Supplementary note to "Simple Introduction to Computational Quantum Mechanics"

A user on Reddit helpfully pointed out my "hint" for why the wave functions are (nearly) zero at the boundaries was pretty bad. Here I try to provide a clearer explanation.

First, we note that something has to happen at the boundaries of our limited universe. This "something" is called, quite logically, a "boundary condition": a description of what happens at the end of the world.

Consider the matrix equation we derived in the main article. The general element is (in units of \( \hbar = m = 1 \) )

\begin{align} E\psi _n = \frac{-1}{2(dx)^2}\bigg( \psi _{n+1}-2\psi _n + \psi _{n-1} \bigg) \end{align}

Now, if we take the first element ("the boundary"), we note an awkward scenario, namely the fact that \( \psi _{n-1} \) is poorly defined. If \( n=1 \), then what do we mean with the element \( n-1 = 0\)? In our system, we chose to limit the x-interval to values between 0 and 4, and the \( \psi _{n-1} \) is not explicitly visible in the matrix multiplication.

What does it mean that that \( \psi _{n-1} \) is not present in our matrix multiplication for the first term? It, of course, means that it is zero. That is to say, we make the claim that "before" our first point, the wave function was exactly zero. That is the boundary condition that we are working with. (The first point is therefore not exactly zero, but very close, and gets closer as you increase the number of points).

Now, what options do we have for the boundary condition? One very important choice is \( \psi _{0} = \psi _N \), that is, saying the wave function "before" our first point had the same value as it does at the end of our interval. This is called the periodic boundary condition, and is important in studying solids. Then the equation for our first point in the wave function reads as

\begin{align} E\psi _1 = \frac{-1}{2(dx)^2}\bigg( \psi _{2}-2\psi _1 + \psi_{N} \bigg) \end{align}

We similarly add a term also to the end point (Hamiltonians are always symmetric). A more thorough treatment of one-dimensional solids is one of the things I'm working ont for this site ;)