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Imagine an area of space far from a black body with temperature $T$ that has emitted perfect black body radiation. So that an area of space has random photons.

Can we construct a "useful" wavefunction that could represent this? To get an exact wavefunction would presumable require us to know the exact entanglement between each pair of photons. But can we make an approximation? Let's ignore the polarisation so we approximate the electromagnetic field with a scalar field , $\phi$

My guess would be something like:

$$|\psi\rangle= \exp\left(\int\int\phi(x) e^{ix.k}\left( \frac{ k^3}{e^{k/T} - 1} \right) dx^3 dk^3\right) |0\rangle $$

using Plank's formula for black body radiation. And I'm using the exponential as a way to sum over 1, 2, 3,.. photons.

Is this anywhere close?

Qmechanic
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  • Not the answer, but are you familiar with thermal states of light? In a given radiation mode, they are convex combinations of different numbers of photons, weighted by Boltzmann factors (ie probability of $n$ photons is proportional to $\lambda^n$ and then the state is normalized) – Quantum Mechanic May 08 '23 at 19:15

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Not the wave function, but the density matrix - since we are dealing with a thermodynamic state. In fact, this density matrix is our starting point in modern description of the Black body radiation: \begin{align} \hat{\rho}&=Z^{-1}e^{-\beta H_{ph}},& Z&=tr[e^{-\beta H_{ph}}]& H_{ph}&=\sum_{\mathbf{k},\lambda}\hbar\omega_{\mathbf{k},\lambda} a_{\mathbf{k},\lambda}^\dagger a_{\mathbf{k},\lambda} \end{align} See How does radiation become black-body radiation? for further details.

Roger V.
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