1 00:00:02,210 --> 00:00:04,600 Welcome back everyone, to exploring Quantum physics. 2 00:00:04,600 --> 00:00:05,700 I'm Charles Clark. 3 00:00:05,700 --> 00:00:08,390 And today you're going to follow the bouncing ball. 4 00:00:10,380 --> 00:00:16,740 So just to remind you we're working with special functions and today, 5 00:00:16,740 --> 00:00:21,440 this part we're going to especially be. involve with the ari function. 6 00:00:21,440 --> 00:00:27,310 So I encourage you to make this reference available to you. 7 00:00:27,310 --> 00:00:30,120 Though, everything I'll show you is self-contained, but 8 00:00:30,120 --> 00:00:33,374 this is a, a good source of additional information. 9 00:00:35,190 --> 00:00:38,650 It's also appropriate for me to remind you the short course. 10 00:00:38,650 --> 00:00:40,310 Crash course on mathematics 11 00:00:42,510 --> 00:00:46,840 reference on the solution of Schrodinger equation, known Schrodinger equations, and 12 00:00:46,840 --> 00:00:53,310 access to numerical, simple numerical integrator that runs on a spreadsheet. 13 00:00:56,550 --> 00:01:01,680 In the previous part, we started out, we treated the first of these three simple 14 00:01:01,680 --> 00:01:07,860 one dimensional examples. And we're now going to today this part 15 00:01:09,930 --> 00:01:11,330 look at the linear of potential. 16 00:01:11,330 --> 00:01:14,500 And I think, I think actually we'll just we will have a 17 00:01:14,500 --> 00:01:19,060 very brief discussion about the Quadratic at the end of the linear potential. 18 00:01:19,060 --> 00:01:23,060 Does sort of, is the best thing to focus on, because it gives one 19 00:01:23,060 --> 00:01:26,430 a very good framework for competence of 20 00:01:26,430 --> 00:01:28,750 understanding the solution of one dimensional Schrodinger equation. 21 00:01:33,120 --> 00:01:36,210 Now, in the previous part we say for this constant potential, and 22 00:01:36,210 --> 00:01:42,335 we found that we could transform the Schrodinger equation into a standard form. 23 00:01:42,335 --> 00:01:46,310 [SOUND] 24 00:01:46,310 --> 00:01:49,200 And the key point about that standard form. 25 00:01:49,200 --> 00:01:51,520 Well, there were two variants of it. 26 00:01:51,520 --> 00:01:52,610 Difference in sides. 27 00:01:53,690 --> 00:01:57,620 whether, according to weather E, what was greater 28 00:01:57,620 --> 00:02:03,230 than or less than the constant V naught. 29 00:02:03,230 --> 00:02:07,460 but in both cases we can identify two solutions. 30 00:02:08,620 --> 00:02:11,370 Now I should have brought this point up in the, 31 00:02:11,370 --> 00:02:18,230 in the previous part, but here is really good tool for helping you to 32 00:02:18,230 --> 00:02:23,660 make decisions about choices of the alternative pairs 33 00:02:25,840 --> 00:02:30,780 of wave functions that you will always face, in solving the Schrodinger equation. 34 00:02:30,780 --> 00:02:36,570 This is the so-called Wronskian, the Wronskian of two functions is given by 35 00:02:37,780 --> 00:02:42,400 fg prime, right, so, so, g prime equals dg 36 00:02:44,500 --> 00:02:44,710 dx. 37 00:02:44,710 --> 00:02:50,370 And this is a very good comparison tool, tell you decision making. 38 00:02:50,370 --> 00:02:54,650 So, the in line quiz, I'm going to invite you to discover 39 00:02:54,650 --> 00:02:58,759 for yourself, the key property, the Wronskian operant, why it's so useful. 40 00:03:00,880 --> 00:03:04,660 So I guess you can see that in these two 41 00:03:04,660 --> 00:03:08,510 cases you can calculate the Wronskian in these two functions easily. 