45:46,040 It's the Schrodinger Hamiltonian where X's, and P's, or spins 610 00:45:46,040 --> 00:45:49,700 and everything has become Heisenberg. 611 00:45:49,700 --> 00:45:53,300 So the equality of the two Hamiltonians 612 00:45:53,300 --> 00:45:58,110 is a very funny condition on the Schrodinger Hamiltonian, 613 00:45:58,110 --> 00:46:02,010 because this is supposed to be equal to the Schrodinger 614 00:46:02,010 --> 00:46:05,900 Hamiltonian, which is of x, p, and t. 615 00:46:11,710 --> 00:46:16,420 So you have a function of x, p, and t. 616 00:46:16,420 --> 00:46:19,430 And you put X Heisenberg P Heisenberg, 617 00:46:19,430 --> 00:46:22,040 and somehow the whole thing is the same. 618 00:46:27,030 --> 00:46:34,470 So this is something very useful and we'll need it. 619 00:46:34,470 --> 00:46:38,170 One more comment, expectation values. 620 00:46:38,170 --> 00:46:41,300 So this is three. 621 00:46:41,300 --> 00:46:47,400 Comment number four on expectation values, which 622 00:46:47,400 --> 00:46:51,220 is something you've already-- it's sort of the way 623 00:46:51,220 --> 00:46:57,770 we began the discussion and wanted to make sure it's clear. 624 00:46:57,770 --> 00:47:02,032 So four, expectation values. 625 00:47:05,430 --> 00:47:09,670 So we started with this with alpha and beta, 626 00:47:09,670 --> 00:47:12,610 two arbitrary states, matrix elements. 627 00:47:12,610 --> 00:47:17,170 Take them equal and to be equal to psi of t. 628 00:47:17,170 --> 00:47:26,120 So you would have psi t As psi t is, in fact, 629 00:47:26,120 --> 00:47:39,080 equal to psi 0 A Heisenberg psi 0. 630 00:47:39,080 --> 00:47:43,790 Now that is a key equation. 631 00:47:43,790 --> 00:47:47,920 You know you're doing expectation value at any given 632 00:47:47,920 --> 00:47:52,640 time of a Schrodinger operator, turn it into Heisenberg 633 00:47:52,640 --> 00:47:55,320 and work at time equals 0. 634 00:47:55,320 --> 00:47:58,170 It simplifies life tremendously. 635 00:47:58,170 --> 00:48:00,370 Now this is the key identity. 636 00:48:00,370 --> 00:48:03,810 It's the way we motivated everything in a way. 637 00:48:03,810 --> 00:48:08,180 And it's written in a way that maybe it's 638 00:48:08,180 --> 00:48:14,460 a little too schematic, but we write it this way. 639 00:48:14,460 --> 00:48:16,620 We just say the expectation value 640 00:48:16,620 --> 00:48:23,640 of As is equal to the expectation value of AH. 641 00:48:28,140 --> 00:48:31,660 And this, well, we save time like that, 642 00:48:31,660 --> 00:48:33,770 but you have to know what you mean. 643 00:48:33,770 --> 00:48:36,290 When you're computing the expectation 644 00:48:36,290 --> 00:48:37,980 value for a Schrodinger operator, 645 00:48:37,980 --> 00:48:40,464 you're using time dependent states. 646 00:48:40,464 --> 00:48:42,005 When you're computing the expectation 647 00:48:42,005 --> 00:48:44,590 value of the Heisenberg operator, 648 00:48:44,590 --> 00:48:48,360 you're using the time equals 0 version of the states, 649 00:48:48,360 --> 00:48:49,680 but they are the same. 650 00:48:49,680 --> 00:48:53,530 So we say that the Schrodinger expectation value 651 00:48:53,530 --> 00:48:55,680 is equal to the Heisenberg expectation value. 652 00:48:55,680 --> 00:49:01,760 We right it in the bottom, but we mean the top equation. 653 00:49:01,760 --> 00:49:06,140 And we use it that way. 654 00:49:06,140 --> 00:49:12,320 So the Heisenberg operators, at this moment, 655 00:49:12,320 --> 00:49:14,640 are a little mysterious. 656 00:49:14,640 --> 00:49:18,920 They're supposed to be given by this formula, 657 00:49:18,920 --> 00:49:22,940 but we've seen that calculating U can be difficult. 658 00:49:22,940 --> 00:49:25,860 So calculating the Heisenberg operator 659 00:49:25,860 --> 00:49:28,010 can be difficult sometimes. 660 00:49:28,010 --> 00:49:31,740 So what we try to do in order to simplify 661 00:49:31,740 --> 00:49:35,950 that is find an equation that is satisfied by the Heisenberg 662 00:49:35,950 --> 00:49:40,270 operator, a time derivative equation. 