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Jackson's electrodynamics problem 7.30: Number of photons in the superposition of plane waves

  • #1

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?
 
Last edited:

Answers and Replies

  • #2
nrqed
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What is the equation for the energy density in terms of electric and magnetic fields?
 
  • #3
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##
 

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