I want to solve bound states (in fact only base state is needed) of time-independent Schrodinger equation with a 2D finite rectangular square well V(x,y)={0,|x|≤a and |y|≤b V0,otherwise.
However, how to determine Vx and Vy in the 2D space? A definitely wrong method is making Vx={0,|x|≤aV1,|x|>a and Vy={0,|y|≤bV2,|y|>b.

Then, I find that a variable-separable bound state for finite 2D square well does not exist. Although analytical solutions exist in each region with a constant potential, problems occur when matching boundary conditions to keep the continuity of ψ(x,y). Unlike matching boundary condition at descrete points in 1D, in 2D we have to match boundary conditions along lines, e.g., f1(a)g1(y)=f2(a)g2(y)
Then, the question is, beyond separating-variable method, how to solve this problem?
BTW: Does anyone know that what kind (shape) of 2D well is solvable for bound states and how? (Potential with circular symmetry is excluded, because I know how to solve it. I want to find another shape of 2D well which is solvable.)
No comments:
Post a Comment