 Given that particles travel as waves and are confined in atoms as standing waves, it followed that a generalized wave equation was needed to describe them. Building on the works of Planck, Einstein, Rutherford, Bohr, De Broglie, and others, Erwin Schrödinger, an Austrian physicist, developed just such an equation, now bearing his name. Here's a simple sine wave in water. It's described by a wave function. The function tells us the displacement of every water molecule in the wave at any time t. If we take the change in a particle's displacement with respect to time, we get its velocity. And if we take the change in a particle's velocity with respect to time, we get its acceleration. For example, if we take a look at the particle 7 meters down the line and take a snapshot at the 11 second mark, we see that it is just above the line, heading up rapidly and slowing down slightly. A generalization called the wave equation describes how a wave function evolves over time. Schrödinger used the fundamental relationships between energy and wave frequency and quantized momentum to develop a quantum-mechanical equivalent of the wave equation. Importantly, it had one critical difference with the classical model, that it did not produce a location for a particle. In fact, it did not represent an observable physical quantity at all. Instead, it produced a probability curve for particle location. Under free particles, the square of the wave function gives us the probability of experimentally finding the particle at a particular location at a particular time. For example, suppose we had a particle moving from left to right at a specific speed. From Newton's equations, the distance is equal to the speed times the time. After 24 seconds, we would say that the particle is here, but because the particle moves as a matter wave, we need to use Schrödinger's equations. So when you touch the wave at a particular time, it collapses into a particle. Where classical physics says it is here, quantum mechanics says here is the most likely place. But there is a smaller probability that it might be here, or an even smaller probability that it's here. In fact, there is a chance that it may be anywhere along the probability curve, with the probabilities dropping rapidly as we move away from the most probable point.