Quantum Polar Filter

Let's see how light behaves going through polarising filters!

You can try this yourself, use polarised lenses from sunglasses or a science supply shop (don't use circular polarisers that are common on cameras).

Polarization

Light is normally free to vibrate in any direction at right angles to its path.

But polarized light vibrates in one plane only:

unpolarized vs polarized
Light gets polarized when passing through a polarizing filter.

Photon

Before the filter we can write the state of the photon like this (see Unit Circle):

cos(θ) + sin(θ)

Where

quantum up-down left-right

 

First Encounter

quantum up-down left-right

 

What happens when the photon meets a polarising filter?

Let us say the filter is aligned in the left-right direction.

After passing through the filter the photon is either blocked or emerges as

with a probability of cos2(θ)

Probability

One of the basic rules in quantum mechanics is that the probability equals the amplitude magnitude squared, in other words:

Probability = |Amplitude|2

The || means magnitude of a vector, not absolute value.

This example may help:

quantum up-down left-right

Example θ = 45°

At 45° we have

cos(45°) + sin(45°)

cos(45°) = 1√2, and sin(45°) = 1√2 (see Unit Circle), so:

1√2 + 1√2

The probability of each state is the amplitude magnitude squared:

(1√2)2 = 12

Which makes sense: 12 + 12 = 1 (the sum of the probabilities must equal 1, right?)

Let's try another angle just to be sure, how about 30°?

cos(30°) + sin(30°)

cos(30°) = √32 and sin(30°) = 12, so:

√32 + 12

The probability of each state is the amplitude magnitude squared:

(√32)2 = 34 and (12)2 = 14

And 34 + 14 = 1

OK, enough examples, back to our filtering.

We are currently polarised in the left-right direction, like this:

100% probability left-right, 0% probability up-down.

Next Filter!

The next filter we use is up-down polarised.

But we currently have 0% probability of up-down.

So too bad. All gone. And the result is blackness.

quantum up-down left-right

 

 

But What If We Add a 45° In Between?

Now we place a third filter in between the other two, and orient it at 45 degrees.

Our "intuition" says that adding more filtering should block the light even more, making for a blacker black, right?

Well, let's work through the mathematics!

After the first (left-right) filter we have (as before):

Now the photon faces the middle filter at 45°

quantum up-down left-right

 

We have already seen an example of what happens at 45°. Well, the photon doesn't care what orientation our nice graph is at, so this works just as well:

quantum up-down left-right

The result is:

1√2 + 1√2

and faces a 1/2 chance of being blocked, and if it gets through it is now at:

 

Now the photon faces the final filter at 45°

Sorry? Isn't that 90°? To us maybe, but from the photon's current point of view it is another 45°. Like this:

quantum up-down left-right

The result is:

1√2 + 1√2

And again there is a 12 chance of being blocked, or getting through at:

The total for the last two filters is 12 × 12 = 14

Meaning that a photon that got through the first filter has a 1-in-4 chance of getting through the next two filters. So there is a modest chance that a photon can get through all 3 filters!

And it looks like this:

quantum up-down left-right

You can see that 0°⇒90° is black (lower center triangle), but 0°⇒45°⇒90° (upper center triangle) actually lets some light through. Adding that middle filter at 45° lets more light through.

Wow, mathematics rules!

What We Learned Here

In Quantum Physics our "common sense" view can be wrong, but we can use mathematics to get results that match what we actually observe.

We can use this special symbol to mean "in the x direction": x 

Photons behave according to probability, and

Probability = |Amplitude|2