Mathematical Description Of Polarization

Sunday, November 18, 2018

Mathematical Description Of Polarization

Polarization

Let's look at a very simple system in quantum mechanics in the polarization of a photon (spin is mathematically the same as polarization). This system is very simple as there are only two orthogonal states for the polarization of a photon.

Light is an electromagnetic wave consisting of an electric field with a magnetic field oscillating perpendicular to it (electric + magnetic therefore we call them electromagnetic waves). A plane polarized wave is one where the electric field oscillates parallel to direction of polarizer and the magnetic field is perpendicular.

A polarizer lets through all light with its electric field aligned with the polariser, blocks out perpendicular electric fields and has a probability of letting anything between through.
This means when using the particle model for light, there are only 2 possible outcomes of a polarization experiment and these are probabilistic.

Let's define to two orthogonal states of polarization.

Light polarized in x direction = |x> = column vector (1,0)
Light polarized in y direction = |y> = column vector (0,1) (Blogger does not let me type column vectors and so you will just have to imagine these as column vectors!)

These are 2 distinct measurable states for the photon and 1 experiment can distinguish these states. They therefore can be viewed as basis vectors in a 2D space where the 2 dimensions correspond to 2 mutually exclusive polarization states.

This along with the properties of the inner product operation tells us that
<y|x> = <x|y> = 0
<y|y> = <x|x> = 1

In the states |x> and |y>, the photon is in a definite state and is not in some weird superposition of both. This means these states are eigenvectors of the hermitan operator corresponding to the polarization of light.

We know the eigenvectors so now we want to decide what eigenvalues to assign and from there find the hermitian operator.

There must be an Hermitian operator corresponding to the polarization of light. Let's call it P.
If we were doing a polarization experiment, we might write +1 in our results if the photon is horizontally polarized and -1 if it is vertically polarized. The way we would be able to tell is if the photon passes through a horizontal polarizer then we know its horizontally polarized and so we can assign it +1 and if it doesn't -1. Our operator should therefore have the property that P is +1 if the polarization is horizontal and -1 if it is vertical.

P|x> = +1|x>
= |x>

P|y> = -1|y>
= -|y>

We can now define our operator as the matrix and rewrite our equations in terms of vectors and matrices.

What about a 45° polarized photon? If we place a polarizer 45°, we know all the photons that are let through it are 45° polarized photons. If we then try and pass these through a vertical polarizer then intuitively, we can say it has a 50% chance of going through. But how can we describe the state of the photon polarized at 45° using our bra-ket notation? Our basis vectors are |x> and |y> and so we can describe the photon in terms of these.

Intuitively the photon has equal components in the x and y directions and so a guess might be

|x> + |y>

This description won't work however because it is not normalized and so it can give probabilities above 1.

1/√2 |x> + 1/√2 |y>

is the correct description of a 45° photon polarized. This is because for normalized vectors, the sum of the squares of the coefficients = 1.

A photon in this state has a 50% chance of going through a vertical or horizontal polarizer. It is in a superposition of the two original states with equal coefficients. Let's call this state |/> and we can describe it using column vectors or our bra ket notation.

There is also an orthogonal state to this and this is a photon polarized at -45° (perpendicular to the 45° polarizer).

This state is

|\> = 1/√2 |x> - 1/√2 |y>

We can check this is orthogonal to the first state by taking the inner product

|\> x |/>*

= </|\>

= 0

Therefore we can also describe photons polarized at 45° angles.

We can now also mathematically show that a photon polarized at 45° has a 50% chance of passing a vertical or horizontal polarizer.

We take the inner product of the photon with the configuration that we are testing for and then square the result.

| <y|/> |²

= ( <y| (|x>/√2 + |y>/√2) )²

= ( <y|x>/√2 + <y|y>/√2) )²

= ( 0/√2 + 1/√2) )²= ( 1/√2 )²
= 1/2

Polarization at a 45° angle is something you can measure and so it also corresponds to an hermitian operator.

It is a rotation of the original observable we had for polarization horizontally and vertically.

We can assign the eigenvalues +1 for the state |/> and -1 for |\>.

We will call our new observable P'

P'|/> = +1|/>

= |/>

P'|\> = -1|\>

= -|\>

We can again write this in terms of matrices and vectors. (The matrix would be the same matrix as the matrix describing P).

Eigenvectors are states for which the observable has a definite value. Note that there are no vectors which are eigenvectors of both the 45° polarization and the x y polarization. This means there are no states for which the photon is definitely polarized at the 45° angle and in the x or y direction. The two polarization's are therefore incompatible. There cannot be one experiment which can say both the x y polarization and the 45° polarization of a photon at the same time.

In the next post we will try to start generalizing these results and consider how we would describe a photon polarized at an arbitrary angle and how we can use this to work out the chance of a photon going through any polarizer.

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