1 00:00:00,499 --> 00:00:01,350 PROFESSOR: OK. 2 00:00:01,350 --> 00:00:07,490 Let's do another application. 3 00:00:07,490 --> 00:00:08,880 I leave one thing. 4 00:00:08,880 --> 00:00:13,280 So this is going to be-- but it's called Landau levels. 5 00:00:13,280 --> 00:00:15,310 And they're pretty interesting. 6 00:00:20,820 --> 00:00:27,740 So it's the problem of solving for the motion of an electrode 7 00:00:27,740 --> 00:00:28,805 in a magnetic field. 8 00:00:34,810 --> 00:00:40,900 So it's called Landau levels for the physicist Lev Landau 9 00:00:40,900 --> 00:00:48,340 from Russia that discovered or did this calculation first. 10 00:00:48,340 --> 00:00:54,580 So you have a mass m charge q, and the magnetic field 11 00:00:54,580 --> 00:00:56,340 b in the z direction. 12 00:01:08,810 --> 00:01:10,680 We will solve it. 13 00:01:10,680 --> 00:01:15,120 And I think you will find it pretty interesting. 14 00:01:15,120 --> 00:01:18,590 Well, you will be left with a little of an uneasy feeling, 15 00:01:18,590 --> 00:01:21,110 I think, at the end of the lecture, 16 00:01:21,110 --> 00:01:26,240 because the problem of gauging variance 17 00:01:26,240 --> 00:01:31,190 is so dramatic that the physics will look a little strange 18 00:01:31,190 --> 00:01:34,460 and a bit unrecognizable. 19 00:01:34,460 --> 00:01:38,840 So let me remind you that in classical physics, 20 00:01:38,840 --> 00:01:41,300 if you have a magnetic field, b, you 21 00:01:41,300 --> 00:01:47,660 can have electrons that perform circular orbits, 22 00:01:47,660 --> 00:01:51,230 and the main property of those orbits 23 00:01:51,230 --> 00:01:55,100 is that they all run at the same frequency 24 00:01:55,100 --> 00:02:02,210 called cyclotron frequency, qb over mc. 25 00:02:02,210 --> 00:02:12,970 So this is just Newton's law, and the force being v over c 26 00:02:12,970 --> 00:02:17,360 cross b should be a couple of lines 27 00:02:17,360 --> 00:02:22,550 for you to remember that particle in the magnetic field 28 00:02:22,550 --> 00:02:25,610 goes in circles with that angular frequency. 29 00:02:30,880 --> 00:02:39,130 So OK, there's lots of comments we could make about this, 30 00:02:39,130 --> 00:02:41,920 but let's assume we're going to solve 31 00:02:41,920 --> 00:02:43,780 for the motion of this particle. 32 00:02:43,780 --> 00:02:45,820 We're going to ignore spin. 33 00:02:45,820 --> 00:02:48,290 Sometimes spin is important. 34 00:02:48,290 --> 00:02:52,210 And when you have spin, you have a Zeeman effect, 35 00:02:52,210 --> 00:02:54,040 spin in a magnetic field. 36 00:02:54,040 --> 00:02:59,300 So that can change the energy levels by some quantities. 37 00:02:59,300 --> 00:03:01,030 Let's ignore it for the moment. 38 00:03:01,030 --> 00:03:02,930 It is sometimes relevant. 39 00:03:02,930 --> 00:03:07,190 Sometimes it's not relevant, and you can easily take care of it. 40 00:03:07,190 --> 00:03:12,190 So again, we're faced with having to represent 41 00:03:12,190 --> 00:03:14,330 the magnetic field. 42 00:03:14,330 --> 00:03:19,176 So you have the magnetic field [INAUDIBLE] of a, 43 00:03:19,176 --> 00:03:25,540 and we'll take a solution in which a this time is minus 44 00:03:25,540 --> 00:03:29,440 by 0 and 0. 