Why Should Amplitude Remain Constant?

The amplitude of the wave, $A_n$ is perpendicular to both $\lambda_n$ and $a_{\psi\,n}$.  If there are forces changing $\lambda_n$ and $a_{\psi\,n}$, these forces will have no effect on $A_n$.

So, because there is no force acting on $A_n$, $A_n$ remains unchanged and the system loses energy as the particle transit from a higher $a_\psi$ to a lower $a_\psi$.

$\lambda_n$ and $a_{\psi\,n}$ changes when we set $n_i \rightarrow n_f$, but this by itself does not change $A_n$.