I'm currently simulating a one-dimensional double Gaussian well potential numerically, and have been asked to find parameters corresponding to the overlap in potential and wave functions from my determined bound energy states.
Using the time-independent Schrödinger equation along with a potential of the form
$$ V(x) = -V_0 \left( e^{-(x-x_0)^2} + e^{-(x+x_0)^2} \right), $$
for some $V_0$ and $x_0$, I'm successfully able to simulate the system and get the bound states, but I've also been asked to use a supposed generalisation of the Hamiltonian in matrix form:
$$\begin{pmatrix} \epsilon_0 - A & W\\ W & \epsilon_0 - A\end{pmatrix}, $$
(where $\epsilon_0$ is the bound state energy at infinite separation, $A$ represents the overlap in potential wells, and $W$ represents the overlap of the wave functions in the two wells), with eigenvalues $\epsilon_{1,2} = \epsilon_0 - A \pm W$.
My task is to determine $W$. I assumed at first it was as easy as taking the difference of the first two bound states (ground and first excited) and dividing that by two, but supposedly $W$ is supposed to vary with energy states (which makes sense, since higher energy wave functions have larger width), but I honestly do not understand how to determine this as a function of energy level - for the system I'm analysis, there are a total of three bound states, and my supposed method would only yield the answer for two of them. Could the first two be the same, and the third different?