# The Schrödinger Equation

**Episode #10 of the course “Equations that changed the world”**

For nearly 300 years, theorems and laws of calculus were used and expanded upon to describe changes in a system over time. As physics expanded to include quantum mechanics, waves, and the movements of small particles, mathematicians found that these environments behaved differently. Their traditional, Newtonian calculus equations could not sufficiently explain the behaviors of these new discoveries. In 1926, Austrian physicist Erwin Schrödinger published an equation that describes how the quantum state of a physical system changes over time. His work would earn him the Nobel Prize for Physics in 1933.

The Schrödinger equation states basically that given a large number of events in a dynamic system, the behaviors of the parts of that system can be predicted, using probability and consideration of the system’s wave function. It is one of several equations and mathematical approaches that can be used to predict the behavior of molecular, atomic, or subatomic systems, and it remains widely in use throughout physics today. In addition to quantum behaviors and quantum physics, the Schrödinger equation predicts the behaviors of parts of dynamic macrosystems as well. It is sometimes seen in astrophysics as well.

There are multiple parts to each Schrödinger equation, because each measures multiple parts in a system. Depending on the dynamics of the system the mathematician is analyzing, he or she will use a “time-dependent” equation, of which there are two, or a “time-independent” equation, of which there are also two. Regardless of whether the system includes time as a factor, there are separate equations to determine general behaviors in the system and the behaviors of single non-relativistic particles.

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