45 00:03:29,440 --> 00:03:37,990 Remember, that b is dx ay minus dy ax. 46 00:03:37,990 --> 00:03:39,980 And therefore this works out. 47 00:03:39,980 --> 00:03:44,620 It has an ax component, the derivative with respect to y 48 00:03:44,620 --> 00:03:49,570 gives you the magnetic field b. 49 00:03:49,570 --> 00:03:53,020 So what happens to your Hamiltonian? 50 00:03:53,020 --> 00:04:07,740 Your Hamiltonian is now 1 over 2m px minus q over c times ax 51 00:04:07,740 --> 00:04:15,495 plus 1 over 2m py squared plus 1 over 2m pz squared. 52 00:04:19,970 --> 00:04:24,850 I will not consider z motion. 53 00:04:24,850 --> 00:04:26,770 It's too simple. 54 00:04:26,770 --> 00:04:28,360 It's not too interesting. 55 00:04:28,360 --> 00:04:32,620 It's just motion, plane waves in the z direction. 56 00:04:32,620 --> 00:04:35,290 It's not interesting. 57 00:04:35,290 --> 00:04:38,380 All the physics is really happening in the plane. 58 00:04:38,380 --> 00:04:40,840 It's the idea that we can constrain 59 00:04:40,840 --> 00:04:43,670 ourselves to have orbits of electrons 60 00:04:43,670 --> 00:04:45,220 that go like this in circles. 61 00:04:48,080 --> 00:04:48,810 OK. 62 00:04:48,810 --> 00:04:52,060 So our Hamiltonian is here. 63 00:04:52,060 --> 00:04:57,840 Let's write it once again, 2m px plus 64 00:04:57,840 --> 00:05:07,725 q by over c squared plus 1 over 2m py squared. 65 00:05:23,560 --> 00:05:29,230 In order to solve a system like that, 66 00:05:29,230 --> 00:05:39,700 it's convenient to see what is conserved, and px is conserved. 67 00:05:39,700 --> 00:05:42,160 px commutes with the Hamiltonian, 68 00:05:42,160 --> 00:05:47,110 because the Hamiltonian has no x dependence. 69 00:05:47,110 --> 00:05:51,310 So h commutes with px. 70 00:05:56,240 --> 00:06:02,800 It doesn't commute with py, because there is a y in there. 71 00:06:02,800 --> 00:06:07,120 And, OK, so if it commutes with px, 72 00:06:07,120 --> 00:06:10,720 we can hope for eigenstates, energy eigenstates that 73 00:06:10,720 --> 00:06:15,100 are also eigenstates of px. 74 00:06:15,100 --> 00:06:21,190 So I will write my wave function of x and y 75 00:06:21,190 --> 00:06:29,740 as a wave function that depends on y times e to the i kx x. 76 00:06:29,740 --> 00:06:33,770 And it's already a little strange. 77 00:06:33,770 --> 00:06:37,210 We're looking for circular orbits maybe, 78 00:06:37,210 --> 00:06:42,310 and the x dependence is really a little funny here. 79 00:06:42,310 --> 00:06:45,430 It's almost like plane waves in the x direction. 80 00:06:47,960 --> 00:06:50,250 OK, well, it's a fact. 81 00:06:50,250 --> 00:06:53,110 It's true. 82 00:06:53,110 --> 00:06:55,480 You could not do that if you would have 83 00:06:55,480 --> 00:06:59,872 chosen a more symmetric gauge. 84 00:06:59,872 --> 00:07:04,060 You see, if you had chosen a more symmetric gauge, 85 00:07:04,060 --> 00:07:09,820 in which a doesn't have just an x component, but a y component. 86 00:07:09,820 --> 00:07:13,150 The y component would have depended on x, 87 00:07:13,150 --> 00:07:16,690 and then x would-- px and py, neither one 88 00:07:16,690 --> 00:07:18,940 would have commuted with the Hamiltonian. 89 00:07:18,940 --> 00:07:20,840 So things would have been different. 90 00:07:20,840 --> 00:07:25,420 But this gauge is easy to solve equations. 91 00:07:25,420 --> 00:07:28,000 It's called a Landau gauge. 