1 00:00:00,780 --> 00:00:08,070 OK, so we'll come back in the previous lecture, we have determined the EIGEN system for a particle 2 00:00:08,070 --> 00:00:10,890 in the box where the potential was infinitely large. 3 00:00:11,640 --> 00:00:15,570 So here we had some advantages that we typically don't have. 4 00:00:16,200 --> 00:00:23,190 So what we could consider here is we could consider a box which only have a width, has a width of a 5 00:00:23,250 --> 00:00:25,170 so starting from zero going to eight. 6 00:00:25,680 --> 00:00:29,850 And at the left edge, we had the wave function that was zero. 7 00:00:30,870 --> 00:00:38,850 And also, we had a very clear and very well-defined condition how our ion functions should look like. 8 00:00:39,240 --> 00:00:47,130 So we had here the wave function that goes back to zero so we could use a condition for our second while 9 00:00:47,130 --> 00:00:54,060 loop where we could say the wave function has to be close to zero so that while loop stops. 10 00:00:55,380 --> 00:01:00,660 Now, when we use some other potential, we cannot use these features anymore. 11 00:01:00,660 --> 00:01:03,700 And this is what I want to show you in this demonstration. 12 00:01:03,720 --> 00:01:06,720 So this is about the harmonic oscillator. 13 00:01:07,350 --> 00:01:15,990 And actually, I'm showing you this example because it is actually very difficult to determine the ion 14 00:01:15,990 --> 00:01:18,270 energies of this system analytically. 15 00:01:19,110 --> 00:01:24,180 And even more difficult is the determination of the EIGEN functions. 16 00:01:24,870 --> 00:01:33,120 So in fact, in one of the next main sections, I will show you how you can solve the harmonic oscillator 17 00:01:33,270 --> 00:01:34,080 analytically. 18 00:01:34,860 --> 00:01:38,970 But in this lecture, we are only able to solve the ion energies. 19 00:01:39,390 --> 00:01:44,370 And still, it's very difficult to do it, and it will take quite a long time. 20 00:01:44,580 --> 00:01:51,510 We have to write on many, many formulas and have to transform our trading equation until we finally 21 00:01:51,510 --> 00:01:53,880 end up with the solvable form. 22 00:01:54,750 --> 00:01:58,800 So here I want to show you how you can solve this system numerically. 23 00:01:59,640 --> 00:02:05,490 I wouldn't say it's easy because it is actually more difficult than the previous example, but still, 24 00:02:05,850 --> 00:02:10,530 you can really manage to do it and you will see the eigenvalues, A.I. and functions. 25 00:02:11,730 --> 00:02:19,590 So let us start with the first part of our previous program because actually, we don't have to change 26 00:02:19,590 --> 00:02:20,670 so much here. 27 00:02:21,480 --> 00:02:28,460 So what we have to change is, first of all, the potential so previously we had here in our trading 28 00:02:28,490 --> 00:02:32,580 equation to potential V, which was zero inside of the box. 29 00:02:33,450 --> 00:02:39,840 So now what we have is we have V Times X Square. 30 00:02:40,200 --> 00:02:43,140 So this is exactly the same as in classical mechanics. 31 00:02:43,560 --> 00:02:51,090 You have some force that is some constant times minus the change of the coordinate, which we call X.. 32 00:02:51,780 --> 00:02:57,960 So the corresponding potential would be x squared times the prefect. 33 00:02:58,920 --> 00:03:00,240 So now we cannot use any. 34 00:03:00,480 --> 00:03:02,070 He are prefecture of zero. 35 00:03:02,580 --> 00:03:06,300 But instead we will use a prefecture of actually. 36 00:03:07,430 --> 00:03:12,890 The potential would be one half times this constant times square. 