1
00:00:00,570 --> 00:00:07,110
Now, of course, we don't only want to have one single eigenvalue, we also want to determine the next

2
00:00:07,110 --> 00:00:07,920
eigenvalues.

3
00:00:08,700 --> 00:00:10,650
So let's try another well, you.

4
00:00:11,250 --> 00:00:13,920
Let's take five point nine.

5
00:00:15,510 --> 00:00:18,540
OK, now the way function goes way beyond zero.

6
00:00:18,810 --> 00:00:21,150
So maybe go even higher.

7
00:00:22,410 --> 00:00:25,980
OK, looks still like a sign function as it should.

8
00:00:27,180 --> 00:00:29,520
OK, we need to go way higher, actually.

9
00:00:29,850 --> 00:00:31,230
Thirteen point nine.

10
00:00:32,070 --> 00:00:33,120
17.

11
00:00:33,720 --> 00:00:34,140
OK.

12
00:00:37,180 --> 00:00:40,060
OK, that looks kind of OK, I guess.

13
00:00:40,420 --> 00:00:47,050
So now we have another wave function that goes back to zero for the Position X is equal to A, which

14
00:00:47,050 --> 00:00:48,280
is in our case, one.

15
00:00:49,240 --> 00:00:51,160
So this would be the second eigenvalue.

16
00:00:52,210 --> 00:00:56,530
And actually, do you remember what the analytical result was?

17
00:00:56,540 --> 00:01:03,880
So this was the first eigenvalue, which corresponds to one squared where this one is the quantum number

18
00:01:03,880 --> 00:01:06,430
and then we make it to two.

19
00:01:06,880 --> 00:01:07,780
We get this one.

20
00:01:08,470 --> 00:01:09,100
Three.

21
00:01:09,820 --> 00:01:11,020
OK, good.

22
00:01:11,470 --> 00:01:15,130
But that's of course, cheating because here we know the analytical result.

23
00:01:15,130 --> 00:01:18,820
But we of course, want to do it without knowing the analytical result.

24
00:01:19,660 --> 00:01:28,210
So what we want to do is we want to scan the energy until we get the proper values so that the way function

25
00:01:28,210 --> 00:01:29,470
returns back to zero.

26
00:01:30,370 --> 00:01:36,340
And actually something that I also want to mention is we know actually that this is two third eigenvalue

27
00:01:36,340 --> 00:01:39,430
because of the zeros here inside of the box.

28
00:01:40,390 --> 00:01:47,890
So you remember from the analytical result, the eigen level of four and equal one had no zero for an

29
00:01:47,890 --> 00:01:50,950
equal to we had one zero and four and equal three.

30
00:01:50,950 --> 00:01:53,030
We had two zeros, as you can see here.

31
00:01:53,680 --> 00:02:01,390
And this is true in every single example where we have a bound state in the potential like X Square

32
00:02:01,390 --> 00:02:03,640
or such a particle in the box.

33
00:02:03,880 --> 00:02:04,990
So that's always true.

34
00:02:05,020 --> 00:02:11,530
So we know that there is no other eigenvalue between the one, the values that we have determined when

35
00:02:11,530 --> 00:02:12,850
we just check the zeros.

36
00:02:14,500 --> 00:02:21,040
OK, but if we are now scan our energy and increase the energy just by small amounts, there's also

37
00:02:21,520 --> 00:02:24,510
no difficulty in finding these eigenvalues.

38
00:02:24,520 --> 00:02:26,200
So we don't really have a problem here.

39
00:02:27,100 --> 00:02:31,810
And actually scanning the ion energy is exactly what we are going to do next.

40
00:02:32,290 --> 00:02:36,220
So we use this auger rhythm here, which works quite well.

41
00:02:36,580 --> 00:02:41,770
Now we can delete this one, and now we are going to modify this rhythm.

42
00:02:42,250 --> 00:02:46,690
So as I told you already, we are going to scan the energy.

43
00:02:47,770 --> 00:02:58,750
So what we will do here is we will actually start with, um yeah, let's see energy equal to zero.

44
00:02:59,950 --> 00:03:05,050
And now we will use another loop over this loop.

