Questions on Plane Wave Superposition

[andrew chen]

Homework Statement

Starting with the expression for the total energy of arbitrary superposition of plane electromagnetic waves in otherwise empty space, Show that the total number of photons (defined for each plane wave of wave vector $\vec{k}$ and polarization $\vec{\epsilon}$ as its energy divide by $\hbar ck$) is given by the double integral $N=\frac{\epsilon_0}{4\pi^2\hbar c}\int d^3x\int d^3x'\frac{\vec{E}(\vec{x}, t)\cdot\vec{E}(\vec{x}', t)+c^2\vec{B}(\vec{x}, t)\cdot\vec{B}(\vec{x}', t)}{|\vec{x}-\vec{x}'|^2}$.

Relevant Equations

$\vec{E}(\vec{x},t)=\mathbf{\mathfrak{E}}e^{ik\vec{n}\cdot\vec{x}}$, $\vec{B}(\vec{x},t)=\mathbf{\mathfrak{B}}e^{ik\vec{n}\cdot\vec{x}}$, and $\mathbf{\mathfrak{B}}=\sqrt{\mu\epsilon}\vec{n}{\times}\mathbf{\mathfrak{E}}$
$\frac{1}{|\vec{x}-\vec{x}'|}=\int d^3k\frac{e^{i\vec{k}\cdot(\vec{x}-\vec{x}')}}{k^2}$

First, I have a question about supereposition of the plane waves - whether the direction of all such plane wave is same, i.e. $\vec{n}$ is in some direction. If not, I think it would be $\vec{E}(\vec{x}, t)=\int\mathbf{\mathfrak{E}}(\vec{k}')e^{i\vec{k}'\cdot\vec{x}-i\omega t}d^3k$. Besides, how to express the energy of such plane wave. I think it would certainly be $\int d^3x$ $u(\vec{x}, t)$. However, I can't figure out how to relate each specific plane wave's energy to the superposition, for energy isn't linear quantity and $\omega$ also relates to wave vector $k$ by $\omega = kc$. I think that deriving the equation Jackson wants may require such formula $\frac{1}{|\vec{x}-\vec{x}'|}=\int d^3k\frac{e^{i\vec{k}\cdot(\vec{x}-\vec{x}')}}{k^2}$. Could someone give me some hints to solve it, tks?

 

[nrqed]

What is the equation for the energy density in terms of electric and magnetic fields?

 

[andrew chen]

What is the equation for the energy density in terms of electric and magnetic fields?

It would be $\frac{\epsilon_0}{2}|\vec{E}|^2$ & $\frac{1}{2\mu_0}|\vec{B}|^2$