37 00:03:12,900 --> 00:03:14,340 So let's use one half. 38 00:03:16,260 --> 00:03:26,130 Um, and the next step we also have to consider it is this constant a here because we do not have a 39 00:03:26,130 --> 00:03:27,570 well-defined box anymore. 40 00:03:28,230 --> 00:03:34,470 Now we would really have to solve the wave function in all of the one dimensional space. 41 00:03:34,620 --> 00:03:41,700 So basically, we would have to propagate our wave function starting from minus infinity going to positive 42 00:03:41,700 --> 00:03:42,330 infinity. 43 00:03:43,080 --> 00:03:45,060 But obviously, we cannot do this. 44 00:03:45,540 --> 00:03:54,510 So we would have we have to select some value, which is quite small and then have to propagate our 45 00:03:54,510 --> 00:03:56,790 system until the value is quite large. 46 00:03:58,510 --> 00:04:08,050 So what I'm going to do now is I will actually use this one here, so I will say, OK, our starting 47 00:04:08,050 --> 00:04:19,690 point is, I don't know, maybe let's use let's use as a starting point minus 10 and then the increment 48 00:04:20,920 --> 00:04:25,330 let's let's make it make it like this, OK? 49 00:04:26,020 --> 00:04:31,180 We could also use eight times this, but OK, let's let's just let's just ride it like this because 50 00:04:31,180 --> 00:04:36,430 maybe later we want to change a and then we would have to change this one as well. 51 00:04:36,730 --> 00:04:40,960 So we have we fix it here to the value of 0.01. 52 00:04:41,020 --> 00:04:41,350 What? 53 00:04:41,410 --> 00:04:42,790 We can change it afterwards. 54 00:04:44,200 --> 00:04:54,370 OK, now we start our propagation at some value minus 10, which is actually for low energy use outside 55 00:04:54,370 --> 00:04:56,470 of this of this parabola. 56 00:04:56,680 --> 00:05:03,370 So the potential sort of solar energy is 10 below the potential and we get a declining wave function, 57 00:05:03,370 --> 00:05:06,160 a declining exponential function for our way function. 58 00:05:07,120 --> 00:05:15,580 So this means the wave function will decrease quite fast because of this exponential behavior. 59 00:05:16,180 --> 00:05:21,130 So I think it's OK if we start from from minus 10, at least for the low energies. 60 00:05:22,180 --> 00:05:29,680 So now let's use for the wave function, some smaller value, because previously we could say, OK, 61 00:05:30,610 --> 00:05:34,270 we start at the left end of the box or we function must be zero. 62 00:05:34,270 --> 00:05:41,560 But as we know also from the finite potential, the wave function is now not zero anymore because the 63 00:05:41,560 --> 00:05:42,790 electron can tunnel. 64 00:05:43,060 --> 00:05:46,780 It can really penetrate into the energy barrier. 65 00:05:46,920 --> 00:05:51,970 And the same is true also for this harmonic oscillator potential. 66 00:05:53,770 --> 00:06:00,090 So we have now a value for wave function, which we could have set basically to any volume. 67 00:06:00,090 --> 00:06:02,260 But let's say, OK, it's quite small. 68 00:06:02,260 --> 00:06:03,820 Let's use 0.01. 69 00:06:04,720 --> 00:06:11,650 And for deep sigh, I would say, because we have this exponential increase would not be such a good 70 00:06:11,650 --> 00:06:18,120 idea to use one because then our wave function would inflate drastically and we would strike. 71 00:06:18,130 --> 00:06:20,290 We would get gigantic values. 72 00:06:20,810 --> 00:06:27,610 So let's use zero here because even in the first step, the wave function may not change. 73 00:06:27,760 --> 00:06:34,750 Already, the second step, it will then finally change because we will have it as deep ci as zero plus 74 00:06:34,990 --> 00:06:36,450 this second derivative. 