I am working on a simple example of how to numerically solve the time-independent Schrodinger Equation for the infinite square well. I've used the Euler Method to find values of the wave function, $\psi (x)$, but now I've just realized something - I have no clue how to determine the energies from this! I know that analytically this comes down to applying the Hamiltonian to the wavefunction, but how do I find the energies numerically??
I can provide more info if needed.
Compute $$E = \int dx \psi^*(x)\left(\frac{-\hbar^2}{2m}\nabla^2 + V(x)\right)\psi(x) $$
For a lot of the questions if you simply want to know about the fundamental energy of the system you don't even have to solve the Schrödinger Equations. Just simple dimensional analysis will tell you the order of the energy of the system. It is much much quicker and you will get a very good intuitive understanding of the system. With Dimensional Analysis, you can construct energies with simple factors like $\bar h$ and $f$. This is much easier than trying to solve the SE for complicated systems.
