1 00:00:00,760 --> 00:00:04,449 PROFESSOR: SHO algebraically. 2 00:00:08,940 --> 00:00:17,300 And we go back to the Hamiltonian, p squared over 2m 3 00:00:17,300 --> 00:00:23,988 plus 1/2 m omega squared x hat squared. 4 00:00:27,130 --> 00:00:31,430 And what we do is observe that this some sort 5 00:00:31,430 --> 00:00:45,550 of sum of squares plus p squared over m-- 6 00:00:48,959 --> 00:00:53,615 p squared over m squared omega squared. 7 00:01:01,960 --> 00:01:03,540 So the sum of two things squared. 8 00:01:06,350 --> 00:01:12,740 Now, the idea that we have now is 9 00:01:12,740 --> 00:01:16,652 to try to vectorize the Hamiltonian. 10 00:01:17,900 --> 00:01:25,970 And what we call vectorizing is when you write your Hamiltonian 11 00:01:25,970 --> 00:01:33,410 as the product of two vectors, V times W. Well actually, 12 00:01:33,410 --> 00:01:35,330 that's not quite the vectorization. 13 00:01:35,330 --> 00:01:42,510 You want kind of the same vector, and not even that. 14 00:01:42,510 --> 00:01:48,365 You sort of want this to be the Hermitian conjugate of that. 15 00:01:48,365 --> 00:01:52,970 And if there is a number here, that's OK. 16 00:01:52,970 --> 00:01:58,220 Adding numbers to a Hamiltonian doesn't change the problem 17 00:01:58,220 --> 00:01:59,380 at all. 18 00:01:59,380 --> 00:02:02,250 The energies are all shifted, and it's 19 00:02:02,250 --> 00:02:05,920 just how you're defining the zero of your potential, 20 00:02:05,920 --> 00:02:08,850 is doing nothing but that. 21 00:02:08,850 --> 00:02:14,860 So vectorizing the Hamiltonian is writing it in this way, 22 00:02:14,860 --> 00:02:22,580 as V dagger V. And you would say, why V dagger V? 23 00:02:22,580 --> 00:02:27,180 Why not VV dagger or VV or V dagger V dagger? 24 00:02:27,180 --> 00:02:32,640 Well, you want the Hamiltonian to be Hermitian. 25 00:02:32,640 --> 00:02:35,730 And this thing is Hermitian. 26 00:02:35,730 --> 00:02:40,532 You may recall that AB dagger. 27 00:02:40,532 --> 00:02:45,880 The Hermitian conjugate of AB dagger is B dagger A dagger. 28 00:02:48,684 --> 00:02:51,895 So the Hermitian conjugate of this product 29 00:02:51,895 --> 00:02:56,940 is V dagger times the dagger of V dagger. 30 00:02:56,940 --> 00:03:01,390 A dagger of a dagger is the same operator, when you dagger it 31 00:03:01,390 --> 00:03:03,040 twice, you get the same. 32 00:03:03,040 --> 00:03:05,260 So this is Hermitian. 33 00:03:05,260 --> 00:03:09,420 V dagger times V is a Hermitian operator, 34 00:03:09,420 --> 00:03:12,440 and that's a very good thing. 35 00:03:12,440 --> 00:03:15,520 And there will be great simplifications. 36 00:03:15,520 --> 00:03:20,830 If you ever succeed in writing a Hamiltonian this way, 37 00:03:20,830 --> 00:03:27,130 you've gone 90% of the way to solving the whole problem. 38 00:03:27,130 --> 00:03:31,510 It has become infinitely easier, as you will see in a second, 39 00:03:31,510 --> 00:03:36,910 if you could just write this vectorization. 40 00:03:36,910 --> 00:03:43,220 So if you had x minus-- 41 00:03:43,220 --> 00:03:47,050 x squared minus this, you would say, oh, clearly that's-- 42 00:03:47,050 --> 00:03:51,230 A squared minus B squared is A minus B time A plus B, 43 00:03:51,230 --> 00:03:53,440 but there's no such thing here. 44 00:03:53,440 --> 00:03:59,660 It's almost like A squared plus B squared. 45 00:03:59,660 --> 00:04:02,230 And how do you sort of factorize it? 46 00:04:02,230 --> 00:04:05,450 Well, actually, since we have complex numbers, 47 00:04:05,450 --> 00:04:10,460 this could be A minus IB times A plus IB. 48 00:04:14,490 --> 00:04:18,540 That is correctly A squared plus B squared, 49 00:04:18,540 --> 00:04:24,130 and complex numbers are supposed to be friends in quantum 50 00:04:24,130 --> 00:04:30,670 mechanics, so having Is, there's probably no complication there. 