45
00:03:05,290 --> 00:03:09,160
So let's add another loop here, another while loop.

46
00:03:09,520 --> 00:03:18,100
And this time we have a condition which is actually, let's say psi is so we should end when psi is

47
00:03:18,100 --> 00:03:19,060
equal to zero.

48
00:03:19,960 --> 00:03:22,030
So how are we going to do this?

49
00:03:22,030 --> 00:03:31,870
We will set the absolute value of psi should be smaller than, let's say, zero zero one some value.

50
00:03:31,870 --> 00:03:33,940
We have to test this maybe if it doesn't work.

51
00:03:34,660 --> 00:03:41,560
So this means he keeps on doing the loop as long as this value is larger than this.

52
00:03:41,740 --> 00:03:44,890
So it stops when it's smaller and this is exactly what we want.

53
00:03:46,090 --> 00:03:49,510
OK, now you have caused this, we don't need here.

54
00:03:49,510 --> 00:03:59,200
So we we start from equal to zero and now we increase our energy in increments of Delta E and that's

55
00:03:59,320 --> 00:04:00,830
that's put Delta.

56
00:04:02,080 --> 00:04:04,600
Maybe not too small, because then it will take forever.

57
00:04:04,600 --> 00:04:12,170
Because remember, um, the computer will do this this loop for many, many times.

58
00:04:12,190 --> 00:04:18,160
So just from going from zero to five, he will do it now 500 times, so it could take a while.

59
00:04:19,000 --> 00:04:20,860
So we don't want to do it too small.

60
00:04:21,880 --> 00:04:28,030
Now what we also need here now is we need to define these two things.

61
00:04:28,840 --> 00:04:29,440
So

62
00:04:32,260 --> 00:04:38,620
let's do this before we change the energy, because then we can first do it for equals zero.

63
00:04:38,620 --> 00:04:48,820
And then the next time we do it for D e um, then I would say we don't actually need all of these lists.

64
00:04:49,360 --> 00:04:54,070
So so we don't need to store all the size and all the excess.

65
00:04:54,070 --> 00:04:59,920
We only need to store it for the correct wave function where we have the eigen function and the eigenvalue.

66
00:05:00,370 --> 00:05:07,180
So I will put it inside of the loop so that it will be overwritten every time the result was not good.

67
00:05:07,660 --> 00:05:12,220
And then in the end, we can only access the last time this was written here.

68
00:05:12,550 --> 00:05:17,620
So this means for the case where the EIGEN system was actually determined.

69
00:05:18,370 --> 00:05:20,740
So this is why I'm putting it inside of the loop here.

70
00:05:21,940 --> 00:05:23,450
OK, what else do we need?

71
00:05:23,470 --> 00:05:26,110
We need to go back to zero.

72
00:05:26,740 --> 00:05:29,410
So, yeah, just put it here.

73
00:05:30,670 --> 00:05:31,240
OK.

74
00:05:31,690 --> 00:05:36,040
Um, yeah, we have this one here.

75
00:05:36,980 --> 00:05:37,790
That looks good.

76
00:05:38,510 --> 00:05:41,630
And, OK, now we could test if it works.

77
00:05:42,080 --> 00:05:44,510
I'm not really sure if it does, but let's test.

78
00:05:46,830 --> 00:05:51,270
OK, maybe let us not put this in the while loop.

79
00:05:52,050 --> 00:05:56,910
Let's put it outside and also let's plot the energy.

80
00:05:57,870 --> 00:05:59,670
OK, cool.

81
00:05:59,940 --> 00:06:12,540
OK, now it's the energy is equal to zero, which means it stops to loop right away because the passive

82
00:06:12,540 --> 00:06:15,380
function is actually zero.

83
00:06:15,630 --> 00:06:23,160
That's probably because we start here with PSI function, which is zero because we have previously solved

84
00:06:23,160 --> 00:06:23,620
this case.

85
00:06:23,620 --> 00:06:30,280
So let's just put it to some random value and then it will be redefined inside of the loop.

86
00:06:30,300 --> 00:06:30,600
OK.

87
00:06:31,980 --> 00:06:32,310
Yeah.

88
00:06:32,430 --> 00:06:32,900
OK.

