Let's say there is an unknown, polarized light source shown in yellow, green and blue that passes
through a vertically polarized filter.
Now, in this particular video, we will mainly discussed.
The filter polarized state.
In four manners.
One, a word equally polarized state of the filter like this.
Horizontally polarized state like this.
Diagonally, polarized state.
Which is what he represented as a forward slash.
And another diagonally polarized state, which is represented as a backward slash, so henceforth,
if I speak forwards and backwards, slash, consider it to be the state of the light, which is diagonally
polarized.
So let's see, the yellow light after emerging from vertically polarized filter again passes through
vertically polarized filter and the pass rate.
Is 100 percent that is 100 percent affordance, get through 100 percent of the light passes through
in the case of green light after vertical polarization.
Let's say it's passed through a horizontal filter and you love so there is zero percent foredoomed that
gets through, that is all the light is blocked.
In the case of blue light.
After passing through vertically polarized fielder.
It is parsed through an angle of 45 degree.
You will absorb 50 percent of the light gets through.
And these experiments are totally verified.
Now, let's study and apply quantum mechanical equations to it, let's say our horizontal line is represented
as a gateway vector one zero and a vertical light with a vector zero one.
Now, these are both getting vectors to convert a vector to Abrar Vector.
You just need to do the transpose of the complex conjugate of the same vector.
And the kid gets converted to a Brabec the.
Will require this a lot when we do the inner product, as you see over here.
Off to Vector's.
In a general sense, if I want to explain what this equation or what is dead in this angle brackets,
if I want to denote in plain English.
How it looks like.
Here, why is the state of the light?
You remember in the earlier slide, we discussed unipolarity, source of light.
So here, just consider why is a certain speed of light?
And why is being parsed via a filter that filter speed is X, X is called as a filter or a polarizers
data.
So we need to find out what exactly is the probability of Y passing through X Fulda?
This equation that is the inner product of X and Y gives you the probability amplitude.
If we do the squaring off it.
We get.
The total probability.
Hmm.
Can you recollect something we discussed about probability in our earlier lectures that is probability
of finding a quantum particle is state, say?
Multiplied by its conjugate Psystar, which is SII Square in the same the inner product of the vector
Y with Brax Vector X multiplied by its conjugate is the probability.
So now you have an idea of the inner product, how it should be done.
And I told you that if you want to convert a great vector to a Brabec, do you just need to do the transpose
of the complex conjugate of the original vector and you get the Ravicher?
I've shown you an example.
So let's say you have a column vector X.
Which has elements A plus A, B and C plus I'll.
If you want to get the projector off it, you just do the transpose of the complex conjugate of each
element, so the column vector becomes the rule vector as shown here.
So the gravity X is equal to A minus A, B and C minus 80.
So Collum gets converted.
So if you see the gate vector X.
Is represented as one zero.
And get a better way as zero one.
So ideally, a vertical light cannot pass through a horizontal filter.
Hence, the probability is zero or zero percentage.
Similarly.
Or horizontal light passing through a horizontal filter.
It probably is one 100 percent percentage.
And a vertical light passing through a vertical filter, its probability is 100 percentage, which is
equal to one over here.
When we calculated by doing the inner product of Y with a Y eyebrow vector Y with thought Gateway Dawai
answer is one.
Back to our magic equation.
Here.
Consider it to be an apparatus which will measure the state.
We know it is the commission matrix.
In this case are measurable equipment, which is used to observe or measure the state.
Kate, Victor, it is the state of the light.
And LAMDA is the result of Zol or in mathematical terms, we call it eigenvalue.
I have selected a commission matrix, which.
Denoted by a plus symbol.
It means it can measure horizontal and vertical polarized like.
So the commission matrix I have selected is one zero zero minus one.
And the state vector is X, which is a horizontal line.
Represented as one zero.
So if we multiply it with St. Victor in.
The answer is one.
Multiplied by the same state vector, so over here, one is the result or the igun value.
Don't worry about how I have written this commission matrix, folks will see that in our future videos
at the moment just.
I assume that it is given.
Same is the case to measure vertical line, if I apply the same equation over here, Lambda is equal
to minus one.
Which is the result of equalitarian eigenvalue.
So as we got to reserves or eigenvalues earlier, plus one and minus one.
Let's hear plus, one represents a red light and minus one represent a green light.
So if I pass the horizontal line and measure using it, you know what is edge of a commission matrix?
I get a red light, which is plus one multiplied by the horizontal vector, and if I pass a vertical
line and measure using each our commission matrix, I get a green light or a minus one multiplied by
the vertical vector.
Classically, if the UN polarized light is passed through a vertical filter and then if a filter is
tilted at 45 degree with vertical axis, ideally 50 percent of light gets through here, i.e. represents
intensity.
So the output would be a square meter.
If theta is equal to forty five, then square meter is equal to half, which is 50 percent.
So the forward slash state of the light you would think could be a mixture or addition of horizontal
plus vertical vector like this.
This doesn't work, Asper quantum equation of a state passing through the polarizer.
So if forward slash passes through forward slash, ideally it should be one or a hundred percent.
That is the probability.
Should be one.
Ideally, we should get the probability probabilities one.
So now let's see, based on the same equation of inner product, what exactly is the probability that
we get when we pass a forward slash state of the light through the forward slash polarizer or diagonally
polarized?
Here is how it looks like.
If you calculate the probability we would get four or 400 percentage, which is ideally wrong, is no
maximum, the percentage should be 100.
So the answer to this puzzle is normalization.
We need to normalize the vectors and then use it in the same equation.
Let's check it out.
So a forward slash state is equal to get vector X plus get vector while divided by square root of two.
And a backwards state is equal to get vector X minus get vector way divided by a square root of two.
So this is how the forward slash state looks like.
In the column vector, we get the values as one Beirut to one Beirut to similarly, we can derive the
value for backwards state as well.
So as you see over here, this is how the backward slash state looks like.
Now, let's see if I want to pass the forward slash state of the light.
Why are Polarizer, which is in backward slash?
Then let's check out the probability of the passing rate or the photons passing through the polarizer
or the filter.
As you know, they are both distant apart.
Classically opposite, so the probability actually should be zero and we get the probability as zero.
So by now, we are crystal clear how to do the inner product in a product is very much similar to our
matrix multiplication.
We just multiply Abrar vector, we get a vector.
Otherwise, the same rule of matrix multiplication applies over here.
And that is how we calculate the inner product.
As we get the answer for the inner product, we square it.
That's basically the probability.
Off the pass through rate.
That is the photons passing through the polarizer.
So you know how a vertically polarized state of light looks like, and here the polarizer that we are
using after the vertically polarized state is there diagonally polarized filter.
So classically, we have seen that 50 percent of the light passes through.
Let's check it out by applying the same equation.
So here you see that our vertical state of light passing through or diagonally polarized filter.
We check out the inner product of it to the square of the answer and we get one by two, which is equal
to 50 percentage.
So if we take over same magic equation and let's say we have the commission matrix or the measurable
like this, which can allow all forward and backward slash state of light that is diagonally polarized,
like fully here is how it can be applied to a one state vector and the result or the eigenvalue we get
is one.
So now I hope the commission matrix is clear.
State of the vector is clear.
Eigenvalue is clear.
Cheers.