51 00:04:30,670 --> 00:04:33,370 So let's try that. 52 00:04:33,370 --> 00:04:34,280 I'll write it. 53 00:04:34,280 --> 00:04:41,220 So here we have x squared plus p squared 54 00:04:41,220 --> 00:04:43,720 over m squared omega squared. 55 00:04:43,720 --> 00:04:52,232 And I will try to write it as x minus i p hat over m omega 56 00:04:52,232 --> 00:04:59,030 times x plus I p hat over m omega. 57 00:05:01,690 --> 00:05:04,390 Let's put the question mark before we 58 00:05:04,390 --> 00:05:06,015 are so sure that this works. 59 00:05:09,030 --> 00:05:11,100 Well, some things work. 60 00:05:14,670 --> 00:05:17,720 The only danger here is that these are operators 61 00:05:17,720 --> 00:05:18,910 and they don't commute. 62 00:05:18,910 --> 00:05:24,120 And when we do this, in one case, in the cross-terms, 63 00:05:24,120 --> 00:05:28,995 the A is to the left of B, but the other problem the B 64 00:05:28,995 --> 00:05:32,970 is to the left of A. So we may run into some trouble. 65 00:05:32,970 --> 00:05:36,810 This may not be exactly true. 66 00:05:36,810 --> 00:05:38,125 So what is this? 67 00:05:40,795 --> 00:05:43,300 This x with x, fine. 68 00:05:43,300 --> 00:05:45,930 x squared. 69 00:05:45,930 --> 00:05:49,320 This term, p with ps, correct. 70 00:05:49,320 --> 00:05:53,410 Plus p squared over m squared omega squared. 71 00:05:53,410 --> 00:05:59,540 But then we get plus i over m omega, 72 00:05:59,540 --> 00:06:08,740 x with p minus p with x, so that x, p commutator. 73 00:06:08,740 --> 00:06:13,490 So vectorization of operators in quantum mechanics 74 00:06:13,490 --> 00:06:17,600 can miss a few concepts because things don't commute. 75 00:06:17,600 --> 00:06:21,310 So the cross-terms give you that, 76 00:06:21,310 --> 00:06:30,500 and this x, p is I h bar, so this whole term will give us 77 00:06:30,500 --> 00:06:32,010 the following statement. 78 00:06:36,100 --> 00:06:45,710 What we've learned is that what we wanted, 79 00:06:45,710 --> 00:06:52,990 x squared plus p squared over m squared 80 00:06:52,990 --> 00:06:57,910 omega squared is equal to-- 81 00:06:57,910 --> 00:07:03,250 so I'm equating this line to the top line-- 82 00:07:03,250 --> 00:07:10,990 is equal to x hat minus i p hat over m omega times 83 00:07:10,990 --> 00:07:16,150 x hat plus i p hat over m omega. 84 00:07:18,890 --> 00:07:27,040 And then, from this whole term, i with i is minus, 85 00:07:27,040 --> 00:07:32,290 so it's h bar over m omega. 86 00:07:32,290 --> 00:07:36,521 So I'll put it in-- 87 00:07:36,521 --> 00:07:39,400 it's a minus h bar over m, [INAUDIBLE]. 88 00:07:39,400 --> 00:07:46,650 So here is plus h bar over m omega times 89 00:07:46,650 --> 00:07:49,030 a unit vector, if you wish. 90 00:07:55,356 --> 00:07:55,855 OK. 91 00:07:58,990 --> 00:08:02,950 So this is very good. 92 00:08:02,950 --> 00:08:14,150 In fact, we can call this V dagger and this V. Better 93 00:08:14,150 --> 00:08:17,150 call this V first and then ask, what 94 00:08:17,150 --> 00:08:20,640 is the dagger of this operator? 95 00:08:20,640 --> 00:08:24,250 Now, you may remember that, how did we define daggers? 96 00:08:24,250 --> 00:08:29,010 If you have phi with psi and the inner product-- 97 00:08:29,010 --> 00:08:31,920 with an integral of five star psi-- 98 00:08:31,920 --> 00:08:36,280 if you have an A psi here, that's 99 00:08:36,280 --> 00:08:41,710 equal to A dagger phi psi. 100 00:08:43,530 --> 00:08:46,810 So an operator is acting on the second wave 101 00:08:46,810 --> 00:08:51,030 function, moves as A dagger into the first wave function. 