89
00:06:32,910 --> 00:06:37,350
So now it says energy is equal to 4.9.

90
00:06:37,350 --> 00:06:39,710
That's actually very close to what we had determined here.

91
00:06:39,720 --> 00:06:42,780
That was, yeah, there was this one.

92
00:06:43,260 --> 00:06:43,560
OK.

93
00:06:44,010 --> 00:06:50,460
So we have now four point nine three and this is the corresponding wave function.

94
00:06:52,560 --> 00:06:58,080
Now, that's pretty good, because now we have automatically determined the first eigenvalue, and now

95
00:06:58,080 --> 00:07:01,740
we can also determined to next eigenvalues.

96
00:07:02,310 --> 00:07:05,910
And for this, we will again copy of a working algorithm.

97
00:07:05,910 --> 00:07:09,930
And you guessed it, we will add another loop.

98
00:07:10,290 --> 00:07:22,200
This time the loop will be again a while loop and we will have some, some counter which we will increase

99
00:07:22,200 --> 00:07:25,050
whenever we have found an eigen energy.

100
00:07:26,130 --> 00:07:34,920
And let's say we were to find some and Maxwell U, which let's let's use three for the beginning.

101
00:07:35,880 --> 00:07:43,290
And OK, now this one we will need here inside of the loop, actually.

102
00:07:44,190 --> 00:07:45,270
Um.

103
00:07:46,170 --> 00:07:47,630
OK, what else do we need?

104
00:07:49,640 --> 00:07:53,930
We OK, this one, we can leave here with this when we can leave here.

105
00:07:54,350 --> 00:08:00,980
OK, we need a counter, so let's say a counter will be starting with.

106
00:08:02,710 --> 00:08:05,110
With one, OK, yeah.

107
00:08:05,890 --> 00:08:09,400
And let's not forget this one, you and.

108
00:08:10,820 --> 00:08:13,880
Now, after we have done these loops.

109
00:08:16,450 --> 00:08:21,880
We will increase the counter counter will be counter +1.

110
00:08:22,180 --> 00:08:30,400
OK, now what we have is we start with a counter off of one counter counter.

111
00:08:31,180 --> 00:08:35,080
Um, then we will search for the first eigenvalue.

112
00:08:36,919 --> 00:08:39,770
We will have to save it in some way, which is not included yet.

113
00:08:40,130 --> 00:08:43,280
And then we will increase to counter by one.

114
00:08:44,540 --> 00:08:50,870
Suzanne Counter will be two and at some point when the counter is large, it and and max, then he will

115
00:08:50,870 --> 00:08:51,320
stop.

116
00:08:51,320 --> 00:08:55,220
And then we have determined three eigenvalues and eigen functions.

117
00:08:55,910 --> 00:08:56,250
OK.

118
00:08:56,600 --> 00:09:01,430
As you noticed, we also have to do is we will have to save our eigen function.

119
00:09:02,360 --> 00:09:08,340
So we will have a list of functions.

120
00:09:09,500 --> 00:09:11,690
So let's define this list

121
00:09:14,720 --> 00:09:15,350
of functions.

122
00:09:15,350 --> 00:09:16,760
Let's make it just an empty list.

123
00:09:16,760 --> 00:09:26,650
And here we use again our append function and we will add the list of the corresponding eigenvalue.

124
00:09:26,670 --> 00:09:28,010
So basically this one here?

125
00:09:29,150 --> 00:09:38,630
No, I think 40 x value, we don't necessarily have to do it because it will always be the same x value.

126
00:09:38,630 --> 00:09:43,700
But OK, let's let's let's do it anyway.

127
00:09:44,570 --> 00:09:45,290
Um.

128
00:09:47,570 --> 00:09:52,370
So I can Functions X list.

129
00:09:54,120 --> 00:10:03,480
Let's put it like this next list is Append X list.

130
00:10:04,320 --> 00:10:13,650
And then we also need to save our icon energies, I think, and and or GS.

131
00:10:15,150 --> 00:10:15,960
So.

132
00:10:18,040 --> 00:10:23,230
Let's if I can again and urges dont append.

133
00:10:26,230 --> 00:10:31,720
And then we have here our value, exactly.

134
00:10:34,070 --> 00:10:38,960
OK, now let's test the code.

135
00:10:44,230 --> 00:10:44,660
OK.

136
00:10:46,330 --> 00:10:49,060
Let's see, I get and or chiefs.

137
00:10:51,950 --> 00:10:58,220
And and they're just OK, so we have no.

138
00:10:59,270 --> 00:10:59,710
OK.

139
00:10:59,870 --> 00:11:02,900
Yeah, that's that's that's that's unfortunate.

140
00:11:03,470 --> 00:11:06,860
So basically, we have not determined in indeed three eigenvalues.

141
00:11:07,250 --> 00:11:15,470
We have four point nine and we have nineteen point seven, which is actually true because we can use

142
00:11:15,470 --> 00:11:18,530
here nineteen point seven.

143
00:11:20,240 --> 00:11:20,570
OK.

144
00:11:20,630 --> 00:11:26,420
That's actually our second eigenvalue, because here the system also determined another eigenvalue,

145
00:11:26,420 --> 00:11:28,520
which is very close to this one here.

146
00:11:29,090 --> 00:11:34,580
This is because the true eigenvalue would probably somewhere between these two values.

147
00:11:34,580 --> 00:11:39,020
But since we had our condition, you too large, it determined both of them.

148
00:11:39,410 --> 00:11:42,080
So now we have two solutions for this.

149
00:11:42,080 --> 00:11:48,980
We could either decrease this, this value here to make the position higher, but actually this would

150
00:11:49,280 --> 00:11:52,400
then take much longer for the system to compute.

151
00:11:53,090 --> 00:11:58,670
But instead, what I'm going to try is I will use this one here.

152
00:11:59,240 --> 00:12:01,880
Let's say, OK, maybe we could do anything.

153
00:12:02,870 --> 00:12:10,490
So what this does is once we have determined the eigen energy, we will say OK as a next step and just

154
00:12:10,490 --> 00:12:18,230
increase the energy by some large enough factor so that you don't get the same eigenvalue twice.

155
00:12:19,520 --> 00:12:27,020
The only problem that could arise here is that we would not be able to register another eigenvalue in

156
00:12:27,020 --> 00:12:27,680
this range.

157
00:12:27,830 --> 00:12:31,760
So if two eigenvalues would be very close, we could not determine them.

158
00:12:32,600 --> 00:12:39,410
But since we know what our solution looks like, and yeah, because here is just an educational example,

159
00:12:39,410 --> 00:12:40,520
I will use it like this.

160
00:12:41,030 --> 00:12:43,130
So we are not missing out on anything.

161
00:12:43,610 --> 00:12:45,500
So I hope now this should work.

162
00:12:47,240 --> 00:12:50,520
OK, so we have this one, this one, and this we can test.

163
00:12:50,540 --> 00:12:54,560
OK, let's use forty four point three.

164
00:12:56,090 --> 00:12:56,990
OK, very good.

165
00:12:57,920 --> 00:12:58,880
So let's see.

166
00:12:59,360 --> 00:13:02,630
Could we go higher and Max Equal five?

167
00:13:06,060 --> 00:13:07,350
Actually, it's pretty fast.

168
00:13:08,340 --> 00:13:10,230
I know it's not some seconds.

169
00:13:11,010 --> 00:13:12,840
OK, let's let's try this one.

170
00:13:13,710 --> 00:13:14,790
What does it look like?

171
00:13:17,940 --> 00:13:18,420
OK.

172
00:13:18,450 --> 00:13:18,870
Nice.

173
00:13:18,900 --> 00:13:22,860
So this is our fifth eigenvalue and I can function.

174
00:13:22,860 --> 00:13:25,680
So it should have five zeros.

175
00:13:26,430 --> 00:13:27,870
Should have four zeros.

176
00:13:28,260 --> 00:13:30,030
One two three four.

177
00:13:30,570 --> 00:13:30,900
OK.

178
00:13:31,680 --> 00:13:39,840
So we have now determined our first five I can energies and I can functions automatically.

179
00:13:41,460 --> 00:13:45,270
So what we could do now is we could plot the ideal functions.

