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: Ex2/a2 + Ey2[/SUP]/ b2 − 2ExEycos 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
Ex = acos(ωt) and Ey = bcos(ωt - δ)
E = Ex + Ey
The Attempt at a Solution
Ex = acos(ωt)
Ey = bcos(ωt)
Ex/a = cos(ωt)
Ey/b = cos(ωt - δ)
I assume I can use Euler formula and say eiΘ = cosΘ + isinΘ
So I get
Ex/a = ei(ωt)
Ey/b = ei(ωt - δ) = eiωt / eiδ
Ey eiδ/b = eiωt
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.