42 00:03:08,510 --> 00:03:12,930 It's, that's plus or minus one depending on how you do it. 43 00:03:12,930 --> 00:03:18,650 And here it's plus or minus 2, I guess. 44 00:03:18,650 --> 00:03:20,220 Depending on, on how you do it. 45 00:03:21,900 --> 00:03:26,000 but perhaps, perhaps if you think about it, you'll see 46 00:03:26,000 --> 00:03:29,420 that if you have if you have, 47 00:03:33,300 --> 00:03:40,150 if you have divergent behavior, In this, in i, in this 48 00:03:40,150 --> 00:03:47,080 case, of one function, then you should be able to find another wane that converges. 49 00:03:48,130 --> 00:03:54,020 Or, maybe it's a easier to say that if you have oscilatory 50 00:03:54,020 --> 00:03:58,640 behavior of one fuinction, in this region, and you pick another 51 00:03:58,640 --> 00:04:03,370 one, Then that's also a solution its also going 52 00:04:03,370 --> 00:04:06,440 to be isolating in order to keep the Wronskian 53 00:04:06,440 --> 00:04:11,190 constant so that's the whole real important is that 54 00:04:11,190 --> 00:04:15,510 these alternative pairs of solutions the Wronskian must be constant. 55 00:04:18,970 --> 00:04:21,590 Now the solution to the linear potential. 56 00:04:24,020 --> 00:04:26,320 Schrodinger's Equation for a linear potential. 57 00:04:26,320 --> 00:04:31,260 Some very nice properties, so I suppose that these 58 00:04:31,260 --> 00:04:34,720 are, I suppose, the area functions must be less 59 00:04:34,720 --> 00:04:37,970 familiar to everybody, then must be less familiar than 60 00:04:37,970 --> 00:04:42,840 the exponential or the trigonometric functions, functions to everybody. 61 00:04:42,840 --> 00:04:44,810 but if you, this is your first exposure to 62 00:04:44,810 --> 00:04:49,020 them, let me point out that this case the linear 63 00:04:49,020 --> 00:04:55,060 potential gives us two universal solutions that combine both 64 00:04:55,060 --> 00:05:01,730 the oscillatory behaviour of the trigonometric functions. 65 00:05:01,730 --> 00:05:05,120 And the exponential, converging, diverging or 66 00:05:05,120 --> 00:05:08,860 converging behavior of the exponential function. 67 00:05:08,860 --> 00:05:09,950 So, there's basically, 68 00:05:12,190 --> 00:05:15,150 a kind of universal function for each type that, 69 00:05:17,290 --> 00:05:21,130 that covers behavior in the allowed 70 00:05:23,490 --> 00:05:25,550 Versus forbidden 71 00:05:29,900 --> 00:05:30,640 regions. 72 00:05:30,640 --> 00:05:32,900 And that is if you have, if you have a 73 00:05:32,900 --> 00:05:37,320 linear potential, then eventually depending on the choice of v1, 74 00:05:39,360 --> 00:05:45,240 Eventually, eventually the, the potential eventually gets larger than the energy. 75 00:05:45,240 --> 00:05:48,660 Or eventually gets smaller than the energy, either way around. 76 00:05:48,660 --> 00:05:50,270 So in this sense, linear potential. 77 00:05:51,950 --> 00:05:53,840 very, is a very nice, tool for 78 00:05:53,840 --> 00:05:58,180 understanding the behavior of quantum mechanical wave functions. 79 00:05:58,180 --> 00:06:04,330 So again, in this case, there's a demarkation point. 80 00:06:04,330 --> 00:06:08,880 Well, these, of course these are two particular choices of function and 81 00:06:08,880 --> 00:06:14,290 you can find their definition in this chapter. 82 00:06:14,290 --> 00:06:19,490 And they're both oscillate for, when the arguments less than 0. 83 00:06:19,490 --> 00:06:24,310 And then the AI converges for row, greater than 0, 84 00:06:24,310 --> 00:06:29,390 and VI diverges, and these two have implementations in 85 00:06:29,390 --> 00:06:30,320 physical systems. 