# 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:

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

- → 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 we have:

\frac{1}{√2}→ + \frac{1}{√2}↑

So, is the probability of passing \frac{1}{√2} ?

Not quite, because total probability should equal 1

But with a little help from pythagoras we have:

(\frac{1}{√2})^{2} + (\frac{1}{√2})^{2} = 1^{2}

\frac{1}{2} + \frac{1}{2} = 1

Each probability is \frac{1}{2}. At 45° that makes sense, right?

We can use pythagoras each time, or simply remember:

The probability of each state is the amplitude magnitude squared:

(\frac{1}{√2})^{2} = \frac{1}{2}

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**.

So 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 there is **a modest chance** that a photon can get through all 3 filters!

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. 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}