Breaking the PDE
To solve \(\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\), we assume the solution is a product of two functions: \(u(x, t) = X(x)T(t)\).
\[X T' = \alpha X'' T \implies \frac{T'}{\alpha T} = \frac{X''}{X} = -\lambda\]
Since the left depends only on \(t\) and the right only on \(x\), they must both equal a constant \(\lambda\). This gives us two ODEs instead of one PDE!
Worked Examples
Example 1: The Decay Solution
The solution for the time part is \(T(t) = e^{-\alpha \lambda t}\). This shows that heat always decays over time. The spatial part \(X(x)\) gives the shape of the temperature distribution (usually sines and cosines).
The Bridge to Quantum Mechanics
This is the exact same method we use to solve the Schrödinger Equation. We separate \(\psi(x, t) = \phi(x) e^{-iEt/\hbar}\). The spatial part \(\phi(x)\) gives us the Stationary States (the orbitals), and the time part gives us the Phase Rotation. This separation is why we can talk about "Energy Eigenstates" as if they were frozen in time, even though the particle is technically a dynamic wave.