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).

Photon

Here is a nice illustration of a photon:

photon illustration

It tries to show the idea of an energetic fuzzy wave moving at high speed, but is more artwork than science. There aren't any good photos of photons mostly because we need many photons to make photos.

We can write down the state of the photon like this:

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 are currently at 0% up-down.

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 with 3 filters (left-right, 45°, up-down) there is a modest chance that a photon can get through!

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.

Wow, mathematics rules!