Electro-Magnetic Field

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Electric and magnetic fields obey Maxwell's equations: those equations determine how the electric field $\vec{E}(\vec{r},t)$ and the magnetic field $\vec{B}(\vec{r},t)$ at each point $\vec{r}$ in space vary in time $t$ given a (time- and space-dependent) distribution of electric charges and currents.

Amazingly, and very fortunate for the world we live in, there are nontrivial solutions to Maxwell's equations even if there are no charges and no currents. Those solutions turn out to propagate at the speed of light and indeed describe light waves. Each solution can be expanded in plane waves, for instance

$\vec{E}(\vec{r},t)=\int{\rm d}{\vec{k}}\sum_{\vec{\epsilon}\perp\vec{k}} A_{\vec{k},\vec{\epsilon}}\exp(i\vec{k}\cdot\vec{r}-i\omega t) +{\rm complex}\,\,{\rm conjugate}.$

There is an integral here over wave vectors $\vec{k}$ , which determines the propagation direction of each plane wave. There is also a sum over 2 orthogonal polarization vectors $\vec{\epsilon}$ both perpendicular to $\vec{k}$ .

When we quantize the electro-magnetic field both $\vec{E}$ and $\vec{B}$ become operators acting on a complex Hilbert space. The Hilbert space is infinite-dimensional: more precisely, there are infinitely many copies of infinite-dimensional Hilbert spaces, as for each mode, i.e. for each combination of wave vector $\vec{k}$ and polarization $\vec{\epsilon}$ , there is one such Hilbert space. Each one of those Hilbert spaces is spanned by states $|n\rangle$ with $n\geq 0$ a nonnegative integer. The number $n$ is interpreted as the number of photons in that particular mode.

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Categories: Concept | Quantum Optics

Electro-Magnetic Field

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