Sudden Changes
Physics isn't always smooth. Sometimes a light is switched on (Heaviside Step Function) or a hammer hits a ball (Dirac Delta Function).
- \(u_c(t)\): Zero for \(t < c\), One for \(t \geq c\).
- \(\delta(t-c)\): Infinitely tall, infinitely thin spike at \(t=c\) with an area of 1.
Worked Examples
Example 1: The Delta Property
The Laplace transform of the Dirac Delta is the simplest of all: \(\mathcal{L}\{\delta(t-c)\} = e^{-cs}\). If the impulse is at \(t=0\), the transform is just 1. This means the Delta function contains all frequencies at once.
The Bridge to Quantum Mechanics
The Dirac Delta was actually invented by Paul Dirac for Quantum Mechanics! It is used to describe a particle that is at one exact point in space: \(\psi(x) = \delta(x-x_0)\). The Delta Potential Well is a standard model for an atom that is so small it can be treated as a single point. Understanding the math of the Delta function is the only way to handle Normalization for particles that aren't trapped in a box—it is the bridge between discrete particles and continuous waves.