Dear people in the field of Quantum mechanics,
There are many online explanations about the semi-analytical solution for the problem of particle in a 1D finite potential well as well analytical solution for particle in a 2D box with infinite potential on barriers.
https://en.wikipedia.org/wiki/Finite_potential_well
Basically, the continuity of wavefunction determines the eigen energies in such problems. However, online resources seems to be empty when it comes to the problem of particle in 2D finite potential well. I have only found a question in a physics forum which I have found the given answers unclear.
https://physics.stackexchange.com/questions/73571/how-to-solve-bound-states-of-2d-finite-rectangular-square-well
Anybody can recommend textbook (or a robust explanation ) where I can follow a semi-analytical solution of bounding states in a 2D rectangular (Not the square) finite potential well?
I do know that the first bonding states are not far from bounding states in a 2D box with infinite potential on barriers. However, here I require very accurate eigen energies and I need a semi-analytical answers {if it exists as a general answer} and not interested in numerical approaches.
thanks in advance.
Are you interested in the potential drawn in your figure, or in the usual finite square well, defind by
V(x, y) = -V_0 (max(|x|, |y|))
Schrodinger equation in a 2D rectangular well has variables separating. It can be considered as two problems for one-dimensional wells. Difficult (computational, but not conceptual) problem might arise if the boundary condition does not separate in the same rectangular coordinates and some caution must be exercised about that.
You may be right, your picture looks like two orthogonal channels crossing.
The only solution I know is cover it all with a mesh, and later diagonalize the matrix of the tight binding equations. This problem does have bound states.
Regions with higher potential energy, automatically give you lower amplitude
for probability density. Its been worked on before, known in the field of nanostructures.
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