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Four Ways to Solve the Schrödinger Equation for the 1D Infinite Square Well
Oh, you are taking quantum mechanics? That’s great. In just about every single version of this course, the students are going to find an expression for the wave function in an infinite 1 dimensional square well using Schrödinger equation. So, how do you solve this problem? Well, you are in luck. I’m going to solve this equation 4 different ways. I’ll start with the plain basic method used most textbooks. After that, I’m going to show you 3 ways to solve this with Python. It’s going to be great.
Analytical Solution
What is a 1D infinite square well? Imagine that you have a particle that can only move in the x-direction. It is also confined to move between x = 0 and x = a by two infinite potential walls (this makes the “well”).
We can find the wave function for this situation using the 1D Schrödinger. It looks like this.
Don’t worry about where this comes from or what it means (that’s a whole other story), just know that it’s a differential equation and we want to find an expression for Ψ(x,t). Oh, what about the other stuff? Well, here’s a list:
