# 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:

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

- → means left-right direction, and
- ↑ means up-down direction:

### First Encounter

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 cos ^{2}(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}

This example may help:

### Example θ = 45°

At 45° we have

cos(45°)→ + sin(45°)↑

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

\frac{1}{√2}→ + \frac{1}{√2}↑

The probability of each state is the amplitude magnitude squared:

(\frac{1}{√2})^{2} = \frac{1}{2}

Which makes sense: \frac{1}{2} + \frac{1}{2} = 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°) = \frac{√3}{2} and sin(30°) = \frac{1}{2}, so:

\frac{√3}{2}→ + \frac{1}{2}↑

The probability of each state is the amplitude magnitude squared:

(\frac{√3}{2})^{2} = \frac{3}{4} and (\frac{1}{2})^{2} = \frac{1}{4}

And \frac{3}{4} + \frac{1}{4} = 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**.

Too bad. All gone. And the result is **blackness**.

### 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°

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:

The result is:

\frac{1}{√2}↗ + \frac{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:

The result is:

\frac{1}{√2}↓ + \frac{1}{√2}→

And again there is a \frac{1}{2} chance of being blocked, or getting through at:

↓

### The total for the last two filters is \frac{1}{2} × \frac{1}{2} = \frac{1}{4}

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:

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!