75 00:06:36,460 --> 00:06:38,650 So that's really not an issue here. 76 00:06:38,800 --> 00:06:41,560 I think we can do this now. 77 00:06:41,890 --> 00:06:43,780 We will use some energy. 78 00:06:44,260 --> 00:06:48,160 Let's go back to one as we had previously. 79 00:06:49,240 --> 00:06:54,910 And I think now we can just propagate this whole system. 80 00:06:56,380 --> 00:07:00,370 OK, then let me restart the kernel and run all cells. 81 00:07:03,370 --> 00:07:08,020 OK, now welcome to the world of numerics. 82 00:07:08,770 --> 00:07:16,630 This is our resulting wave function, actually for a equal to attend the wave function goes to tend 83 00:07:16,630 --> 00:07:18,600 to the power of th8. 84 00:07:19,330 --> 00:07:21,220 So this is why it looks so strange. 85 00:07:21,820 --> 00:07:26,020 So this is, I would say, some numerical artifact. 86 00:07:26,470 --> 00:07:33,100 Maybe the wave function looks physical in this area, but then at the end, it just explodes and goes 87 00:07:33,100 --> 00:07:37,480 to such large values so that we cannot use it for anything. 88 00:07:38,380 --> 00:07:45,520 And the reason that I found out is that what is happening here is we are starting from a to small value 89 00:07:45,520 --> 00:07:46,030 of A. 90 00:07:46,420 --> 00:07:47,680 So we are starting. 91 00:07:49,460 --> 00:07:56,750 Very far for the case where the wave function, by the way, function evolves exponentially for quite 92 00:07:56,750 --> 00:08:00,970 a long distance of X, so we start way too far on the left. 93 00:08:00,980 --> 00:08:03,470 So let's decrease to value of and try again. 94 00:08:04,280 --> 00:08:06,110 OK, still pretty large. 95 00:08:06,200 --> 00:08:07,910 10 to the power of six. 96 00:08:09,860 --> 00:08:10,310 OK. 97 00:08:10,430 --> 00:08:11,930 Yeah, this looks now much better. 98 00:08:11,930 --> 00:08:17,900 So now we have really already here some kind of physical wave function. 99 00:08:17,900 --> 00:08:20,490 But as you can see, um. 100 00:08:21,590 --> 00:08:25,650 It it does not go back to a similar value to zero. 101 00:08:25,670 --> 00:08:31,820 So we know that when we approach X equals two or three, it should go back to a value that is similar 102 00:08:31,820 --> 00:08:38,000 to our starting value, which was zero point zero one, but instead you can see here that we have a 103 00:08:38,000 --> 00:08:38,659 zero. 104 00:08:39,080 --> 00:08:45,020 So this means actually we are already past our first eigen energy. 105 00:08:45,770 --> 00:08:55,070 So what we have to do now is we have to try the lower test energy case now we are here at minus two 106 00:08:55,670 --> 00:08:56,570 at this position. 107 00:08:57,500 --> 00:09:06,770 So you see already you need some some um yeah, physical knowledge already to be able to solve something 108 00:09:06,770 --> 00:09:10,160 like this, which makes it already already quite difficult. 109 00:09:10,190 --> 00:09:15,650 OK, now you see the value decreases a bit and OK. 110 00:09:16,220 --> 00:09:17,370 Oh, that was lucky. 111 00:09:17,390 --> 00:09:20,360 OK, now we we have the energy with zero point five. 112 00:09:21,470 --> 00:09:27,020 So you can see here we have now a reasonable wave function, which is actually symmetric to the position 113 00:09:27,020 --> 00:09:28,190 X equal to zero. 114 00:09:28,610 --> 00:09:34,460 And the energy so looks like to be zero point five. 115 00:09:35,820 --> 00:09:36,180 OK. 116 00:09:36,600 --> 00:09:45,420 And the next step, we will automatically generate these energies and I can functions for the different 117 00:09:45,420 --> 00:09:46,200 quantum numbers. 118 00:09:46,650 --> 00:09:51,210 So here we have the first state and we will determine the second, third and so on.