How would the Schrödinger equation be solved for curved barriers which change as a function of time, e.g., a paraboloid potential barrier with maximum height, V changing with time into a Hyperboloid potential barrier (with the same constant height, V, at its saddle point), which further changes into an ellipsoidal barrier. What would be the mathematical tools required for analysis?
Mathematical formulation:
Consider a n-dimensional Schrödinger equation of the form:
[n∑k=1∂n∂x2k−V(x,t)]ψ(x,α)=λ(α)ψ(x,α)
where the potential
V(x,t) depends on the column vector
x belonging to the n-dimensional complex space
Cn
Now let the elliptic potential be: the 2-gap Lamé potential
Ve(x,t)=2℘(x−x1(t))+2℘(x−x2(t))+2℘(x−x3(t))
Now this potential varies with time and changes into a hyperbolic potential of the form:
Vh(x,t)=aV0coth(αx)+bV1coth2(αx)−cV2cosech(αx)+d−cos(αt)
where
a,b,c,d and
V0,V1,V2 are constants.
How would the 3-d graph of the Lamé potential look like?
How do I handle this system, as I want the change of the potential functions(as a function of time)to be continuous, I.e., the elliptic 2-gap lamé potential changing to the hyperbolic potential and further to a parabolic potential with the wavefunction being continuous in every ϵ part of the barrier for every δ change in time. How would I solve such a system of time varying potential?