Emission Spectrum Simplified

Unfortunately in this scheme of things, we have $\psi$ balls of various sizes, $n$ in collisions.

When 2 particles of size $n=1$ coalesce 2 photons $E_{1\rightarrow2}$ are emitted, and when 2 particles of sizes $n=1$ and $n=2$ coalesce 2 different photons $E_{1\rightarrow 3}$ and $E_{2\rightarrow 3}$ are emitted.

This is not the simple scheme where emissions as the result of hops from energy level to energy level has equal intensity, but in the example above, $E_{1\rightarrow 2}$ has twice the intensity of $E_{1\rightarrow 3}$ and $E_{2\rightarrow 3}$.

The plot below shows ((x+1)^(1/3)-x^(1/3))/(((x+1)*x)^1/3), ((x+2)^(1/3)-x^(1/3))/(((x+2)*x)^1/3) and ((x+3)^(1/3)-x^(1/3))/(((x+3)*x)^1/3).

where particle of size $n=1$, $n=2$ and $n=3$ coalesce with particle of size $n=x$.

We see that the highest energy transition occurs with $E_{1\rightarrow n=large}$, when a basic particle $a_{\psi\,c}$ (ie. $n=1$) coalesces with a large particle $n\rightarrow 77$.  Two photons are released $E_{1\rightarrow n=large}$ and $E_{large\rightarrow large+1}$.

Emission occurs in bands as $n=x$ increases and such bands narrows with increasing $n=x$.

In the graph above, the top most three horizontal lines maroon, red and blue correspond to $x=1$.  The next band of maroon, red and blue correspond to $x=2$.  Each band progressively narrows as $x$ increases.

As $x$ increases, all graph approaches asymptotically to zero, ie as $x$ increases all emissions due to the coalescence of particles of various sizes, $n$ approaches zero.

Good night.