\[fig:stat\]. The incoming states of the final scattering are almost real photon and electron and have large momentum spreads. These photon and electron are described by superpositions of momentum states of large momentum uncertainties with a suitable energy weight.

![Final Thomson scattering in which one photon and one electron interact and the photon in the final state is detected. The black box shows many-particle states. ](fig2.eps)

\[fig:stat\]

### Statistical model

The weight of the superposition of the initial states at the final scattering is determined using its previous scatterings and identical particle effects. From the facts that the waves have large momentum spreads after the macroscopic distance and that the amplitude of the Thomson scattering in low energy of the range 3000 - 4000 K is constant and spherically symmetric, waves of the photon and electron in the initial state of the final scattering are regarded as spherical waves that are superpositions of plane waves of all orientations.

In the many-particle state, identical particles satisfy either Fermi-Dirac statistics or Bose-Einstein statistics, and the total energy conservation law $$\begin{aligned}
\sum_i^N n_i E_i=E,\end{aligned}$$ is satisfied. An average occupation number of the state ${\vec p}$, i.e., one particle distribution, is given by Bose-Einstein 