102 00:08:51,030 --> 00:08:56,640 And you know that x moves without any problem. 103 00:08:56,640 --> 00:08:58,020 x is Hermitian. 104 00:08:58,020 --> 00:09:01,730 We've discussed that p is Hermitian as well, 105 00:09:01,730 --> 00:09:03,240 moves to the other side. 106 00:09:03,240 --> 00:09:08,540 So the Hermitian conjugate of this operator 107 00:09:08,540 --> 00:09:14,460 is x, the p remains means p, but the i becomes minus i. 108 00:09:14,460 --> 00:09:15,780 So this is correct. 109 00:09:15,780 --> 00:09:20,280 If this second operator is called V, 110 00:09:20,280 --> 00:09:23,310 the first operator should be called V dagger. 111 00:09:23,310 --> 00:09:25,680 That is a correct statement. 112 00:09:25,680 --> 00:09:29,880 One is the dagger of the other one. 113 00:09:29,880 --> 00:09:37,400 So the Hamiltonian is 1/2 m omega 114 00:09:37,400 --> 00:09:42,940 squared times this sum of squares, 115 00:09:42,940 --> 00:09:51,130 which is now equal to V dagger V plus h bar over m omega. 116 00:09:55,280 --> 00:10:00,730 So h hat is now 1/2 m omega squared 117 00:10:00,730 --> 00:10:11,100 V dagger V plus a sum, which is plus 1/2 h bar omega. 118 00:10:16,920 --> 00:10:17,850 So we did it. 119 00:10:17,850 --> 00:10:23,400 We vectorized the Hamiltonian V dagger V, 120 00:10:23,400 --> 00:10:28,510 and this is quite useful. 121 00:10:28,510 --> 00:10:34,360 So the Vs, however, have units. 122 00:10:34,360 --> 00:10:39,300 And you probably are aware that we like things without units, 123 00:10:39,300 --> 00:10:42,440 so that we can see the units better. 124 00:10:42,440 --> 00:10:44,410 This curve is perfectly nice. 125 00:10:44,410 --> 00:10:46,620 It's a number added to the Hamiltonian. 126 00:10:46,620 --> 00:10:49,780 It's h omega, it has units of energy, 127 00:10:49,780 --> 00:10:53,370 but this is still a little messy. 128 00:10:53,370 --> 00:10:59,810 So let's try to clean up those Vs, and the way I'll do it 129 00:10:59,810 --> 00:11:03,631 is by computing their commutator, to begin with. 130 00:11:07,690 --> 00:11:15,520 So let's compute the commutator of V and V dagger 131 00:11:15,520 --> 00:11:18,312 and see how much is that commutator. 132 00:11:21,270 --> 00:11:24,360 It's a simple commutator, because it involves vectors 133 00:11:24,360 --> 00:11:31,040 of x and V. So it's the commutator 134 00:11:31,040 --> 00:11:39,180 of x plus ip over m omega, that's V, 135 00:11:39,180 --> 00:11:43,537 with x minus ip over m omega. 136 00:11:47,530 --> 00:11:54,110 So the first x talks only to the second piece, 137 00:11:54,110 --> 00:11:59,200 so it's minus i over m omega x, p. 138 00:12:02,960 --> 00:12:05,970 And for the second case, you have plus 139 00:12:05,970 --> 00:12:09,180 i over am omega p with x. 140 00:12:12,996 --> 00:12:20,960 This is i h bar, and this is minus i h bar. 141 00:12:20,960 --> 00:12:25,050 Each term will contribute the same, i times minus i 142 00:12:25,050 --> 00:12:32,610 is plus, so h bar over and, omega times the 2. 143 00:12:32,610 --> 00:12:40,340 That is V dagger V. VV dagger, I'm sorry. 144 00:12:40,340 --> 00:12:43,512 2 h bar over m omega. 145 00:12:47,290 --> 00:12:52,040 So time to change names a little bit. 146 00:12:52,040 --> 00:12:53,980 Let's do the following. 147 00:12:53,980 --> 00:13:01,880 Let's put square root of m omega over 2 h bar V. 148 00:13:01,880 --> 00:13:07,971 Have a square root of m omega over 2 h bar V dagger, 149 00:13:07,971 --> 00:13:09,740 commute to give you 1. 150 00:13:09,740 --> 00:13:12,190 That's a nice commutator. 