# This is to clarify Landau's treatment of polarization in his classical

• Mindscrape
In summary, Landau's treatment of polarization in his classical fields book involves representing a monochromatic plane wave as a vector b, which can be expressed as a sum of two orthogonal vectors b1 and b2. These vectors are real and their components must be expressed in the same basis as the original plane wave. By factoring out a common phase alpha, we can write out the real and imaginary parts of the original plane wave and use them to determine the components of b1 and b2. This approach allows for the representation of elliptical polarization, but it is not necessary for b1 and b2 to be parallel to the basis vectors e1 and e2.
Mindscrape
This is to clarify Landau's treatment of polarization in his classical fields book.

Say we have a monochromatic plane wave
$$\mathbf{E}=\mathbf{E_0}e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}$$
and we want to make a vector
$$\mathbf{b}=\mathbf{b_1}+i\mathbf{b_2}$$
where
$$\mathbf{b_1}\cdot\mathbf{b_2}=0$$
such that
$$\mathbf{E_0}=\mathbf{b}e^{-i\alpha}$$
meaning that
$$\mathbf{E}=\mathbf{b}e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t - \alpha)}$$

I'm having trouble seeing how the original plane wave variables relate to the new plane wave representation. Specifically, what are the original components E1, E2, with their phases in terms of b1, b2, and alpha.

I started breaking E_old into unit vectors and projecting it onto b1 and b2 vectors respectively, but didn't really seem to be getting anywhere.

If E0 is written in some basis e1,2 then b1,2 have to be expressed in the same basis. Given that a common phase alpha is factored out, the vectors b1,2 should be real. Write out the real and imaginary parts of E0 from the equation after "such that" and take it from there.

Why do they have to be expressed on the same basis, i.e. you are implying that b1 is parallel to e1 and and e2 parallel to b2? That seems to be the approach Landau takes to arrive at the equation for elliptical polarization, but was a constraint imposed afterwards. Can't b1 and b2 really be in any direction though (I suppose they could be expressed by the rotation matrix of some angle in terms of e1 and e2)?

This approach to polarization seems so simple and makes me so feel so stupid for not being able to fill in the gap.

## What is Landau's treatment of polarization in his classical theory?

Landau's treatment of polarization in his classical theory involves considering the motion of a charged particle in an external electric field. He showed that the motion of the particle is affected by the polarization of the medium in which it is moving.

## Why is Landau's treatment important?

Landau's treatment of polarization is important because it provides a way to understand the behavior of charged particles in a medium. This is useful for studying a wide range of physical phenomena, from the behavior of electrons in a solid to the interaction of light with materials.

## What is the difference between longitudinal and transverse polarization?

In longitudinal polarization, the direction of polarization is parallel to the direction of propagation of the wave. In transverse polarization, the direction of polarization is perpendicular to the direction of propagation of the wave.

## How does Landau's treatment relate to quantum mechanics?

While Landau's treatment is based on classical mechanics, it can be extended to include quantum effects. In fact, Landau's treatment provides a starting point for understanding more complex quantum phenomena related to polarization, such as the quantum Hall effect.

## What are some applications of Landau's treatment of polarization?

Landau's treatment of polarization has numerous applications in various fields of physics, including condensed matter physics, optics, and plasma physics. It is used to understand the behavior of materials, the propagation of electromagnetic waves, and the dynamics of charged particles in plasmas.

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