00:01
In this problem, we're dealing with an electron confined in an infinite well.
00:06
The length of that well is 61 picometers to 10 to minus 12 meters.
00:11
They're told it's in a superposition of these three energy eigenstates.
00:18
Psi -3, side 7, side 5, with certain amplitudes, which leads to certain probabilities.
00:27
Before we go on, though, just to make my life a little easier writing, i'm going to call a coefficient a3, psi 3, i should say.
00:37
A minus 0 .56 will be b, not the i.
00:42
I'll leave the i explicit, and g is still there.
00:46
So i'm going to be writing a, psi 3, plus i, b, size 7, plus g, psi, 5.
00:57
Actually, the first goal, the problem is to find g.
01:01
So that's what i'm going to be using.
01:03
Now, it talks about it being normalized.
01:05
What does that mean in this case? if you were to have 1 ,000 measurements, say that for side 3, get 350, size 7, get 400.
01:21
And for side 5, you get 250.
01:26
You're not going to get anything other than these three.
01:30
Notice, 350 plus 400 plus 250 is 1 ,000.
01:35
So if somebody asked you, what's the percentage that you got, you know, between the three.
01:44
100%.
01:47
So when we normalize si in this case, that's really what we're specified, that we're going to get, we're going to get only these, we're going to get one of these guys.
01:59
We're not going to get psi two.
02:02
We're going to get psi three, side, side five, state.
02:06
That's it.
02:07
One of them.
02:08
Which one is a different issue? that's a different question.
02:11
That's a probability issue we'll get to later.
02:13
But for the si when we normalize it that is that is indicating that it is 100 % certain that you will find it in one of these.
02:27
Like i said 1 ,000 out of a thousand were one of these three.
02:33
So let's not do the normalization.
02:36
Let me give myself some room here.
02:37
One and i should mention, well i'll put it in here.
02:42
C -o -l sye star -sy -t -x.
02:47
Forget, they mentioned that these eigenstates, these wave functions, psi -3, side of, side 5, are orthormal.
02:57
So that means they are already normalized, and they are orthogonal to each other.
03:03
And we'll make use of that right now.
03:07
So 0 to l, a, psi -3 star, minus i -b, psi -7 star, plus g -star, side.
03:22
You might say, why isn't there a star on the a or the b? well, because i've chosen that they're just numbers.
03:28
They're not complex numbers.
03:29
They're just numbers.
03:31
But i did have to complex conjugate the i.
03:36
It becomes minus i.
03:37
I've taken care of this minus sign up here.
03:39
Don't worry about that.
03:40
That's part of b.
03:41
B is minus 0 .506.
03:45
And now let's write to the si itself, a, si, 3, plus ib, si, 7, plus g, psi, 5, dx.
03:56
That's the normalization condition.
04:00
Now it's just a matter of expanding everything out.
04:02
I'm going to do the first term with all three, the second term with all three, so on.
04:06
So i'll get one with side 3 across the top here, then the next line.
04:10
You'll see it.
04:12
So, a squared, a squared, zero to l, psi three star, side 3 d x and this is 1.
04:25
It's normalized.
04:28
That's where the normal, orthonormal.
04:30
That's where the normal comes in.
04:31
They're 1.
04:35
Plus i, a, b, 0 to l, psi 3 star, psi 7, dx, this is zero.
04:49
Orthogynum.
04:51
That's where the ortho comes in.
04:53
And then we have a gene, 0 to l, psi 3 star, psi 5, d -x, again, orthogonal, gone.
05:09
Okay, now to the second term with each of the three, minus i -b -a, 0 to l, psi -7 star, psi -3, d -x, and i think as you realize by now, anything but 3 -3 -7 -5, all going to be zero.
05:33
So this is gone.
05:35
Plus, now i got minus i times i.
05:41
I squared is minus 1, so that becomes just b squared.
05:45
B squared, zero to l, size 7 star, size 7, dx, this is 1.
05:57
Then we have minus i, b, g, 0 to l, si 7 star, psi 5, dx, and this again, 0, 0.
06:12
So that's gone.
06:19
Now, notice we're kind of repeating what i was talking about.
06:23
350 out of 1 ,000, that's 35 % of them, would have been side 3, 400 out of 1 ,000, 40 % would be that.
