Shifty Spectra

Given equal probabilities that an emission or absorption occurs at all energy levels to all energy levels, why does not $-E_{1\rightarrow n}$, an emission line and $E_{n\rightarrow 1}$, where $n_2=n_1$, an absorption line, cancels?

When $n_2\lt n_1$, emission plots become absorption plots but the single absorption plot does not become an emission plot.  For the case of $n_1=1$ there cannot be a lower $n_2$, so there is no emission lines from $n_1$ to a lower energy level.  This does not mean that there is no emission lines  that would cancel the absorption lines from $n_1=1$.  A transition $E_{2\rightarrow 1}$ would generate a emission line that cancels the absorption line of $E_{1\rightarrow 2}$.

Do the forward transition and its the reversed transition cancels?  Only with numerical calculations can tell.

but we see that,

because the emission plots are not parallel for values of $n_2$ smaller than $n_2$ at the minimum point.  And in the following case its is not possible to tell graphically whether the forward transition and its reversed transition cancels,

Notice that $E_{2\rightarrow 3}$ is an emission line and $E_{3\rightarrow 2}$ is an absorption line.  All transitions to the left of the final energy state plot minimum point $n_{2\,\,min}$ have the opposite sign.  The corresponding reversed transition is at the minimum point and move downwards vertically to the final energy state.

It is also noted that for the plot $n_1=5$ another zero x-axis intercept occurs at $n_2=2$. And that for the plot $n_1=2$ another zero x-axis intercept occurs at $n_2=5$.

Both plots are coincidental.  This suggests that moving from $n_1=2$ to $n_2=5$ and in reverse, requires no net energy.  In moving from $n_1=$ to $n_2=5$, the loss in energy due to a decrease in $a_{\psi}$ at $n_2=5$ is made up for by the increase in orbital energy there, and vice versa.

It is possible that there is an energy gradient as the spectra line observations are being made.  When energy of the system is increasing, small particles tend to form and $n_1\gt n_2$.  When energy of the system is decreasing, big particles tend to form on separation after a collision and $n_1\lt n_2$.  In this way, $n_1\ne n_2$ and the emission lines are not coincidental with the absorption lines.  Emission and absorption that cancel do not occur with equal probability.

Which means, with no energy input to the system, the spectra line would disappear, but a different sets of lines reappears as the system cools.

Note: The difference between $\Delta E_{1\rightarrow 2}$ and $E_{1\rightarrow 2}$ is the overall change in energy state as demarcated by the definition of one single process and one energy state change of many in a more prolonged process.

In the case where separation follows coalescence,

$\Delta E_{1\rightarrow 2}\ne h.f$

instead there are both an emission line,

$E_{1\rightarrow n}$

and an absorption line,

$E_{n\rightarrow 2}$

As the small particle grows bigger its energy drops due to a larger $a_{\psi}$.