# Boundary conditions for two dimensional problems in Quantum mechanics

1. Oct 25, 2009

### PRB147

I am stuck at the problems of Boundary conditions for two dimensional problem in QM.
iIf we have a two-dimensional domain,
along the boundary, we can define two directions, one is tangential, the other is normal,
assuming that there is no current flowing in and out along the normal direction.
How can we define the boundary conditions?
To be specific, we have the following wavefunction in the domain
$$\psi({\vec r})=e^{i{\vec k}_i\cdot {\vec r}}+r e^{-i{\vec k}_f\cdot {\vec r}}$$
while outside the domain, we have
$$\psi({\vec r})=e^{i{K}_s S-K_n N}$$
$$K_s, K_n$$ is the tangential and normal components of the momentum $${\vec K}$$ outside the domain.
S,N are the coordinates of the position vector $${\vec R}$$

Last edited: Oct 25, 2009