663 00:49:40,270 --> 00:49:43,170 So let's try to find an equation that 664 00:49:43,170 --> 00:49:47,830 is satisfied by the Heisenberg operator rather than a formula. 665 00:49:47,830 --> 00:49:50,150 You'll say, well, this is better. 666 00:49:50,150 --> 00:49:54,996 But the fact is that seldom you know U. And even if you know U, 667 00:49:54,996 --> 00:49:59,010 you have to do this simplification, which is hard. 668 00:49:59,010 --> 00:50:03,080 So finding a differential equation for the operator 669 00:50:03,080 --> 00:50:04,090 is useful. 670 00:50:04,090 --> 00:50:11,300 So differential equation for Heisenberg operators. 671 00:50:14,730 --> 00:50:16,300 So what do we want to do? 672 00:50:16,300 --> 00:50:28,775 We want to calculate ih bar d dt of the Heisenberg operator. 673 00:50:32,670 --> 00:50:35,710 And so what do we get? 674 00:50:35,710 --> 00:50:40,160 Well, we have several things. 675 00:50:40,160 --> 00:50:42,050 Remember, the Schrodinger operator 676 00:50:42,050 --> 00:50:45,160 can have a bit of time dependence. 677 00:50:45,160 --> 00:50:48,220 The time dependence would be an explicit time dependence. 678 00:50:48,220 --> 00:50:51,360 So let's take the time derivative of all this. 679 00:50:51,360 --> 00:50:53,680 So you would have three terms. 680 00:50:53,680 --> 00:51:06,670 ih bar dU dagger dt As U plus U dagger 681 00:51:06,670 --> 00:51:21,455 As dU dt plus-- with an ih bar-- U dagger ih bar dAs minus dt. 682 00:51:26,050 --> 00:51:40,670 dAs dt and U. 683 00:51:40,670 --> 00:51:44,060 Well, you have these equations. 684 00:51:44,060 --> 00:51:45,910 Those were the Schrodinger equations 685 00:51:45,910 --> 00:51:49,100 we started with today. 686 00:51:49,100 --> 00:51:53,600 The derivatives of U, or the derivatives of U dagger. 687 00:51:53,600 --> 00:51:56,790 so what did we have? 688 00:51:56,790 --> 00:52:11,800 Well, we have that ih bar dU dt was HU-- H Schrodinger times U. 689 00:52:11,800 --> 00:52:20,160 And therefore ih bar dU dagger dt. 690 00:52:20,160 --> 00:52:21,770 I take the dagger of this. 691 00:52:21,770 --> 00:52:23,510 I would get a minus sign. 692 00:52:23,510 --> 00:52:25,290 I would put it on the other side. 693 00:52:25,290 --> 00:52:32,310 Is equal to U dagger Hs with a minus here. 694 00:52:32,310 --> 00:52:37,020 And all the U's are U's of t and t0. 695 00:52:37,020 --> 00:52:41,900 I ran out of this thick chalk. 696 00:52:41,900 --> 00:52:43,890 So we'll continue with thin chalk. 697 00:52:46,500 --> 00:52:48,050 All right, so we are here. 698 00:52:51,890 --> 00:52:54,290 We wrote the time derivative, and we 699 00:52:54,290 --> 00:52:58,870 have three terms to work out. 700 00:52:58,870 --> 00:53:00,450 So what are they? 701 00:53:00,450 --> 00:53:04,000 Well we have this thing, ih bar this. 702 00:53:04,000 --> 00:53:16,060 So let's write it. ih bar d d dt of As-- of A Heisenberg, 703 00:53:16,060 --> 00:53:27,310 I'm sorry-- Is equal to that term is minus U dagger Hs A 704 00:53:27,310 --> 00:53:31,830 Schrodinger U. 705 00:53:31,830 --> 00:53:37,940 The next term plus ih bar dU dt on the right. 706 00:53:37,940 --> 00:53:55,840 So we have plus U dagger As Hs dU dt, so U. Well, 707 00:53:55,840 --> 00:53:57,020 that's not bad. 708 00:53:57,020 --> 00:53:59,250 It's actually quite nice. 709 00:53:59,250 --> 00:54:03,830 And then the last term, which I have very little to say, 710 00:54:03,830 --> 00:54:08,600 because in general, this is a derivative of a time 711 00:54:08,600 --> 00:54:11,370 dependent operator. 712 00:54:11,370 --> 00:54:16,470 Partial with respect to time, it would be 0 if As depends, 713 00:54:16,470 --> 00:54:21,160 just say, on X, on P, on Sx, or any of those things, 714 00:54:21,160 --> 00:54:23,320 has to have a particular t. 715 00:54:23,320 --> 00:54:36,800 So I will just leave this as plus ih bar dAs dt Heisenberg. 716 00:54:36,800 --> 00:54:40,760 The Heisenberg version of this operator using 717 00:54:40,760 --> 00:54:44,790 the definition that anything, any operator that we have 718 00:54:44,790 --> 00:54:47,880 a U dagger in front, a U to the right, 719 00:54:47,880 --> 00:54:52,620 is the Heisenberg version of the operator. 720 00:54:52,620 --> 00:54:58,460 So I think I'm doing all right with this equation. 721 00:54:58,460 --> 00:55:00,850 So what did we have? 722 00:55:00,850 --> 00:55:08,230 Here it is. ih bar d dt of A Heisenberg of t. 723 00:55:08,230 --> 00:55:14,850 And now comes the nice thing, of course. 724 00:55:14,850 --> 00:55:16,670 This thing, look at it. 725 00:55:16,670 --> 00:55:21,150 U dagger U. This turns everything here 726 00:55:21,150 --> 00:55:22,460 into Heisenberg. 727 00:55:22,460 --> 00:55:25,170 H Heisenberg, A Heisenberg. 728 00:55:25,170 --> 00:55:30,160 Here you have A Heisenberg H Heisenberg, and what you got 729 00:55:30,160 --> 00:55:32,590 is the commutator between them. 730 00:55:32,590 --> 00:55:39,000 So this thing is A Heisenberg commutator with H Heisenberg. 731 00:55:45,070 --> 00:55:46,330 That whole thing. 732 00:55:46,330 --> 00:55:54,215 And then you have plus ih bar dAs dt Heisenberg. 733 00:56:00,320 --> 00:56:05,280 So that is the Heisenberg equation of motion. 734 00:56:05,280 --> 00:56:11,640 That is how you can calculate a Heisenberg operator 735 00:56:11,640 --> 00:56:12,290 if you want. 736 00:56:14,950 --> 00:56:18,020 You tried to solve this differential equation, 737 00:56:18,020 --> 00:56:20,060 and many times that's the simplest 738 00:56:20,060 --> 00:56:23,320 way to calculate the Heisenberg operator. 739 00:56:23,320 --> 00:56:25,840 So there you go. 740 00:56:25,840 --> 00:56:28,400 It's a pretty important equation. 741 00:56:28,400 --> 00:56:34,390 So let's consider particular cases immediately 742 00:56:34,390 --> 00:56:36,560 to just get some intuition. 743 00:56:36,560 --> 00:56:40,175 So remarks. 744 00:56:43,450 --> 00:56:55,700 Suppose As has no explicit time dependence. 745 00:56:55,700 --> 00:56:58,765 So basically, there's no explicit t, 746 00:56:58,765 --> 00:57:02,310 and therefore this derivative goes away. 747 00:57:02,310 --> 00:57:11,090 So the equation becomes ih bar dAh, of course, 748 00:57:11,090 --> 00:57:21,406 dt is equal to Ah Heisenberg sub h of t. 749 00:57:21,406 --> 00:57:24,150 And you know the Heisenberg operator 750 00:57:24,150 --> 00:57:27,110 is supposed to be simpler. 751 00:57:27,110 --> 00:57:27,850 Simple. 752 00:57:27,850 --> 00:57:30,730 If the Schrodinger operator is time independent, 753 00:57:30,730 --> 00:57:34,810 it's identical to the Schrodinger Hamiltonian. 754 00:57:34,810 --> 00:57:39,020 Even if the Schrodinger operator has time dependence, 755 00:57:39,020 --> 00:57:43,620 but they commute, this will become the Schrodinger 756 00:57:43,620 --> 00:57:44,470 Hamiltonian. 757 00:57:44,470 --> 00:57:46,040 But we can leave it like that. 758 00:57:46,040 --> 00:57:50,870 It's a nice thing anyway. 759 00:57:50,870 --> 00:57:53,470 Time dependence of expectation value. 760 00:57:53,470 --> 00:57:59,841 So let me do a little remark on time dependence of expectation 761 00:57:59,841 --> 00:58:00,340 values. 762 00:58:00,340 --> 00:58:05,310 So suppose you have the usual thing that you want to compute. 763 00:58:05,310 --> 00:58:11,620 How does the expectation value of a Schrodinger operator 764 00:58:11,620 --> 00:58:12,880 depend on time? 765 00:58:15,590 --> 00:58:20,050 You're faced with that expectation value of As, 766 00:58:20,050 --> 00:58:22,670 and it changes in time, and you want 767 00:58:22,670 --> 00:58:24,530 to know how you can compute that. 768 00:58:24,530 --> 00:58:30,100 Well, you first say, OK, ih bar d dt. 769 00:58:30,100 --> 00:58:41,960 But this thing is nothing but psi 0 A Heisenberg of t psi 0. 770 00:58:41,960 --> 00:58:46,230 Now I can let the derivative go in. 771 00:58:46,230 --> 00:58:59,690 So this becomes psi 0 ih bar dAh dt psi 0. 