Numerical integration of the Schrödinger

Numerical integration of the Schrödinger equation ***Please read the file under additional files. 6.1 The Problem Solve for the stationary states of an electron in a ramped infinite square well. 6.2 Introduction A basic problem in quantum mechanics is to find the stationary states (“energy levels”) of a bound system, using the time-independent Schrödinger equation (TISE) (-h^2(bar)/2m)(d^2/dx^2)(psi(x)) + V(x)*psi(x) = E*psi(x) In this activity, we will look at the numerical integration of the differential equation. The problem we face here is not that of the initial value problem of mechanics, that we dealt with earlier. We can’t just start from some initial value, use numerical integration, and end up somewhere after some time. Instead, solving the TISE is a boundary value problem. There is an unknown quantity (E) in the differential equation, and we need to solve for this, subject to constraints on the boundary conditions of the wave function. How to go about this? How do we tell if we have a correct value for E (called an eigenvalue)? We have to look at the properties of the required solution. We find that if E is an energy eigenvalue, the wave function ψ(x) has the expected behaviour at the boundaries. If we choose the wrong value for E, then the wave function will not have this property. Let us consider a specific example. Suppose that we have an electron in a ramped infinite square well with walls at x = 0 and x = a: