Are quasiclassical minima isolated
I start to find astonishing that derivative for \langle \phi_z \rangle
at \phi_x=0
is not 0
. According to quasiclassics, if minima are well isolated there should be a sqare root dependance (\langle \phi_z \rangle = \sqrt{1-r^2}
).
If I draw quasiclassical energy for the parameters we have used for numerics I see that minima are not deep:
par = {Lz -> 4, Lr -> 1.3, Iz -> 0.32, Ir -> 0.24};
expr = (phir - r)^2/(2 Lr) + phiz^2/(2 Lz) -
Sqrt[Ir^2 phir^2 + Iz^2 phiz^2] ;
ex2 = expr /. par
Manipulate[
ContourPlot[
ex2 /. {phir -> x, phiz -> z, r -> rp}, {x, 0, +5}, {z, -5, +5},
Contours -> 100], {rp, 0, 5}]
Perhaps that makes wavefunctions interacting. In the toy example above I see that if one increases L_z
one gets a deeper minima. Would be nice to see what effect that would make on the plot.
To understand the behaviour there are three things which could be done:
- Evaluate
\Delta E/\hbar \omega
to see if it can be tuned as a large number. - Do numerics for a larger
L_z
or other parameters - Look if we can understand the numerics assuming a double well potential
- Plotting of
P(\phi_z)