Stored Energy
If the work done by a force depends only on the start and end points (and not the path), the force is Conservative. We can define a Potential Energy (\(V\)) such that:
\[F = -\frac{dV}{dx}\]
The total energy \(E = T + V\) is conserved.
Worked Examples
Example 1: The Potential Well
A particle in a potential \(V(x) = x^4 - 2x^2\) is "trapped" in the valleys. To escape, it must have enough kinetic energy to "climb" the walls. These valleys are called Potential Wells.
The Bridge to Quantum Mechanics
Quantum Mechanics is almost entirely defined by Potential Energy Wells. An atom is a spherical potential well created by the nucleus. An electron is a wave trapped in that well. Because the electron is a wave, it cannot have any arbitrary energy; its waves must "fit" perfectly inside the well. This requirement is what leads to the Quantization of energy—the fact that electrons can only exist in specific shells.