772 00:58:59,690 --> 00:59:09,335 And using this, assuming that A is still no time dependence, 773 00:59:09,335 --> 00:59:18,700 A has no explicit time dependence, 774 00:59:18,700 --> 00:59:21,910 then you can use just this equation, 775 00:59:21,910 --> 00:59:31,935 which give you psi 0 Ah Hh psi 0. 776 00:59:34,940 --> 00:59:38,760 So all in all, what have you gotten? 777 00:59:44,840 --> 00:59:53,100 You've gotten a rather simple thing, the time derivative 778 00:59:53,100 --> 00:59:54,930 of the expectation values. 779 00:59:54,930 --> 00:59:57,900 So ih bar d dt. 780 00:59:57,900 --> 01:00:00,420 And now I write the left hand side 781 01:00:00,420 --> 01:00:06,780 as just expectation value of H Heisenberg of t. 782 01:00:10,588 --> 01:00:15,330 And on the left hand side has to the A Schrodinger expectation 783 01:00:15,330 --> 01:00:18,360 value, but we call those expectation values 784 01:00:18,360 --> 01:00:22,080 the same thing as a Heisenberg expectation value. 785 01:00:22,080 --> 01:00:30,350 So this thing becomes the right hand side is the expectation 786 01:00:30,350 --> 01:00:37,050 value of A Heisenberg H Heisenberg like that. 787 01:00:40,354 --> 01:00:47,690 And just the way we say that Heisenberg expectation 788 01:00:47,690 --> 01:00:50,140 values are the same as Schrodinger expectation values, 789 01:00:50,140 --> 01:00:52,760 you could as well write, if you prefer, 790 01:00:52,760 --> 01:00:59,600 as d dt of A Schrodinger is equal to the expectation 791 01:00:59,600 --> 01:01:02,756 value of A Schrodinger with H Schrodinger. 792 01:01:06,990 --> 01:01:09,720 It's really the same equation. 793 01:01:09,720 --> 01:01:14,240 This equation we derived a couple of lectures ago. 794 01:01:16,830 --> 01:01:19,570 And now we know that the expectation values 795 01:01:19,570 --> 01:01:22,450 of Schrodinger operators are the same as the expectation 796 01:01:22,450 --> 01:01:25,050 value of their Heisenberg counterparts, 797 01:01:25,050 --> 01:01:28,890 except that the states are taking up time equals 0. 798 01:01:28,890 --> 01:01:34,590 So you can use either form of this equation. 799 01:01:34,590 --> 01:01:38,590 The bottom one is one that you've already seen. 800 01:01:38,590 --> 01:01:43,340 The top one now looks almost obvious from the bottom one, 801 01:01:43,340 --> 01:01:48,140 but it really took quite a bit to get it. 802 01:01:48,140 --> 01:01:50,453 One last comment on these operators. 803 01:01:54,850 --> 01:01:56,900 How about conserved operators? 804 01:01:56,900 --> 01:01:58,950 What are those things? 805 01:01:58,950 --> 01:02:10,440 A time independent As is set to be conserved 806 01:02:10,440 --> 01:02:12,400 if it commutes with a Schrodinger Hamiltonian. 807 01:02:16,390 --> 01:02:27,190 If As commutes with As equals 0. 808 01:02:27,190 --> 01:02:41,830 Now you know that if As with Hs is 0, Ah with Hh is 0, 809 01:02:41,830 --> 01:02:45,950 because the map between Heisenberg and Schrodinger 810 01:02:45,950 --> 01:02:49,650 pictures is a commutator that is valued at the Schrodinger 811 01:02:49,650 --> 01:02:52,540 picture is valued in the Heisenberg picture by putting 812 01:02:52,540 --> 01:02:53,650 H's. 813 01:02:53,650 --> 01:02:58,020 So what you realize from this is that this thing, 814 01:02:58,020 --> 01:03:05,690 this implies Ah commutes with Hh. 815 01:03:05,690 --> 01:03:12,260 And therefore by point 1, by 1, you 816 01:03:12,260 --> 01:03:20,810 have to dAh dt is equal to 0. 817 01:03:20,810 --> 01:03:25,150 And this is nice, actually. 818 01:03:25,150 --> 01:03:32,530 The Heisenberg operator is actually time independent. 819 01:03:32,530 --> 01:03:34,590 It just doesn't depend on time. 820 01:03:34,590 --> 01:03:39,230 So a Schrodinger operator, it's a funny operator. 821 01:03:39,230 --> 01:03:41,020 It doesn't have time in there. 822 01:03:41,020 --> 01:03:47,790 It has X's, P's, spins, and you don't know in general, 823 01:03:47,790 --> 01:03:52,530 if it's time independent in the sense of conserve 824 01:03:52,530 --> 01:03:55,430 of expectation values. 