\(x=e^{ -\cfrac {A( x-1) }{ n^{ 2 } } }\)
where \(x=1\) is an immediate solution. It is a unit circle onto which \(n\) cycles of wavelength \(\lambda\) can be packed, where \(n\in Z^{+}\), \(0\lt n\lt n_{\infty}\), and \(n_{\infty}\rightarrow\infty\).
\(2\pi.x=2\pi=n\lambda\)
Although not a "natural" solution for a particle of unit radius (1 m in SI units), it is a valid solution to the equation for \(\psi\). If a particle can be sufficiently isolated (1 m radius), then it can be a high frequencies, ultra-broadband, discrete frequency resonator.
And be made to resonate at,
\(f=\cfrac{nc}{2\pi}\)
for \(n=1,2,3...\)
Very interesting.