180
00:13:45,750 --> 00:13:49,440
So I will use that using a loop.

181
00:13:50,220 --> 00:13:52,440
So just plodding down on top of each other.

182
00:13:54,660 --> 00:13:58,950
So we know we have and max eigenvalues, which in our case is five.

183
00:13:59,520 --> 00:14:07,890
So people use plot and then we have here our next list.

184
00:14:10,800 --> 00:14:14,610
Um, I think and functions.

185
00:14:17,950 --> 00:14:18,350
OK.

186
00:14:18,490 --> 00:14:22,840
This is a bit bit of a difficult syntax in Python, I think so

187
00:14:25,630 --> 00:14:27,540
it's a bit hard to follow along, I admit.

188
00:14:27,550 --> 00:14:34,330
But OK, it's not so important is just for just for plotting counter.

189
00:14:35,170 --> 00:14:37,840
We have to, of course, increase our counter.

190
00:14:37,840 --> 00:14:39,370
And now it should.

191
00:14:39,730 --> 00:14:40,120
OK.

192
00:14:40,390 --> 00:14:40,720
Yeah.

193
00:14:41,830 --> 00:14:47,800
So now we have here in blue the first iron function, which has no zero apart from these boundaries.

194
00:14:48,340 --> 00:14:56,620
Then we have second and orange, then we have the green one, the red one and the purple one.

195
00:14:57,310 --> 00:15:01,420
And also what you can see is that all of these functions here have the same slope at the beginning.

196
00:15:01,810 --> 00:15:09,550
This is because we have always defined deep side is equal to one at a position zero.

197
00:15:09,970 --> 00:15:18,430
So we have strictly enforced that the slope is equal to one at this position because of course not true.

198
00:15:18,430 --> 00:15:24,820
It does not give us to correct eigen functions because as you can tell, the absolute square of this

199
00:15:24,820 --> 00:15:29,830
blue function will be much, much larger when you integrate over the whole box compared to the other

200
00:15:29,830 --> 00:15:30,400
functions.

201
00:15:30,820 --> 00:15:32,830
So this is what I mentioned at the very beginning.

202
00:15:33,520 --> 00:15:35,790
These functions are not normalized yet.

203
00:15:36,400 --> 00:15:38,320
We have to normalize these functions now.

204
00:15:41,920 --> 00:15:46,750
So I think now you have already learned what this shooting method is all about, and I think it's quite

205
00:15:46,750 --> 00:15:47,260
impressive.

206
00:15:47,680 --> 00:15:55,630
We have determined the eigen energies and also the principle, the principle shape of the wave function.

207
00:15:56,410 --> 00:16:02,080
So what was following now is the normalization of these wave functions and we will report them.

208
00:16:02,440 --> 00:16:04,840
So this is actually not so important.

209
00:16:05,140 --> 00:16:08,470
But still, I want to show you how you can do it using Python.

210
00:16:08,890 --> 00:16:15,580
But I think that's a bit just tedious work because of some, maybe some syntax

211
00:16:18,190 --> 00:16:18,730
problems.

212
00:16:20,170 --> 00:16:26,340
But I have, of course, done it already before, so I know we're now here.

213
00:16:26,370 --> 00:16:32,380
What are the difficulties, for example, using these indices here, these lists?

214
00:16:33,580 --> 00:16:47,590
So what I've shown you here is we are actually calculating to some over the absolute square of the wave

215
00:16:47,590 --> 00:16:50,860
function and we multiply it by our increment.

216
00:16:51,370 --> 00:16:56,440
So what this actually is is the integral over the absolute square of our wave function.

217
00:16:57,130 --> 00:16:59,890
And so we have here just index number two.

218
00:16:59,920 --> 00:17:06,040
So since Python is starting from zero, this is actually for the third wave function one our norm.

219
00:17:06,700 --> 00:17:08,980
OK, now something went wrong.

220
00:17:11,329 --> 00:17:13,040
I can function.

221
00:17:15,530 --> 00:17:17,300
OK, I can functions.

222
00:17:17,810 --> 00:17:21,740
OK, so this is now our norm for the third, I can function.

223
00:17:22,550 --> 00:17:29,600
So now we have to divide our function by the individual norm.

