2D Schrödinger time-independent finite difference and eigenvalues

Question

I'm learning about numerical methods to obtain the eigenvalues of a system. I have to find the eigenvalues for the time-independent Schrödinger equation but I'm having some difficulties understanding the real problem.

I was told to consider a 2D potential box (what's this? never heard about this before). The domain is $\Omega = (0,0)\times(1,1)$ and inside the domain the potential $V=0$ and outside $V=\infty$.

With that in mind the Schrödinger equation looks like $\displaystyle E\Psi = \frac{\hbar}{2m}\Delta\Psi$, is this right?

Now from my linear algebra class I know that the eigenvalues (I don't know how to correctly express this, so forgive me if I say something incorrectly) are those values such that the following equation holds $Ax = \lambda x$

Back to the Schrödinger equation, it looks like the previous equation for eigenvalues, where I think $\displaystyle A = \frac{\hbar}{2m}\Delta$ and $\lambda = E$, is this right?

Now if all previous was right, here's what I'm getting confused. If I want to know the function $\Psi$ why do I care to solve for the eigenvalues and find $E$?? I really don't get how the eigenvalues are used to find something else, what's the purpose of this?? I think I'm missing something really important here, if someone can explain me I would be really thankful.

Finally, to find the eigenvalues I have to assembly the matrix $A$. To do this I have to replace the derivatives of the laplacian with a finite difference scheme which gives me something like

$$p\Psi_{i-1,j}+p\Psi_{i+1,j}+q\Psi_{i,j-1}+q\Psi_{i,j+1}-4r\Psi_{i,j} = 0$$

The matrix that represent this will have $N$-rows and $N^2$-columns? Something like

$$A=\pmatrix{ a_{0,0} & a_{0,1} & a_{0,2} & 0 & 0 & ... & 0 & 0 & 0\\ 0 & a_{1,1} & a_{1,2} & a_{1,3} & 0 & ... & 0 & 0 & 0\\ ...& & & & & & ... & & ...\\ 0 & 0 & 0 & 0 & 0 & 0 & a_{N-2,N-2} & a_{N-1,N-1} & a_{N,N} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{N-1,N-1} & a_{N,N} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{N,N} } \cdot \pmatrix{ \Psi_{0,0} \\ \Psi_{1,0} \\ \Psi_{2,0} \\ ... \\ \Psi_{N,0} \\ \Psi_{0,1} \\ \Psi_{1,1} \\ \Psi_{2,1} \\ ... \\ \Psi_{N,N} }$$

I don't see how to assembly the matrix, what are the values of every entry $a_{i,j}$ how to get them from the finite difference equation? In 1D I can clearly see it but in 2D I don't.

UPDATE

I found a question in SE-Math of the importance of eigenvalues and eigenvector to help me understand it. The remaining question is how to assembly the matrix from the difference equation.

Answer 1

Finally got it. After establishing the size of the grid (i.e. the internal nodes) in the x and y directions each row of the matrix will correspond to how much each point contributes to the value $\Psi_{i,j}$. Then you will have a matrix of $N^2$ entries.

If I think of a grid with N = 3 (3x3) then matrix A will be something like

$$A=\pmatrix{ \Psi_{i,j} & \Psi_{i+1,j} & 0 & \Psi_{i,j+1} & 0 & 0 & 0 & 0 & 0 \\ \Psi_{i-1,j} & \Psi_{i,j} & \Psi_{i+1,j} & 0 & \Psi_{i,j+1} & 0 & 0 & 0 & 0\\ 0 & \Psi_{i-1,j} & \Psi_{i,j} & 0 & 0 & \Psi_{i,j+1} & 0 & 0 & 0\\ \Psi_{i,j+1} & 0 & 0 & \Psi_{i,j} & \Psi_{i+1,j} & 0 & \Psi_{i,j+1} & 0 & 0\\ 0 & \Psi_{i,j+1} & 0 & \Psi_{i-1,j} & \Psi_{i,j} & \Psi_{i+1,j} & 0 & \Psi_{i,j+1} & 0\\ 0 & 0 & \Psi_{i,j+1} & 0 & \Psi_{i-1,j} & \Psi_{i,j} & 0 & 0 & \Psi_{i,j+1}\\ 0 & 0 & 0 & \Psi_{i,j+1} & 0 & 0 & \Psi_{i,j} & \Psi_{i+1,j} & 0\\ 0 & 0 & 0 & 0 & \Psi_{i,j+1} & 0 & \Psi_{i-1,j} & \Psi_{i,j} & \Psi_{i+1,j}\\ 0 & 0 & 0 & 0 & 0 & \Psi_{i,j+1} & 0 & \Psi_{i-1,j} & \Psi_{i,j}\\ }$$

The importance of the eigenvalues and vectors can be seen in this thread

Answer 2

Many of your questions (including why it's called the "box potential" or "infinite well") are answered here: http://www.dealii.org/developer/doxygen/deal.II/step_36.html (Some of the answers are given in the "Results" section.)

Disclaimer: I'm one of the authors of this site and the program described there.