Quantum Mechanics: numerical techniques

In this post, I am going to talk about what I learned from my exposure to numerical techniques.

I am taking quantum mechanics this term and during these weeks, we are exposed to numerical techniques to solve Schrodinger’s equation. In this process, I learned how to solve a Schrodinger’s equation numerically and two techiques were involved: the shooting method and the matrix diagonalization method.

The shooting method is a method that starts with knowing the value of the solution and the first derivative of the solution at some point and proceed from that point to get more of the solution. We assume linear relation locally for the solution and the first derive. When we know the first derivative, we can calculate the change of value of the next point from this point. Also, once we know the second derivative, we can calculate the first derivative at the next point by increasing or decreasing from this point. This process keeps going until we gets somewhere we feel sufficient to understand this system. On the other hand, matrix diagonalization method starts from choosing a basis which is the eigenstates of a known system. From that, we can expression the Hamiltonian in that basis and then get the eigenvectors of the Hamiltonian matrix when we choose a finite size of basis from the infinite. By finding out the eigenvectors and the eigenvalues of the matrix, we manage to get the eigenenergies and reconstruct the eigenstates of the system from the basis chosen. A shortcut that we could take is to split the Hamiltonian matrix into two parts: the parts with diagonal values as the eigenvalues of the chosen basis and whatever is left over from that. A good choice enables us to simplify the problem.