86 00:06:30,320 --> 00:06:31,130 The one we're going to look at in 87 00:06:31,130 --> 00:06:35,570 detail here, is the quantum mechanical bouncing ball. 88 00:06:35,570 --> 00:06:38,700 but the second type of airy function becomes important. 89 00:06:38,700 --> 00:06:42,130 In modeling things like scanning time telling microscope. 90 00:06:44,920 --> 00:06:48,800 Now the bouncing ball problem, is, as you'll see, 91 00:06:48,800 --> 00:06:53,560 is important from a, a technical, theoretical perspective and 92 00:06:53,560 --> 00:06:55,030 in term of learning how to solve the short 93 00:06:55,030 --> 00:06:58,080 new equation, but actually it's something that has been. 94 00:06:58,080 --> 00:07:05,120 recently realized a remarkable experiment of the, 95 00:07:05,120 --> 00:07:11,480 bouncing of a neutron confined in a, in a trap and bouncing off a mirror. 96 00:07:11,480 --> 00:07:14,170 and there's actually experiments that measured 97 00:07:14,170 --> 00:07:16,660 the quantum mechanical spectrum of the 98 00:07:16,660 --> 00:07:20,820 Earth subject of the, of the neutron subject to the Earth's graviational field. 99 00:07:20,820 --> 00:07:25,730 There have also been some implentations in ultra cooled, atom systems. 100 00:07:25,730 --> 00:07:30,930 So we're actually going to, solve the problem with reference 101 00:07:30,930 --> 00:07:34,930 to parameters that are germane to this recent experimental realization. 102 00:07:38,740 --> 00:07:42,420 So, here, we're going to describe this amazing neutron experiment. 103 00:07:42,420 --> 00:07:45,770 The starting point is something from freshman physics. 104 00:07:45,770 --> 00:07:49,990 V is equal to MG V is equal to MGH, or V is equal to MGX. 105 00:07:49,990 --> 00:07:56,120 So the potential energy of the neutron, is given by this form, where 106 00:07:56,120 --> 00:08:02,959 X is the distance going vertically. off the surface of the mirror. 107 00:08:04,500 --> 00:08:07,600 G, acceleration due to the Earth's gravity, we'll just 108 00:08:07,600 --> 00:08:10,580 take a nominal value, 9.8 meters per second squared. 109 00:08:11,630 --> 00:08:14,380 M, the mass of the neutron, about the same as the mass of 110 00:08:14,380 --> 00:08:17,650 the proton, here's the nominal value, just going to use a round number value. 111 00:08:17,650 --> 00:08:21,700 And now here is the boundry condition. 112 00:08:22,930 --> 00:08:24,580 for x equal to 0. 113 00:08:24,580 --> 00:08:29,540 So in other words the wave function has to vanish at the surface of the mirror. 114 00:08:29,540 --> 00:08:32,550 That's an idealization that's largely true. 115 00:08:32,550 --> 00:08:34,765 And then it's got to, it's got to, the 116 00:08:34,765 --> 00:08:35,472 [INAUDIBLE] 117 00:08:35,472 --> 00:08:39,990 function has to oscilllate and reach a point of whatever 118 00:08:39,990 --> 00:08:44,050 the energy is The wave function must die off eventually 119 00:08:44,050 --> 00:08:47,650 in order for the neutron to be localized and continue 120 00:08:47,650 --> 00:08:49,790 to bounce back and forth in the way that we expect. 121 00:08:51,630 --> 00:09:01,024 Now, in order to cast this equation into appropriate dimensional form, I just, 122 00:09:01,024 --> 00:09:03,008 we're going to use. 123 00:09:03,008 --> 00:09:08,084 We're going to define you know I, I've worked this problem out in 124 00:09:08,084 --> 00:09:13,442 advance so to define our dimensions coordinate grow in terms of this 125 00:09:13,442 --> 00:09:18,870 transformation and I like you to look at that in the, in video quiz. 126 00:09:18,870 --> 00:09:23,269 Right, well I hope that you found that you could solve the quiz. 127 00:09:25,660 --> 00:09:30,013 Solve the quiz question using dimensional 128 00:09:30,013 --> 00:09:35,810 analysis only because if you just look at e over mg, I mean that, that 129 00:09:35,810 --> 00:09:41,269 tells you the that dimensional analysis of this expression tells you the answer. 