825 01:03:55,430 --> 01:04:00,420 But whenever As commutes with Hs, 826 01:04:00,420 --> 01:04:04,740 well, the expectation values don't change in time. 827 01:04:04,740 --> 01:04:08,100 But as you know, this d dt can be brought in, 828 01:04:08,100 --> 01:04:10,740 because the states are not time dependent. 829 01:04:10,740 --> 01:04:14,420 So the fact that this is 0 means the operator, 830 01:04:14,420 --> 01:04:18,810 Heisenberg operator, is really time independent. 831 01:04:18,810 --> 01:04:22,710 Whenever you have a Schrodinger operator, has no t, 832 01:04:22,710 --> 01:04:25,830 the Heisenberg one can have a lot of t. 833 01:04:25,830 --> 01:04:29,460 But if the operator is conserved, 834 01:04:29,460 --> 01:04:33,640 then the Heisenberg operator will have no t's after all. 835 01:04:33,640 --> 01:04:37,260 It will really be conserved. 836 01:04:37,260 --> 01:04:42,640 So let's use our last 10 minutes to do an example 837 01:04:42,640 --> 01:04:46,140 and illustrate much of this. 838 01:04:46,140 --> 01:04:48,560 In the notes, there will be three examples. 839 01:04:48,560 --> 01:04:51,860 I will do just one in lecture. 840 01:04:51,860 --> 01:04:56,580 You can do the other ones in recitation next week. 841 01:04:56,580 --> 01:05:00,495 There's no need really that you study these things 842 01:05:00,495 --> 01:05:01,990 at this moment. 843 01:05:01,990 --> 01:05:05,410 Just try to get whatever you can now from the lecture, 844 01:05:05,410 --> 01:05:09,840 and next week you'll go back to this. 845 01:05:09,840 --> 01:05:13,050 So the example is the harmonic oscillator. 846 01:05:21,080 --> 01:05:25,130 And it will illustrate the ideas very nicely, I think. 847 01:05:25,130 --> 01:05:28,420 The Schrodinger Hamiltonian is p squared 848 01:05:28,420 --> 01:05:35,400 over 2m plus 1/2 m omega squared x hat squared. 849 01:05:35,400 --> 01:05:41,350 OK, I could put x Schrodinger and p Schrodinger, but that 850 01:05:41,350 --> 01:05:43,880 would be just far too much. x and p 851 01:05:43,880 --> 01:05:46,380 are the operators you've always known. 852 01:05:46,380 --> 01:05:48,470 They are Schrodinger operators. 853 01:05:48,470 --> 01:05:51,980 So we leave them like that. 854 01:05:51,980 --> 01:05:57,150 Now I have to write the Heisenberg Hamiltonian. 855 01:05:57,150 --> 01:06:00,490 Well, what is the Heisenberg Hamiltonian? 856 01:06:07,540 --> 01:06:08,520 yes? 857 01:06:08,520 --> 01:06:09,852 AUDIENCE: It's identical. 858 01:06:09,852 --> 01:06:10,560 PROFESSOR: Sorry? 859 01:06:10,560 --> 01:06:12,140 AUDIENCE: It's identical. 860 01:06:12,140 --> 01:06:14,800 PROFESSOR: Identical, yes. 861 01:06:14,800 --> 01:06:18,280 But I will leave that for a little later. 862 01:06:18,280 --> 01:06:21,670 I will just assume, well, I'm supposed 863 01:06:21,670 --> 01:06:26,340 to do U dagger U. As you said, this 864 01:06:26,340 --> 01:06:28,000 is a time independent Hamiltonian. 865 01:06:28,000 --> 01:06:29,960 It better be the same, but it will 866 01:06:29,960 --> 01:06:35,000 be clearer if we now write what it should be in general. 867 01:06:35,000 --> 01:06:37,650 Have a U dagger and a U from the right. 868 01:06:37,650 --> 01:06:43,550 They come here, and they turn this into P Heisenberg over 2m 869 01:06:43,550 --> 01:06:47,380 plus 1/2 m omega squared x Heisenberg. 870 01:06:50,300 --> 01:06:55,100 OK, that's your Heisenberg Hamiltonian. 871 01:06:55,100 --> 01:06:58,660 And we will check, in fact, that it's time independent. 872 01:06:58,660 --> 01:07:09,130 So how about the operators X Heisenberg and P Heisenberg. 873 01:07:09,130 --> 01:07:10,810 What are they? 874 01:07:10,810 --> 01:07:13,040 Well, I don't know how to get them, 875 01:07:13,040 --> 01:07:16,240 unless I do this sort of U thing. 876 01:07:16,240 --> 01:07:22,170 That doesn't look too bad but certainly would be messy. 877 01:07:22,170 --> 01:07:25,430 You would have to do an exponential of e 878 01:07:25,430 --> 01:07:30,225 to the minus iHt over t with the x operator 879 01:07:30,225 --> 01:07:31,990 and another exponential. 