00:01
In this problem we have two operators a and b.
00:03
We're given their eigenstates and their associated eigenvalues.
00:06
The eigenstates do form an orthonormal set.
00:11
That means the scalar product, the inner product of one of the states with itself is 1 and between different states is 0.
00:22
And it's true also for the phi's also.
00:26
I didn't write it but it's true.
00:27
And they give us the relationship between the psi's and the phi's here.
00:32
Now the first question, it asks if you operate, you take a measurement of a, operator a, you get eigenvalue a1, what is the state after that measurement? well that state, it will collapse into the eigenstate that has eigenvalue a1, psi 1.
00:53
It will stay there forever.
00:54
You can take a million more measurements on that one state and it's going to give you a1 every single time.
01:01
Now part b wants to now take a measurement of b but it wants to know what is the probability of getting b1, what is the probability of getting b2, the two eigenvalues.
01:15
Well let's talk about probability first though.
01:17
Probability, say b1, what you have to do first is find the phi 1 component of psi 1.
01:29
Just like you would if you were given a vector, regular vector, a force vector, and you wanted to know the x component.
01:37
You could just look.
01:37
We can see that the phi 1 component is 3 fifths.
01:42
But how do you get it formally? how would you do it if you were doing a vector operation? dot product, i dot f.
01:49
That's a scalar product, an inner product.
01:51
Do the same thing here.
01:53
Our inner product is phi 1 star psi.
01:59
I'll just write it in one dimensional notation.
02:02
That's going to give me the phi 1 component.
02:10
That's not the probability.
02:11
The probability is take whatever result this is, the complex conjugate, multiply it together, the modulus squared is the probability.
02:20
Or you may be used to direct notation where you'd write like this.
02:27
Same thing.
02:28
Remember the psi is actually psi 1.
02:33
You could just put psi 1 in here.
02:35
But i like to use psi to represent the overall thing and not particularly one of the states.
02:41
So for b1, this would give me, because again it's an orthonormal set, the only thing that would give me is the only inner product that is non -zero is the one psi 1 star psi 1.
02:56
That's the only inner goal that's going to be the non -zero.
03:01
So you get out of this 3 fifths.
03:05
You got to square that.
03:07
So that's 9 25ths.
03:10
So b1, the probability, 9 25ths.
03:20
And i mentioned that at this point, you would have collapsed into phi 1.
03:29
Because you got b1.
03:34
The b2, the probability, we look for fifths, is the component, in this case the phi 2 component.
03:46
So 16 25ths.
03:51
And here you'd collapse into phi 2.
03:57
So there's our two probabilities, 9 25ths.
04:00
Notice it adds to 1.
04:02
It's got to.
04:03
You're going to get b1 or b2.
04:06
You're going to get one of them with certainty.
04:10
You're not going to get a third.
04:12
B1, b2, done deal.
04:14
Which one is a different issue.
04:16
But will you get b1 or b2? the answer is yes.
04:20
100 % certain you're going to get one of those.
04:23
You're going to get one in that set.
04:27
All right, now part c.
04:30
We're going to take another measurement of a at this point.
04:35
And we want to know what is the probability of getting a1 at that point.
04:41
And we'll talk more about that.
04:43
But we need to invert.
04:46
We need to invert these to get the phi 1 and phi 2 in terms of the psi 1 and psi 2.
04:52
So let's just do a little algebra.
04:55
4 phi psi 1, 1 5th, 12 phi 1, plus 16 phi 2.
05:04
What we're trying to do is when we add up these quantities i'm calculating, we're going to add or subtract.
05:10
We're going to only have one phi on the right.
05:15
So 3 psi 2, 1 5th, 12 psi 1, minus 9 phi 2.
05:26
So if we take the subtraction, 4 psi 1 minus 3 psi 2, 1 5th.
05:34
And we get 12 phi 1, plus 16 phi 2, minus 12 phi 1, plus 9 phi 2.
05:49
So the 12s go away.
05:51
This gives me 1 5th, 25 phi 2, 5 phi 2.
05:59
So we have phi 2 is equal to 1 5th, 4 psi 1, minus 3 psi 2.
06:07
There we have it.
06:09
Now we've got to do the same thing to get phi 2.
06:12
Just some simple algebra...