130 00:09:44,060 --> 00:09:44,360 Oh yeah. 131 00:09:44,360 --> 00:09:49,690 So in other words this parameter alpha, has to have the units of energy. 132 00:09:49,690 --> 00:09:58,091 I mean since row is dimensionless as well, you can see that that's quite clear. 133 00:09:58,091 --> 00:10:03,420 Now, in the next brief in-line quiz, just take it a step 134 00:10:03,420 --> 00:10:08,700 further, and this has to do with I, the, relationship between rho and x. 135 00:10:08,700 --> 00:10:10,160 So if 136 00:10:10,160 --> 00:10:15,740 you were to draw a rho axis, where does it go in this mirror, and what, 137 00:10:15,740 --> 00:10:18,280 what you know for example, there's an important, 138 00:10:19,320 --> 00:10:22,970 important issue regarding what does rho equal 0 means? 139 00:10:22,970 --> 00:10:26,750 What is the physical interpretation of 140 00:10:26,750 --> 00:10:29,020 the value that particular mentioned coordinates. 141 00:10:31,930 --> 00:10:37,940 So I hope it was obvious, I hope it's easy for you to find that the 142 00:10:37,940 --> 00:10:42,320 value of rho equals 0. Describes the classical turning point. 143 00:10:42,320 --> 00:10:45,560 That is, when I say the place where the velocity changes sign. 144 00:10:45,560 --> 00:10:52,328 It, it represents the maximum distance, 145 00:10:52,328 --> 00:10:57,968 represents the maximum distance, 146 00:10:57,968 --> 00:11:02,856 maximum value of x for 147 00:11:02,856 --> 00:11:07,744 which is attained in classical 148 00:11:07,744 --> 00:11:12,352 motion. So that means that the function that we 149 00:11:12,352 --> 00:11:17,824 are going to, that we want is Ai because what it attains is maximum 150 00:11:17,824 --> 00:11:23,104 the little bit before rho equal 0 but it's the one that dies off, 151 00:11:23,104 --> 00:11:24,786 as rho gets large. 152 00:11:24,786 --> 00:11:27,006 As rho increases beyond, into the positive 153 00:11:27,006 --> 00:11:30,200 region, which is the classically forbidden region. 154 00:11:30,200 --> 00:11:32,930 So we know that we, we must of the two 155 00:11:32,930 --> 00:11:37,620 solutions that we are going to use, we must use AI. 156 00:11:37,620 --> 00:11:39,720 That is the only acceptable solution. 157 00:11:42,590 --> 00:11:48,290 in the region rho greater than 0, because the only other possible solution there 158 00:11:48,290 --> 00:11:53,200 would involve an ad mixture of Bi, because any solution in 159 00:11:53,200 --> 00:11:56,900 this region can be written as a linear combination of Ai and Bi, 160 00:11:56,900 --> 00:12:03,490 the one unique solution, up to some overall constant multiple factor, is Ai. 161 00:12:03,490 --> 00:12:07,910 So if we take that, we now say, now how, knowing that that is the solution. 162 00:12:07,910 --> 00:12:11,640 How do we satisfy the boundary condition. 163 00:12:13,160 --> 00:12:17,310 That psi of x equals 0 164 00:12:17,310 --> 00:12:21,490 is equal to 0. Well, we're going to 165 00:12:21,490 --> 00:12:25,810 write, we write the wave function in terms of an area function. 166 00:12:25,810 --> 00:12:32,914 So we have an additional parameter avail, available to us, which is the unconcern 167 00:12:32,914 --> 00:12:35,400 offset in the solution. 168 00:12:35,400 --> 00:12:41,010 And so that, that means that we basically, we. 169 00:12:42,060 --> 00:12:47,980 We take this area function and we sort of start pulling it up, until we, 170 00:12:47,980 --> 00:12:54,590 until the node lies at the relevant, at the surface of the mirror. 