224
00:17:29,750 --> 00:17:32,000
So let's do this in the next step.

225
00:17:32,000 --> 00:17:40,220
So again, we need a loop with a counter looping over our way functions.

226
00:17:44,030 --> 00:17:50,630
So first of all, we have to calculate the norm for every individual wave function.

227
00:17:50,640 --> 00:17:55,160
So since we are in the loop now, we are starting with the first wave function.

228
00:17:55,880 --> 00:17:58,400
And here we need this one.

229
00:17:59,000 --> 00:18:03,530
And here as the index, we need to counter.

230
00:18:03,530 --> 00:18:09,980
And since starting to count from zero, we have to write this minus one here and here.

231
00:18:10,760 --> 00:18:17,150
OK, so this is now norm off the wave function and we are looking at at the moment inside of the loop.

232
00:18:17,770 --> 00:18:24,530
And now we have to determine what we have to divide the EIGEN function by does norm.

233
00:18:25,400 --> 00:18:29,150
So let's use again counter minus one.

234
00:18:31,040 --> 00:18:34,490
And this zero is just to get rid of some brackets in the list.

235
00:18:35,180 --> 00:18:37,250
Otherwise, it would not work, I have noticed.

236
00:18:38,150 --> 00:18:41,360
So we can do it like this.

237
00:18:45,360 --> 00:18:51,690
So if you are a Python expert, you will probably even find much more elegant ways to do this, but

238
00:18:51,690 --> 00:18:54,640
I think this is a bit more instructive.

239
00:18:54,660 --> 00:19:02,220
So now we are dividing every single element in the list or in the eigen function that we are currently

240
00:19:02,220 --> 00:19:06,150
looking at by its respective norm.

241
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So, OK, I guess that's it.

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Oh, OK.

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And you must not forget to increase the counter by one.

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OK, so we have determined the norm we have.

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We are looking here at basically every single element in the list that basically holds information about

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this.

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I can function and we are looking now at every element and dividing by the square root of the norm.

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OK, I think that's good.

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Now we have manipulated this, I can function.

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So basically, I think we can now just plot again.

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So just copy this one.

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OK, now good.

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Now, every wave function has the same maximum, and this is something that is not generally true.

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So when you have a general potential, the way functions do not have to have the same maximum and minimum.

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But for this particle in the infinitely potential box, it's actually true.

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And we have learned this also in the analytical lecture where we have determined is Prefect Capital

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Way, which was something like square root of I don't remember too over a or a over two or something

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like that.

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So all of the way function of the same prefecture and this we can see also here.

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Now let's check again if our if our norm is now correct.

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Yeah, OK.

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It's equal to one.

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But also we could have maybe estimated this.

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We have here something like, yeah, we have your length of one and a height of actually the square

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root of two.

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And yeah, so this if you if you draw a triangle, here it is, has approximately integrale of one.

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So that's definitely true.

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OK, now what we can do next is we can do something that I've also done in the analytical case, a different

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way of plotting a whole wave function because you can see here we have all of the wave functions that

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overlap each other.

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So instead, we could plot the individual energy levels and then plot the wave functions on top of this

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energy level.

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So this is something that I will do now.

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And so that's pretty similar to actually to determining the norm.

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So we will again loop over every individual wave function using this counter.

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So norm, we don't need we have done this already.

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And now what we are doing here for every individual element is we take the element, which is called

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acts here and we will add the eigen energies this time and we need to correct one.

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So, OK, you should go like this.

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Let's plot again.

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OK.

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Yeah, cool.

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So as you can see, or as you remember, our first eigenvalue was something like five actually is what

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it was.

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We can look at it again.

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I get it and or cheese was 4.9 three.

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So was this blue one here, then the next one, approximately 20 that we have forty four, seventy eight

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and one hundred twenty three.

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And on top of this, we have some wave functions that have an amplitude of square root of two, which

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is of approximately one point four.

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So we can almost not see this punctuation anymore.

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Of course, we could now reshape the way function, but I think it is really the correct way of plotting

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it.

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And when you have followed along and you have programmed this yourself, you can really be proud of

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yourself that you have solved now also numerically your first quantum mechanical example.