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## Homework Statement

Before diving into the quantum-mechanical superposition principle, let’s get some practice with superposition in classical physics. Consider an electromagnetic wave propagating in the z-direction, which is a superposition of two linearly polarized waves. The electric field vector in the wave is

E = Ex + Ey, where Ex = a cos(kz − ωt), Ey = b cos(kz − ωt + δ). (1) The parameter δ is a real number between −π/2 and π/2, and indicates by how much the two components are out of phase. Look at the behavior of the electric field at some fixed value of z, say z = 0 for simplicity.

a) [2pt] Describe what the electric fields Ex and Ey are doing as a function of time.

b) [4pt] Show that there is a simple relation between Ex and Ey which does not involve t. Namely, you should find the following: E

_{x}

^{2}/a

^{2}+ E

_{y}

^{2}[/SUP]/ b

^{2}− 2E

_{x}E

_{y}

**cos δ/**ab = constant. (2) Express the constant in the right-hand side of (2) in terms of the phase shift δ.

I am trying to do b[/B]

as I found in a) that

E

_{x}= acos(

**ωt) and E**

_{y}= bcos(**ωt -****δ)**

## Homework Equations

E = E

_{x}+ E

_{y}

## The Attempt at a Solution

E

_{x}= acos(

**ωt)**

E

E

E

E

_{y}= bcos(**ωt)**

E

_{x}/a = cos(**ωt)**

EE

_{y}/b = cos(**ωt -****δ)**

I assume I can use Euler formula and say e

So I get

I assume I can use Euler formula and say e

^{iΘ}= cosΘ + isinΘSo I get

E

_{x}/a = e^{i(ωt)}**E**_{y}/b = e^{i(ωt - δ)}= e^{iωt}/ e^{iδ}So

**E**_{y}**/b =****e**^{iδ}**e**

I assume I set them equal to each other but I don't get the terms that I want for the LHS and the RHS becomes 0.^{iωt}I assume I set them equal to each other but I don't get the terms that I want for the LHS and the RHS becomes 0.