171 00:12:54,590 --> 00:12:57,790 So I think you can see that what we're going to get is, 172 00:12:57,790 --> 00:12:59,380 here's the lowest solution will be like that. 173 00:12:59,380 --> 00:13:03,090 Then the next lowest solution Oppss. 174 00:13:05,380 --> 00:13:11,860 The first excited state has a, has a node one node in the wave function and so one. 175 00:13:11,860 --> 00:13:15,400 So basically, the a, the, this curve ai, 176 00:13:15,400 --> 00:13:19,140 shows you a universal set of profile of the 177 00:13:19,140 --> 00:13:21,380 excited state of the bouncing ball and there's 178 00:13:21,380 --> 00:13:24,320 just obtained by displacing the surface of the mirror. 179 00:13:24,320 --> 00:13:29,820 To each successive node in the solution ai. 180 00:13:29,820 --> 00:13:30,880 That will conclude 181 00:13:30,880 --> 00:13:33,760 actually really with just a very brief description 182 00:13:33,760 --> 00:13:38,110 of what goes on in the quadratic system. 183 00:13:38,110 --> 00:13:40,790 We're not going to discuss it in a lot of detail. 184 00:13:40,790 --> 00:13:44,250 The most important case of this is concerns the harmonic oscillate. 185 00:13:44,250 --> 00:13:50,030 Later which we've solved elsewhere. But there's sometimes 186 00:13:50,030 --> 00:13:56,000 things like a harmonic oscillator with dis, impurities and other disruptions, 187 00:13:56,000 --> 00:14:01,210 and so it's, it's nice to have the ability to solve the general second order 188 00:14:01,210 --> 00:14:06,200 equation. And, I'll point out that once again. 189 00:14:08,530 --> 00:14:14,650 it turns out that the, the divergent character of solutions is the generic one. 190 00:14:14,650 --> 00:14:15,660 So for example, 191 00:14:18,150 --> 00:14:23,400 in, in this system, because of the quadratic dependence of x. 192 00:14:23,400 --> 00:14:32,930 You know, eventually v is greater than e. 193 00:14:34,380 --> 00:14:37,170 If you go, if you go sufficiently far away, sufficient large 194 00:14:37,170 --> 00:14:41,250 values of x, positive or negative, you're in the forbidden region. 195 00:14:41,250 --> 00:14:41,750 So, 196 00:14:44,580 --> 00:14:50,810 the the, because you have propagation that, because of the way the equation 197 00:14:50,810 --> 00:14:56,410 in forbidden region, you always get a divergent solution. 198 00:14:56,410 --> 00:15:01,110 but there are special cases, and these are you know, these are exactly the. 199 00:15:01,110 --> 00:15:03,520 eigenvalues in the harmonic oscillator here. 200 00:15:03,520 --> 00:15:05,940 You can see this is a harmonic oscillator looking equation. 201 00:15:08,820 --> 00:15:09,430 that when 202 00:15:11,610 --> 00:15:12,690 n is an integer. 203 00:15:15,280 --> 00:15:20,060 you get, you get bound, eigenstates. So this, this just shows 204 00:15:21,140 --> 00:15:26,240 the, the characteristics of this so called parabolic cylinder function. 205 00:15:28,920 --> 00:15:34,340 And we, we're not going to make practical use of that 206 00:15:34,340 --> 00:15:38,620 in this, in this course but I thought you should see it. 207 00:15:38,620 --> 00:15:42,668 Again, the quadratic equations solutions to the 208 00:15:42,668 --> 00:15:46,452 shortage equation of quadratic potential are part 209 00:15:46,452 --> 00:15:50,324 of this, this systematic family special functions 210 00:15:50,324 --> 00:15:54,460 that's been developed to solve the shortage equations 211 00:15:54,460 --> 00:15:57,734 in general. Okay, that concludes this lecture. 212 00:15:57,734 --> 00:15:59,840 I'll hope to see you in the next one.