Switching Domains
Solving DEs in the "Time Domain" (\(t\)) can be hard. The Laplace Transform converts the problem into the "Frequency Domain" (\(s\)), where derivatives turn into simple algebra.
\[\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) dt\]
Worked Examples
Example 1: Basic Transform
Find the Laplace transform of \(f(t) = 1\).
- Integral: \(\int_0^{\infty} e^{-st} dt = [-\frac{1}{s}e^{-st}]_0^{\infty} = 0 - (-1/s) = 1/s\).
- Result: \(\mathcal{L}\{1\} = 1/s\).
Example 2: The Exponential
Find the Laplace transform of \(f(t) = e^{at}\).
- Integral: \(\int_0^{\infty} e^{(a-s)t} dt = \frac{1}{s-a}\).
- Result: \(\mathcal{L}\{e^{at}\} = \frac{1}{s-a}\).
The Bridge to Quantum Mechanics
The Laplace transform is the "big brother" of the Fourier Transform. In Quantum Mechanics, we use these transforms to switch between Position Space (\(x\)) and Momentum Space (\(p\)). A particle's wavefunction in position space is the transform of its wavefunction in momentum space. This mathematical duality is what allows us to say that position and momentum are "conjugate variables"—you can't fully know one without losing information about the other.