06:35
Notice where when you add up 30 % plus 30 % plus 40 % plus another 25%, it gives you 100%, doesn't it? so all of them.
06:46
It's basically saying, yeah, all these put together, it's 100 % of all measurements are made up with these.
06:53
Now, last one, plus g -star a, 0 to l, psi -5 -star, psi -3, d -x, 0, plus g -star, i -b, 0 -0 to l, psi -5 -star, p -3, d -x, 0, plus g -star, ib, 7 d x equal to 0 now we have we have g times g star or g star times g to b doesn't order doesn't matter that is the modulus of g squared so to l psi 5 star sci 5 d x equal to 1 so notice only along the diagonal only along the diagonal do we have so 1 is equal to a squared plus b squared and plus g squared.
08:05
But in case you'll see it in a couple minutes, but a squared, the probability of finding your measuring psi 3.
08:17
So if you have 1 ,000 measurements, a squared will give you that percentage.
08:23
Likewise, b squared will give you the percentage out of the 1 ,000, and g squared will give you the percentage out of the 1 ,000 for each respective size.
08:32
That's how it is.
08:34
So notice, they add up to one.
08:37
They got two.
08:40
You know, they have to add up.
08:46
So we want the first part of this problem is to get g squared, or actually to get the modules of g.
08:53
So g squared is going to be one minus a squared, minus b squared.
08:59
I'm going to calculate this out for a reason, as you'll see it a couple of minutes.
09:06
So putting in the a and the b now, this gives me.
09:12
That the modules of g squared is 0 .3935.
09:19
So this gives me that the modules of g is what they wanted, the first thing, 0 .627.
09:32
So that's the first thing they asked for.
09:34
What is the modules of g? and there it is.
09:38
But again, understand what's going on here.
09:41
This a squared, again, if you're not clear on this, i'm going to show it to you in a minute.
09:46
This is the probability of getting side three.
09:48
This is the probability of getting side 7.
09:51
It's the probability of getting side 5.
09:54
They have to add to 1.
09:56
We can get nothing else.
10:00
So out of that 1 ,000 measurements, you get 1 ,000 out of 1 ,000 being those 3.
10:07
That 300 divided by 1 ,000, you know, it's 30 % is 0 .3.
10:16
So it'll be 0 .3 plus 0 .4.
10:20
You know, in my case, it would then be 0 .25.
10:23
It's obviously not this number here, but i'm just using my other example.
10:28
So that's an indication.
10:30
Yeah, they cover everything.
10:33
30 % are this, 40 % of that, 25 % are that.
10:38
When you have a large number of measurements, you can correlate that percentage with the probability, not if you have two measurements.
10:46
That's not right.
10:47
But if you've got to, actually it's got to be basically almost, if it has to be infinite, they've got to be really technically precise, but let's not even go there.
10:54
You have to have a large number of measurements, and you can correlate the probability with the percentage that you get of one particular state.
11:03
That's how it works.
11:06
So you see how it adds up.
11:09
It has to add up to one.
11:11
All right.
11:15
Now, it wants the expectation value.
11:20
Now, before we do that, let's go back.
11:24
Let's say we have 500 students taking a test, 15 of them have got 85.
11:32
The weighting of the 85 is 15 divided by 500.
11:37
It's a relative, it's an absolute weighting, not a relative waiting.
11:41
15 is a relative weight.
11:43
15 divided by 500 is an absolute rating.
11:46
Just like, just like 300 divided by 1 ,000 is an absolute probability.
11:55
So 15 divided by of 500 is an absolute weighting of the 85.
12:04
And then if you have somebody, and you got 25 getting 65.
12:08
25 divided by 500 is the weighting for the 65 score.
12:12
And you do that for everything between 0 and 100.
12:15
Inclusive.
12:18
Isn't that going to give you your average? now, you may be more used to basically thinking of dividing everything at the end by the 500, which is the same thing, isn't it? 15 times 85 plus 25 times 65, divided all the, add that up.
12:38
And then divide by 500, but what's different? only difference in terms of abstraction, how i'm viewing it.
12:48
So as a weighting, there you don't think about waiting, you're just thinking of, well, it's an average.
12:53
You know, that's how you do an average.
12:55
Well, let's talk about waiting...