Half wave when \(n=\cfrac{1}{2}\) is just fanciful,
Does it really exist? The point of zero amplitude, where the forward and return path cross is a point from which emission occurs at full amplitude \(A_i\rightarrow A_f\) then \(A_i\rightarrow 0\); where emission occurs when \(\psi\) returns from an elevated state \(A_f\) to \(A_i=0\). \(A_f\) is emitted. In the case of the infinitely small, it may not be noticeable, but on a planetary scale, this big particle will be pulsating when the cross point spins as the big particle spins.
A pulsar!