92 00:07:28,000 --> 00:07:31,450 And let's just explore what this says. 93 00:07:31,450 --> 00:07:33,910 Trust that if you're solving the equations, 94 00:07:33,910 --> 00:07:38,740 and you get a legal solution, it must mean the right thing. 95 00:07:38,740 --> 00:07:50,670 So here, in this solution here, px is just h bar kx, 96 00:07:50,670 --> 00:07:55,390 and though we can restrict the action of the Hamiltonian 97 00:07:55,390 --> 00:08:02,530 to that subspace of momentum, px equal h kx, 98 00:08:02,530 --> 00:08:04,600 we can look at the Hamiltonian, it 99 00:08:04,600 --> 00:08:07,150 is like looking at the Hamiltonian for this wave 100 00:08:07,150 --> 00:08:10,600 function, we already constrain ourselves 101 00:08:10,600 --> 00:08:13,190 to have this kind of momentum. 102 00:08:13,190 --> 00:08:16,540 So the Hamiltonian that acts on the y function 103 00:08:16,540 --> 00:08:22,150 is the Hamiltonian constrained to the momentum kx, 104 00:08:22,150 --> 00:08:30,850 and it's equal to 1 over 2 m py squared, the second term, 105 00:08:30,850 --> 00:08:39,981 plus 1 over 2 m qb over cy plus h kx. 106 00:08:44,250 --> 00:08:51,420 It's the q by over c plus the value of px, which is h kx, 107 00:08:51,420 --> 00:08:52,780 and I should square it. 108 00:08:57,380 --> 00:09:02,150 If you look at it carefully, you see 109 00:09:02,150 --> 00:09:08,380 that this is a simple harmonic oscillator, Hamiltonian. 110 00:09:08,380 --> 00:09:14,250 There's py squared, and there's a y where the origin shifted. 111 00:09:14,250 --> 00:09:16,100 It's not just a y squared, but it's 112 00:09:16,100 --> 00:09:20,470 some sort of y minus y0 at some point squared. 113 00:09:24,400 --> 00:09:27,660 So let's write it like that. 114 00:09:27,660 --> 00:09:33,300 So I have to make it look like a harmonic oscillator, py 115 00:09:33,300 --> 00:09:36,000 squared plus 1 over 2. 116 00:09:36,000 --> 00:09:39,420 For a harmonic oscillator, I should have an m here. 117 00:09:39,420 --> 00:09:42,300 So I'm going to have an m squared in the denominator, 118 00:09:42,300 --> 00:09:43,580 so m. 119 00:09:43,580 --> 00:09:53,600 Let's get the qb mc squared out. 120 00:09:53,600 --> 00:10:00,200 y, I got that out, minus-- 121 00:10:00,200 --> 00:10:02,930 I have to put all these things. 122 00:10:02,930 --> 00:10:08,240 Minus h bar kx over-- 123 00:10:08,240 --> 00:10:12,590 times the c over qb. 124 00:10:12,590 --> 00:10:13,670 Look at all that. 125 00:10:16,800 --> 00:10:22,010 So we factor out qb over c. 126 00:10:22,010 --> 00:10:24,870 With the m squared here, and the m-- 127 00:10:24,870 --> 00:10:27,060 this is the 1 over m there. 128 00:10:27,060 --> 00:10:31,140 y, and I put two minus signs, because I always 129 00:10:31,140 --> 00:10:36,150 like to write y minus some y0, which is the equilibrium 130 00:10:36,150 --> 00:10:38,270 position of the oscillator. 131 00:10:38,270 --> 00:10:41,310 Here, as you pull the c and the qb out, 132 00:10:41,310 --> 00:10:44,500 you get all these things. 133 00:10:44,500 --> 00:10:56,130 So this is a p squared y over 2n, plus 1 over m qb 134 00:10:56,130 --> 00:11:07,950 over mc squared y minus y0 squared with y0 equal minus h 135 00:11:07,950 --> 00:11:11,190 bar k xc over qb. 136 00:11:16,580 --> 00:11:17,760 That's it. 137 00:11:17,760 --> 00:11:21,280 That's the system. 