880 01:07:31,990 --> 01:07:34,290 Sounds a little complicated. 881 01:07:34,290 --> 01:07:41,150 So let's do it the way the equations of the Heisenberg 882 01:07:41,150 --> 01:07:42,740 operators tell you. 883 01:07:42,740 --> 01:07:46,570 Well, X and P are time independent Schrodinger 884 01:07:46,570 --> 01:07:49,950 operators, so that equation that I boxed holds. 885 01:07:49,950 --> 01:07:57,290 So ih dx Heisenberg dt is nothing else 886 01:07:57,290 --> 01:08:02,910 than X Heisenberg commuted with H Heisenberg. 887 01:08:02,910 --> 01:08:05,860 OK, can we do that commutator? 888 01:08:05,860 --> 01:08:07,030 Yes. 889 01:08:07,030 --> 01:08:09,580 Because X Heisenberg, as you remember, 890 01:08:09,580 --> 01:08:11,670 just commutes with P Heisenberg. 891 01:08:11,670 --> 01:08:15,422 So instead of the Hamiltonian, you can put this. 892 01:08:15,422 --> 01:08:20,720 This is X Heisenberg P Heisenberg squared over 2m. 893 01:08:27,352 --> 01:08:30,330 OK well, X Heisenberg P Heisenberg 894 01:08:30,330 --> 01:08:34,240 is like you had X and P. So what is this commutator? 895 01:08:34,240 --> 01:08:36,200 You probably know it by now. 896 01:08:36,200 --> 01:08:39,600 You act with this and these two p. 897 01:08:39,600 --> 01:08:43,350 So it acts on one, acts on the other, gives the same on each. 898 01:08:43,350 --> 01:08:49,100 So you get P Heisenberg times the commutator of X and P, 899 01:08:49,100 --> 01:08:53,830 which is ih bar times a factor of 2. 900 01:08:57,250 --> 01:09:00,760 So we could put hats. 901 01:09:00,760 --> 01:09:03,689 Better maybe. 902 01:09:03,689 --> 01:09:05,960 And then what do we get? 903 01:09:05,960 --> 01:09:09,069 The ih there and ih cancels. 904 01:09:09,069 --> 01:09:11,029 And we get some nice equation that 905 01:09:11,029 --> 01:09:17,960 says dX Heisenberg dt is 1 over m P Heisenberg. 906 01:09:21,430 --> 01:09:23,729 Well, it actually looks like an equation 907 01:09:23,729 --> 01:09:25,310 in classical mechanics. 908 01:09:25,310 --> 01:09:28,109 dx dt is P over m. 909 01:09:28,109 --> 01:09:31,529 So that's a good thing about the Heisenberg equations of motion. 910 01:09:31,529 --> 01:09:34,960 They look like ordinary equations 911 01:09:34,960 --> 01:09:37,890 for dynamical variables. 912 01:09:37,890 --> 01:09:39,560 Well, we've got this one. 913 01:09:39,560 --> 01:09:46,479 Let's get P. Well, we didn't get the operator still, 914 01:09:46,479 --> 01:09:49,569 but we got an equation. 915 01:09:49,569 --> 01:09:53,100 So how about P dP dt. 916 01:09:53,100 --> 01:10:03,060 So ih dP Heisenberg dt would be P Heisenberg with H Heisenberg. 917 01:10:03,060 --> 01:10:06,550 And this time only the potential term in here matters. 918 01:10:06,550 --> 01:10:11,090 So it's P Heisenberg with 1/2 m omega 919 01:10:11,090 --> 01:10:14,730 squared X Heisenberg squared. 920 01:10:14,730 --> 01:10:15,695 So what do we? 921 01:10:15,695 --> 01:10:20,340 We get 1/2 m omega squared. 922 01:10:20,340 --> 01:10:23,870 Then we get again a factor of 2. 923 01:10:23,870 --> 01:10:27,420 Then we get one left over Xh. 924 01:10:27,420 --> 01:10:33,310 And then a P with Xh, which is a minus ih bar. 925 01:10:33,310 --> 01:10:43,302 So the ih bars cancel, and we get dPh dt is equal to-- the h 926 01:10:43,302 --> 01:10:48,380 bar cancelled-- m omega squared Xh. 927 01:10:48,380 --> 01:10:49,250 Minus m. 928 01:10:56,960 --> 01:11:01,616 All right, so these are our Heisenberg equations of motion. 929 01:11:04,570 --> 01:11:08,350 So how do we solve for them now? 930 01:11:08,350 --> 01:11:12,570 Well, you sort of have to try the kind of things 931 01:11:12,570 --> 01:11:14,690 that you would do classically. 932 01:11:14,690 --> 01:11:18,140 Take a second derivative of this equation. 