Equations of Change
A Differential Equation (DE) is an equation that relates a function to its own derivatives. In physics, we usually know how a system changes (the derivative) and want to find the state of the system (the function).
Worked Examples
Example 1: Exponential Growth
The simplest DE is \(\frac{dy}{dt} = ky\). This means the rate of change is proportional to the amount present.
- Solution: \(y(t) = Ce^{kt}\).
- Verify: \(\frac{d}{dt}(Ce^{kt}) = k(Ce^{kt}) = ky\).
- Result: Exponential growth (like bacteria) or decay (like radioactive atoms).
The Bridge to Quantum Mechanics
Quantum Mechanics IS the study of differential equations. The Schrödinger Equation is a differential equation that describes how the wavefunction \(\psi\) changes in space and time. Unlike classical equations that predict a single path, quantum DEs predict a "wave of possibility." Solving these equations is the only way to understand why atoms have specific sizes and why light comes in discrete colors.