Suppose that we have the four-vector potential of the electromagnetic field, [texA^i[/tex](adsbygoogle = window.adsbygoogle || []).push({});

The wave equation is given by $$(\frac {1}{c^2} \frac {\partial^2}{\partial t^2}-\nabla^2) A^i=0$$

Now the solution, for a purely spatial potential vector, is given by

$$\mathbf{A}(t, \mathbf{x})=\mathbf{a_k} \exp{i(\pm \omega_{\mathbf{k}}t-\mathbf{k}.\mathbf{x}}); \mathbf{k}.\mathbf{a}=0$$

To find the general solution we write the superposition as

$$\mathbf{A}(t, \mathbf{x})=\int (\mathbf{f(k)}\exp{i( \omega_{\mathbf{k}}t-\mathbf{k}.\mathbf{x}})+\mathbf{g(k)}\exp{-i( \omega_{\mathbf{k}}t+\mathbf{k}.\mathbf{x}})) \frac {d^3 \mathbf{k}}{(2 \pi)^3}$$

My question is that where this $$\frac {d^3 \mathbf{k}}{(2 \pi)^3}$$

comes from? Shouldn't it be $$d^3\mathbf{x}$$

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# General solution to the wave equation of electromagnetic field

**Physics Forums | Science Articles, Homework Help, Discussion**