138 00:11:24,960 --> 00:11:27,540 One good thing happened already. 139 00:11:27,540 --> 00:11:32,550 While this thing is a little funny or strange, I think, 140 00:11:32,550 --> 00:11:36,360 at first sight, this harmonic oscillator 141 00:11:36,360 --> 00:11:39,900 is resonating with an omega, which 142 00:11:39,900 --> 00:11:43,120 is the cyclotron frequency. 143 00:11:43,120 --> 00:11:44,470 So that's nice. 144 00:11:44,470 --> 00:11:47,460 The harmonic oscillator is that. 145 00:11:47,460 --> 00:11:49,320 And therefore, already, you know you're 146 00:11:49,320 --> 00:11:54,120 going to get levels that are going to be separated by h bar 147 00:11:54,120 --> 00:11:56,520 omega cyclotron. 148 00:11:56,520 --> 00:12:00,120 So that classical cyclotron frequency 149 00:12:00,120 --> 00:12:03,660 is going to separate the levels, and these levels 150 00:12:03,660 --> 00:12:09,500 are called the Landau levels, the various Landau levels. 151 00:12:09,500 --> 00:12:16,740 OK, so let's see what this means, or how it looks like. 152 00:12:16,740 --> 00:12:26,320 For that, let's imagine a solution. 153 00:12:26,320 --> 00:12:29,400 So how does a solution look? 154 00:12:33,190 --> 00:12:34,880 So here is x and y. 155 00:12:41,240 --> 00:12:49,910 And suppose I take a kx, which is negative. 156 00:12:49,910 --> 00:12:54,080 This kx is any number you wish at this moment. 157 00:12:54,080 --> 00:12:59,010 So kx negative means y0 positive. 158 00:12:59,010 --> 00:13:03,140 So here is the y0 is here. 159 00:13:06,900 --> 00:13:12,370 And the wave function is certainly the probability 160 00:13:12,370 --> 00:13:14,820 densities independent of x. 161 00:13:14,820 --> 00:13:19,870 So the wave function sort of has support all over x here. 162 00:13:23,930 --> 00:13:30,220 And it represents an oscillator in the y direction. 163 00:13:30,220 --> 00:13:33,250 In the y direction, you're oscillating, 164 00:13:33,250 --> 00:13:36,790 and well, you're going to oscillate a little. 165 00:13:36,790 --> 00:13:40,660 We know, in particular, is a length scale associated 166 00:13:40,660 --> 00:13:41,800 with the oscillator. 167 00:13:41,800 --> 00:13:45,660 So let's look at that length scale. 168 00:13:45,660 --> 00:13:51,430 The length scale on an oscillator h bar over m omega, 169 00:13:51,430 --> 00:13:52,280 in general. 170 00:13:52,280 --> 00:13:54,820 But now, we have an expression for omega. 171 00:13:54,820 --> 00:14:04,750 So h bar over m, and omega is qb over mc. 172 00:14:04,750 --> 00:14:13,120 So this is square root of hc over qb. 173 00:14:13,120 --> 00:14:16,750 And we're going to call this the magnetic length. 174 00:14:16,750 --> 00:14:22,540 So this is for an oscillator, an arbitrary oscillator 175 00:14:22,540 --> 00:14:23,690 takes these values. 176 00:14:23,690 --> 00:14:31,800 So let's call the magnetic length hc over qb. 177 00:14:31,800 --> 00:14:35,335 This magnetic length. 178 00:14:42,290 --> 00:14:45,040 So that's a nice definition. 179 00:14:45,040 --> 00:14:49,810 And that's roughly the width of the state. 180 00:14:49,810 --> 00:14:51,820 Imagine you're in the ground state, 181 00:14:51,820 --> 00:14:55,990 you oscillate from the harmonic oscillator, the typical length 182 00:14:55,990 --> 00:14:59,540 scale, and in this case, is the magnetic length. 