In the last chapter we learned that |Ψ|2—the square of a system’s wavefunction—is itself a function that gives us the probability of finding a particle at a certain time and position. However, in the last chapter we simply assigned the fancy symbol |Ψ|2 to the probability of finding a particle at a certain place and at a certain time. For example, in the simple case of the double-slit experiment of Figure 101, we just gave the distribution of electron hits on the screen of Tonomura’s electron microscope the name |Ψ|2, but we didn’t calculate what |Ψ|2 would look like before we looked at the results of the experiment.
In 1925, Austrian physicist Erwin Schrödinger figured out a way of calculating the wavefunction of a system, allowing the behavior of an experimental system to be predicted.
Possibly, the simplest problem that we can solve using Schrödinger’s equation is the so-called “particle-in-a-box” problem, in which we determine the probability of finding a particle at a certain position within an ideal box. As shown in Figure 109, a particle of mass m is trapped between two infinitely high walls. Let’s assume the box is in the vacuum of space, away from gravitational pull and any other influence. The walls are perfectly elastic, so they don’t absorb any energy from the particle. This is modeled by setting the potential energy at zero between the two walls, and making it infinite outside the box: