$\psi$ Algebra Don't Add Up, Shaky

$x_z$ is independent of $a_\psi$.  Increasing the frequency of oscillations of $\psi$ changes $a_\psi$ but does not change $x_z$.  Adding $\psi$/charges, changes the peak value of $\psi$ at $x_z$, but does not change $x_z$.

It is likely that,

$x_z=a_\psi$

only for $n=1$.  This is without proof.

What is $x_z$ physically?  If $\psi$ can be detected, it is the radius of the sphere confining $\psi$.  The first level of oscillations $n=1$ is when $a_\psi=x_z$.  $x_z$, the radius of $\psi$ determines this fundamental frequency.

$x_z$ is constrained by the resistance of space and time dimensions to $\psi$.  When $\psi$ is fully manifested, the particle is has no mass.  This occurs when $\psi$ oscillates between one space and one time dimension.  When $\psi$ oscillates between two space dimensions, part of the particle is mass and part of it is $\psi$.

Changing the resistance to $\psi$ changes $x_z$.

When

$x_z=a_\psi$,  $n=1$

changing temperature will change the resistance to $\psi$ and so changes $x_z$ and thus changes $a_\psi$.

The presence of another $\psi$ field changes $x_z$.  $\psi$ due to the particle, gets squeezed.  If the applied $\psi$ field is oscillating then the change in $x_z$ also oscillates and thus when

$x_z=a_\psi$,  $n=1$

the fundamental frequency oscillates as $a_\psi$ oscillates.

So, the application of an oscillating magnetic/electric field will oscillates the fundamental frequency of the particle.

Note:  $\psi$ is energy density, adding densities does not change the total volume; the resulting sum is still energy per unit volume.