Schroedinger’s wave equation is sometimes just called the “wave equation” in quantum mechanics. This can be a source of confusion, because a wave function itself will look like an equation when we write it down. But the Schroedinger wave equation is a very special kind of equation called a differential equation. We know this because of the other symbols that we have yet to define.
First of all, the “total energy term” on the right side contains a mathematical operation known as the derivative of the wave function (Ψ) with respect to time (t):
The derivative in this case is just a way of expressing how fast the wave function is changing in time. The important thing is that it represents a rate of change, which itself could vary in time and space.
The last term to define, the “kinetic energy term,” contains something called the second derivative of the wave function (Ψ) with respect to position (x):
This is a way of representing how fast the wave function changes for different locations in x. Multivariate calculus is, by the way, the kind of advanced mathematics you need to study if you want to really sink your teeth into this.
The derivative of a mathematical function is the rate with which the function changes as you move along the independent variable. It can be represented as the slope, like the dashed lines shown here. The second derivative is a measure of how fast the slope of the original function changes.
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