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 \(A_n\).
