Boundary conditions for two dimensional problems in Quantum mechanics

[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 $${\vec R}$$

 

[azntoon]

outside the domain.The boundary conditions for a two-dimensional problem in quantum mechanics are typically expressed as a combination of continuity of the wavefunction at the boundary and a condition on the derivatives of the wavefunction along the normal and tangential directions. For the example given, the boundary conditions can be written as:In the domain:\psi({\vec r})=e^{i{\vec k}_i\cdot {\vec r}}+r e^{-i{\vec k}_f\cdot {\vec r}}At the boundary:\psi({\vec R})=e^{i{K}_s S-K_n N}Continuity of the wavefunction:\psi(R)=e^{i{\vec k}_i\cdot {\vec R}}+r e^{-i{\vec k}_f\cdot {\vec R}} = e^{i{K}_s S-K_n N}Derivative along the normal direction:\frac{\partial}{\partial N}\psi(R) = -K_n e^{i{K}_s S-K_n N}Derivative along the tangential direction:\frac{\partial}{\partial S}\psi(R) = K_s e^{i{K}_s S-K_n N}

 

[Entropia]

perpendicular and parallel to the boundary, respectively

Boundary conditions in quantum mechanics refer to the constraints placed on the wavefunction at the boundaries of a system. In the case of two-dimensional problems, these boundary conditions are defined along both the tangential and normal directions of the boundary.

In this specific scenario, we have a wavefunction that is defined as a combination of two components, one inside the domain and one outside. The inside component is represented by a plane wave with a wavevector ${\vec k}_i$ and the outside component is represented by a plane wave with a wavevector ${\vec k}_f$. These two components are connected by a reflection coefficient, represented by the variable 'r'.

To define the boundary conditions, we need to consider the behavior of the wavefunction at the boundary. Along the tangential direction, the wavefunction must be continuous, meaning that the value of the wavefunction at the boundary must be the same for both the inside and outside components. This results in the equation:

e^{i{\vec k}_i\cdot {\vec r}}+r e^{-i{\vec k}_f\cdot {\vec r}} = e^{i{K}_s S}

where ${K}_s$ is the tangential component of the wavevector outside the domain and S is the coordinate of the position vector perpendicular to the boundary.

Along the normal direction, we can assume that there is no current flowing in or out of the domain. This means that the normal derivative of the wavefunction must be continuous at the boundary. This results in the equation:

i{\vec k}_i e^{i{\vec k}_i\cdot {\vec r}}-ir{\vec k}_f e^{-i{\vec k}_f\cdot {\vec r}} = i{K}_n e^{i{K}_s S}

where ${K}_n$ is the normal component of the wavevector outside the domain and N is the coordinate of the position vector parallel to the boundary.

These two equations, along with the given wavefunction, define the boundary conditions for this specific two-dimensional problem in quantum mechanics. By solving these equations, we can determine the values of the reflection coefficient 'r' and the wavevectors ${K}_s$ and ${K}_n$ outside the domain. These boundary conditions are essential in accurately describing the behavior of the wavefunction in a two-dimensional system and are crucial in solving many quantum mechanical problems.