00:02
In this prime, you're given a spin state, but we're given it in matrix form.
00:11
And we're also going to need the operators, spin operators, x, y, and z components of those.
00:19
This is for electrons, so it's h -by -over -2 that we have there, two -by -2 matrices.
00:29
The matrices themselves, leaving out the h -by -over -2, these are poly -spin matrices.
00:38
Now, what's nice about when you're given it in matrix form, in other places you've got to go find the eigenstates.
00:49
You're going to have to find, you might want to expand the state in terms of the basic, you know, eigenbasis.
00:59
You need all the coefficients, which you certainly can do even here if you wanted to.
01:05
But what's nice about knowing the state, knowing the operators that you're going to work on, it's just a matrix problem then.
01:14
It's just a matrix problem.
01:18
Part a wants to know the normalization constant a.
01:22
So 1 is equal to the hermission conjugate times the original matrix.
01:29
Spin matrix.
01:33
Remember, transpose, it takes complex conjugation.
01:37
So 1 -2 element becomes 2 -1 element, complex -conjugation.
01:43
So 2 .1 is 1 -2, complex conjugated.
01:48
It's a better way to say it.
01:51
So, a star is just a number, so it's just complex, it's complex conjugated.
01:59
You can think of a being multiplying each one, and you have to complex conjugate each one when you transpose.
02:08
So it becomes 2 minus 3i.
02:12
Remember you have to take the complex conjugate.
02:15
But notice what is this position here is row 2, column 1.
02:26
So 2 -1 ends up in 1 -2, complex conjugate, and there it is.
02:33
And let me just fix this behind here.
02:36
So that is the hermitian conjugate.
02:43
Then we multiply the rest, 2, 3i.
02:52
Now the a star a, that just gives me the modulus of a squared, and we'll forsake a simplicity later on.
03:00
We'll just choose it as real.
03:03
Don't worry about modulus of a real number squared is the number, is the number squared.
03:08
Nothing fancy has to be done.
03:10
All right, now, this is going to end up as a single number.
03:16
You go across the row, down the column.
03:20
This is one by two, two by one, leaving you with one by one.
03:25
One by one is just a number.
03:29
So 2 times 2.
03:31
4.
03:32
Minus 3i times 3i.
03:34
So this is minus 9 times i squared.
03:38
I squared is minus 1 plus 9.
03:45
So modulus is a squared times 13.
03:50
So a, we can choose that a, and i said, like i said, we're going to choose it real.
03:55
We could add in a, we could add a complex part of this, no reason to.
04:02
A is equal to 1 over the square root of 13.
04:08
So our spin state now, spin matrix, really, you know, well, i shouldn't use that because that kind of connotates these.
04:18
Our spin state over square root of 13, 2 over 3i.
04:24
Now it is properly normalized.
04:26
So that was part a to normalize.
04:32
Now part b wants the expectation value of sx, sy, and s.
04:38
And again, if you want to find the probability that you are in the plus 1 half x states, you can do the classic expectation calculation where each value that you can have is weighted by the probability.
05:02
But we don't need to do that here.
05:05
We can continue doing matrix operations.
05:07
So the expectation value of sx is going to be, permission, conjugate, sx times the state, original state.
05:24
Okay, and we'll have the a squared, h bar over 2, that's from the sx, i could just come out, be out of our way, and now we have to deal with the matrices.
05:40
2, minus 3i, 0, 1 ,000, 1 .0.
05:54
2.
05:58
And we can put it in 1 over 13 here, h bar over 2.
06:04
I'm going to write everything pretty much leaving h bar over 2 alone.
06:08
It's a common factor in spin 1 half particles, so i just leave it alone.
06:17
So let's do this as a two -step process.
06:23
So let's multiply the sx times the spin state.
06:29
We're doing this part here.
06:31
All right.
06:33
What do we get? 0 1, multiplying by 2 over 3i.
06:41
So 0 times 2, 1 times 3i.
06:45
3i.
06:48
Then for the bottom element, 1 times 2, 0 times 3i over 2.
06:57
Now we do the same time multiplication we did earlier.
07:01
Notice when we go across and down, that's given me row 1, column 1, the 1, the 1 1 element.
07:11
You go here, now this is row 2, 1, 2 -1, element.
07:19
That's how it works.
07:20
If you're not used to matrix operations, but that's how it works.
07:24
Okay, so now 1 over 13, h -bar over 2, 2 times 3i.
07:33
Oh, i should mention, your problem showed the parentheses indicating the matrix, i use square brackets.
07:41
It cuts down on any confusions with things like this...