933 01:11:18,140 --> 01:11:29,430 d second Xh dt squared would be 1 over m dPh dt. 934 01:11:29,430 --> 01:11:32,550 And the dPh dt would be [INAUDIBLE] 935 01:11:32,550 --> 01:11:39,530 1 over m times minus m omega squared Xh. 936 01:11:39,530 --> 01:11:50,710 So d second Xh dt squared is equal to minus omega squared 937 01:11:50,710 --> 01:11:55,420 Xh, exactly the equation of motion 938 01:11:55,420 --> 01:11:57,050 of a harmonic oscillator. 939 01:12:01,010 --> 01:12:05,060 It's really absolutely nice that you 940 01:12:05,060 --> 01:12:09,350 recover those equations that you had before, 941 01:12:09,350 --> 01:12:14,310 except that now you're talking operators. 942 01:12:14,310 --> 01:12:17,970 And it's going to simplify your life quite dramatically when 943 01:12:17,970 --> 01:12:23,400 you try to use these operators, because, in a sense, 944 01:12:23,400 --> 01:12:28,840 solving for the time dependent Heisenberg operators 945 01:12:28,840 --> 01:12:33,340 is the same as finding the time evolution of all states. 946 01:12:33,340 --> 01:12:35,730 This time the operators change, and you 947 01:12:35,730 --> 01:12:38,020 will know what they change like. 948 01:12:38,020 --> 01:12:39,500 So you have this. 949 01:12:39,500 --> 01:12:48,120 And then you write Xh is equal to A cosine omega t plus B 950 01:12:48,120 --> 01:12:54,910 sine omega t where A and B are some time 951 01:12:54,910 --> 01:12:56,560 independent operators. 952 01:12:56,560 --> 01:13:04,360 So Xh of t, well, that's a solution. 953 01:13:04,360 --> 01:13:07,880 How about what is P? 954 01:13:07,880 --> 01:13:18,590 Ph of t would be m dX m dx dt. 955 01:13:18,590 --> 01:13:31,190 So you get minus m omega sine omega tA plus m omega 956 01:13:31,190 --> 01:13:33,100 cosine omega tB. 957 01:13:38,053 --> 01:13:40,480 OK, so that's it. 958 01:13:40,480 --> 01:13:42,210 That is the most general solution. 959 01:13:44,780 --> 01:13:50,280 But it still doesn't look like what you would want, does it? 960 01:13:50,280 --> 01:13:54,580 No, because you haven't used the time equals 0 conditions. 961 01:13:54,580 --> 01:13:57,430 At time equals 0, the Heisenberg operators 962 01:13:57,430 --> 01:14:00,310 are identical they to the Schrodinger operators. 963 01:14:00,310 --> 01:14:09,070 So at t equals 0, Xh of t becomes A, 964 01:14:09,070 --> 01:14:15,440 but that must be X hat, the Schrodinger operator. 965 01:14:15,440 --> 01:14:21,630 And at t equals 0, Ph of t becomes 966 01:14:21,630 --> 01:14:30,880 equal to this is 0 m omega B. And that 967 01:14:30,880 --> 01:14:35,580 must be equal to the P hat operator. 968 01:14:35,580 --> 01:14:43,220 So actually we have already now A and B. So B from here 969 01:14:43,220 --> 01:14:47,890 is P hat over m omega. 970 01:14:47,890 --> 01:14:55,040 And therefore Xh of t is equal to A, 971 01:14:55,040 --> 01:15:00,370 which is X hat cosine omega t plus B, which 972 01:15:00,370 --> 01:15:06,560 is P hat over m omega sine omega t. 973 01:15:06,560 --> 01:15:12,660 An Ph of t is here. 974 01:15:12,660 --> 01:15:23,780 A is-- Ph of t is m omega B, which is [INAUDIBLE] P. 975 01:15:23,780 --> 01:15:33,563 So it's P hat cosine omega t minus m omega X 976 01:15:33,563 --> 01:15:36,050 hat sine omega t. 977 01:15:42,640 --> 01:15:43,800 So let's see. 978 01:15:43,800 --> 01:15:46,380 I hope I didn't make mistakes. 979 01:15:46,380 --> 01:15:50,360 P hat minus m omega X hat sine omega t. 980 01:15:50,360 --> 01:15:53,000 Yep, this is correct. 981 01:15:53,000 --> 01:15:59,400 This is your whole solution for the Heisenberg operators. 982 01:15:59,400 --> 01:16:04,990 So any expectation value of any power of X and P 983 01:16:04,990 --> 01:16:08,630 that you will want to find its time dependence, 984 01:16:08,630 --> 01:16:11,490 just put those Heisenberg operators, 985 01:16:11,490 --> 01:16:15,630 and you will calculate things with states at time equals 0. 986 01:16:15,630 --> 01:16:18,360 It will become very easy. 987 01:16:18,360 --> 01:16:20,760 So the last thing I want to do is 988 01:16:20,760 --> 01:16:26,730 complete the promise that we had about what 989 01:16:26,730 --> 01:16:30,520 is the Heisenberg Hamiltonian. 