I am looking for all the possible forms in 3D space of the function defined as $$ \Psi(x,t) = \psi(x) e^{-it}$$
There is this funny constraint: $$|\Psi (x,t)|^2 = \psi^{\ast}(x)\psi(x) e^{it} e^{-it}$$ $$|\Psi (x,t)|^2 = |\psi (x)|^2$$
For every $t$, so $|\Psi|$ doesn't depend on t.
So far, I've found one possible form, a function looking like a spring like in this image (but with the x-axis inside of the spring):
As time goes by (yes, I am of the ones that have a very hard time to see time as an extra axis) the spring rotates exactly as a screw does
Do you guys know another form of this function in 3D?
EDIT: After the reply below, I am looking now forward to vizualize what the $\Psi$ function, where $\psi$ has the form $A(x)e^{it}$, would look like.
I am wondering if it looks like the function that is in the two following images... or not:
Here the $A(x)$ would be a sort of gaussian?
$\psi(x) = A(x) e^{i \phi(x) }$
Therefore,
$ \Psi(x) = A(x) e^{i(\phi(x) - t )} = A(x) \cos(\phi(x) - t) + i A(x) \sin(\phi(x) - t) $
Let $y = A(x) \cos(\phi(x) - t) $ , $ z = A(x) \sin(\phi(x) - t) $
To generate the screw in the picture, set $A(x) = A $ a constant, and $ \phi(x) = x $
Then
$(x,y, z) = ( x, A \cos(x - t), A \sin(x - t))$
For a fixed $t$ this is a spiral (helix) along the $x$ axis. As $t$ advances,the spiral rotates.