183 00:14:59,540 --> 00:15:05,646 So in the ground state, this size is lb. 184 00:15:05,646 --> 00:15:09,460 And that's how your orbit looks. 185 00:15:09,460 --> 00:15:15,650 In particular, y0 has a simple interpretation here. 186 00:15:15,650 --> 00:15:27,247 y0 is equal to minus kx times the square 187 00:15:27,247 --> 00:15:33,760 of this magnetic length is a [? right ?] squared. 188 00:15:33,760 --> 00:15:41,740 So it's minus kx times lb squared. 189 00:15:41,740 --> 00:15:45,830 And the units work out because k has units of 1 over length, 190 00:15:45,830 --> 00:15:48,340 and you have a length squared. 191 00:15:48,340 --> 00:15:50,420 So that's y0. 192 00:15:50,420 --> 00:15:58,180 So that's how the orbits look. 193 00:15:58,180 --> 00:16:01,540 Now, you would say OK, so where are the circular orbits? 194 00:16:01,540 --> 00:16:04,810 Well, they're not quite so visible here. 195 00:16:04,810 --> 00:16:08,000 You have to do some work to find them. 196 00:16:08,000 --> 00:16:11,740 And in particular, what is happening here 197 00:16:11,740 --> 00:16:15,370 is a very strange thing. 198 00:16:15,370 --> 00:16:21,100 I think-- well, I'll say one more thing. 199 00:16:21,100 --> 00:16:25,060 If you have this as your harmonic oscillator, 200 00:16:25,060 --> 00:16:33,460 your energies, that may depend of kx and ny, 201 00:16:33,460 --> 00:16:36,400 those are the quantum numbers you have already. 202 00:16:36,400 --> 00:16:39,230 Well, this is just a harmonic oscillator. 203 00:16:39,230 --> 00:16:42,940 So it's h bar, the cyclotron frequency. 204 00:16:42,940 --> 00:16:45,430 Some people call the Landau frequency. 205 00:16:48,610 --> 00:16:53,140 Occupation number for the oscillator times 1/2, 206 00:16:53,140 --> 00:16:56,440 the usual formula for a harmonic oscillator. 207 00:16:59,140 --> 00:17:02,090 And here is the energy. 208 00:17:02,090 --> 00:17:04,440 And well, you had the plane wave. 209 00:17:04,440 --> 00:17:10,420 Why don't we have the p squared over 2m of a plane wave? 210 00:17:10,420 --> 00:17:13,109 It's nowhere to be found. 211 00:17:13,109 --> 00:17:16,500 That doesn't show and doesn't contribute to the energy. 212 00:17:16,500 --> 00:17:19,770 The mathematics was shown for us. 213 00:17:19,770 --> 00:17:25,349 So this kind of dependence doesn't do anything. 214 00:17:25,349 --> 00:17:32,590 It's the absolute degeneracy, the kx. 215 00:17:32,590 --> 00:17:36,240 So you could construct the superposition of states 216 00:17:36,240 --> 00:17:39,840 here, using Fourier transforms in which you somehow 217 00:17:39,840 --> 00:17:44,310 localize this in the x direction, 218 00:17:44,310 --> 00:17:49,080 by superimposing degenerate energy eigenstate. 219 00:17:49,080 --> 00:17:54,210 So this is the most important thing about this Landau levels. 220 00:17:54,210 --> 00:17:57,690 The Landau levels are the different levels of ny, 221 00:17:57,690 --> 00:18:01,860 so ny equals 0 is the lowest Landau level. 222 00:18:01,860 --> 00:18:06,360 Then you go ny equal 1, ny equal 2, ny equal 3. 223 00:18:06,360 --> 00:18:11,490 But each Landau level is infinitely degenerate 224 00:18:11,490 --> 00:18:16,260 because you can put different values of kx, 225 00:18:16,260 --> 00:18:18,871 and the energy doesn't change.