990 01:16:30,520 --> 01:16:34,440 Well, we had the Heisenberg Hamiltonian there. 991 01:16:34,440 --> 01:16:38,300 And now we know the Heisenberg operators 992 01:16:38,300 --> 01:16:43,800 in terms of the Schrodinger one. 993 01:16:43,800 --> 01:16:53,630 So Hh of t is equal to Ph-- 1/2m Ph squared. 994 01:16:53,630 --> 01:17:01,550 So I have P hat cosine omega t minus m omega X hat sine omega 995 01:17:01,550 --> 01:17:15,120 t squared plus 1/2 m omega squared Xh squared. 996 01:17:15,120 --> 01:17:19,940 So X hat cosine omega t plus P hat 997 01:17:19,940 --> 01:17:27,150 over m omega sine omega t squared. 998 01:17:27,150 --> 01:17:30,810 So that's what the Heisenberg Hamiltonian is. 999 01:17:30,810 --> 01:17:36,010 So let's simplify this. 1000 01:17:36,010 --> 01:17:37,790 Well, let's square these things. 1001 01:17:37,790 --> 01:17:47,310 You have 1/2m cosine squared omega t P hat squared. 1002 01:17:47,310 --> 01:17:50,750 Let's do the square of this one. 1003 01:17:50,750 --> 01:17:57,340 You would have plus 1/2m m squared 1004 01:17:57,340 --> 01:18:07,440 omega squared sine squared omega t X squared. 1005 01:18:10,000 --> 01:18:12,340 And then we have the cross product, 1006 01:18:12,340 --> 01:18:17,465 which would be plus-- or actually minus 1/2m. 1007 01:18:20,120 --> 01:18:22,910 The product of these two things. 1008 01:18:22,910 --> 01:18:29,900 m omega sine omega t cosine omega t. 1009 01:18:29,900 --> 01:18:35,670 And you have Px plus XP. 1010 01:18:35,670 --> 01:18:45,000 OK, I squared the first terms. 1011 01:18:45,000 --> 01:18:46,410 Now the second one. 1012 01:18:46,410 --> 01:18:49,120 Well, let's square the P squared here. 1013 01:18:49,120 --> 01:18:51,540 What do we have? 1014 01:18:51,540 --> 01:18:57,950 1/2 m omega squared over m squared 1015 01:18:57,950 --> 01:19:06,120 omega squared sine squared of omega t P squared. 1016 01:19:06,120 --> 01:19:11,740 The x plus 1/2 m omega squared cosine 1017 01:19:11,740 --> 01:19:14,965 squared omega t X squared. 1018 01:19:17,740 --> 01:19:20,880 And the cross term. 1019 01:19:20,880 --> 01:19:37,990 Plus 1/2 m omega squared over m omega times cosine omega t 1020 01:19:37,990 --> 01:19:42,863 sine omega t XP plus PX. 1021 01:19:45,540 --> 01:19:49,740 A little bit of work, but what do we get? 1022 01:19:49,740 --> 01:19:52,330 Well, 1/2 m. 1023 01:19:52,330 --> 01:19:56,170 And here we must have 1/2 m, correct. 1024 01:19:56,170 --> 01:19:58,190 1/2 m. 1025 01:19:58,190 --> 01:20:01,040 Sine squared omega t P squared. 1026 01:20:01,040 --> 01:20:05,680 So this is equal 1/2 m P squared. 1027 01:20:08,370 --> 01:20:14,180 These one's, hall here you have 1/2 m omega squared. 1028 01:20:14,180 --> 01:20:18,440 So it's 1/2 m omega squared cosine and sine 1029 01:20:18,440 --> 01:20:21,930 squared is X hat squared. 1030 01:20:21,930 --> 01:20:26,430 And then here we have all being over 2. 1031 01:20:26,430 --> 01:20:30,970 And here omega over 2, same factors, same factors, 1032 01:20:30,970 --> 01:20:33,350 opposite signs. 1033 01:20:33,350 --> 01:20:35,460 Very good. 1034 01:20:35,460 --> 01:20:37,000 Schrodinger Hamiltonian. 1035 01:20:37,000 --> 01:20:41,340 So you confirm that this theoretical expectation 1036 01:20:41,340 --> 01:20:43,350 is absolutely correct. 1037 01:20:43,350 --> 01:20:45,100 And what's the meaning? 1038 01:20:45,100 --> 01:20:47,900 You have the Heisenberg Hamiltonian 1039 01:20:47,900 --> 01:20:51,000 written in terms of the Heisenberg variables. 1040 01:20:51,000 --> 01:20:57,040 But by the time you substitute these Heisenberg variables in, 1041 01:20:57,040 --> 01:21:01,130 it just becomes identical to the Schrodinger Hamiltonian. 1042 01:21:01,130 --> 01:21:03,530 All right, so that's all for today. 1043 01:21:03,530 --> 01:21:07,420 I hope to see in office hours in the coming days. 1044 01:21:07,420 --> 01:21:13,160 Be here Wednesday 12:30, maybe 12:25 would be better, 1045 01:21:13,160 --> 01:21:16,800 and we'll see you then. 1046 01:21:16,800 --> 01:21:18,350 [APPLAUSE]