151 00:13:12,190 --> 00:13:16,206 It's one number-- or an operator is the same thing. 152 00:13:18,990 --> 00:13:21,470 So I brought the square root into each one. 153 00:13:24,710 --> 00:13:29,810 And we'll call the first term-- 154 00:13:29,810 --> 00:13:31,940 because of reasons we'll see very soon-- 155 00:13:36,080 --> 00:13:43,560 the destruction operator, A square root of m omega over 2 h 156 00:13:43,560 --> 00:13:50,438 bar V. It's called the destruction operator. 157 00:13:50,438 --> 00:13:54,560 And the dagger is going to be A dagger. 158 00:13:54,560 --> 00:13:58,360 Some people put hats on them. 159 00:13:58,360 --> 00:14:03,810 I sometimes do too, unless I'm too tired. 160 00:14:03,810 --> 00:14:05,370 2h bar V dagger. 161 00:14:08,300 --> 00:14:12,600 And those A and A daggers are now unit-free-- 162 00:14:12,600 --> 00:14:15,420 and you can check That-- 163 00:14:15,420 --> 00:14:18,570 Because they have the same units. 164 00:14:18,570 --> 00:14:23,890 And A with A dagger is the nicest commutator, 1. 165 00:14:31,470 --> 00:14:33,880 Is A a Hermitian operator? 166 00:14:36,660 --> 00:14:38,150 Is it? 167 00:14:38,150 --> 00:14:39,060 No. 168 00:14:39,060 --> 00:14:40,410 A is not Hermitian. 169 00:14:40,410 --> 00:14:45,820 A dagger is different from A. A is basically this thing, 170 00:14:45,820 --> 00:14:47,400 A dagger is this thing. 171 00:14:47,400 --> 00:14:49,350 So not Hermitian. 172 00:14:49,350 --> 00:14:52,204 So we're going to work with these operators. 173 00:14:52,204 --> 00:14:53,120 They're non-Hermitian. 174 00:14:53,120 --> 00:14:56,900 I need to write the following equations. 175 00:14:56,900 --> 00:15:01,080 It's very-- takes a little bit of writing, 176 00:15:01,080 --> 00:15:05,370 but they should be recorded, they will always 177 00:15:05,370 --> 00:15:08,000 make it to the formula sheet. 178 00:15:08,000 --> 00:15:12,770 And it's the basic relation between A, A dagger, and x 179 00:15:12,770 --> 00:15:16,110 and p. 180 00:15:16,110 --> 00:15:21,160 A is this, A dagger, as you know, 181 00:15:21,160 --> 00:15:26,760 is x minus ip hat over m omega. 182 00:15:26,760 --> 00:15:30,410 Since I'm copying, I'd better copy them right. 183 00:15:30,410 --> 00:15:33,150 x, on other hand, is the square root 184 00:15:33,150 --> 00:15:39,190 of h bar over 2m omega A plus A dagger, 185 00:15:39,190 --> 00:15:48,320 and p is equal to i square root of m omega h bar over 2 186 00:15:48,320 --> 00:15:50,844 A dagger minus A. 187 00:15:52,280 --> 00:15:55,460 So these four equations, A and A dagger 188 00:15:55,460 --> 00:16:02,390 in terms of x and p and vice versa, are important. 189 00:16:02,390 --> 00:16:05,870 They will show up all the time. 190 00:16:05,870 --> 00:16:07,470 Here are the things to notice. 191 00:16:07,470 --> 00:16:10,290 A and A dagger is visibly clear that 192 00:16:10,290 --> 00:16:14,830 on is the Hermitian conjugate of the other. 193 00:16:14,830 --> 00:16:18,055 Here, x is Hermitian. 194 00:16:18,055 --> 00:16:20,950 And indeed, A plus A dagger is Hermitian. 195 00:16:20,950 --> 00:16:24,780 When you do the Hermitian conjugate of A plus A dagger, 196 00:16:24,780 --> 00:16:27,460 the first A becomes an A dagger. 197 00:16:27,460 --> 00:16:30,970 The second A, with another Hermitian conjugation, 198 00:16:30,970 --> 00:16:34,110 becomes A. So this is Hermitian. 199 00:16:34,110 --> 00:16:38,100 But p is Hermitian, and here we have A dagger minus A. This is 200 00:16:38,100 --> 00:16:40,140 not Hermitian, it changes sign. 201 00:16:40,140 --> 00:16:46,560 Well, the i is there for that reason, and makes it Hermition. 202 00:16:46,560 --> 00:16:50,870 So there they are, they're Hermitian, they're good.