Worked example: Normalise ψ(x) = A·e^{−ax} for x ≥ 0 (ψ = 0 for x < 0), with a > 0. Step 1: Normalization condition: ∫₀^∞ |A|²e^{−2ax}dx = 1. Step 2: Evaluate: |A|² × [e^{−2ax}/(−2a)]₀^∞ = |A|² × (1/2a) = 1. Step 3: So |A|² = 2a, and choosing A real and positive: A = √(2a). Step 4: Normalised wave function: ψ(x) = √(2a)·e^{−ax} for x ≥ 0. Step 5: Check a probability: P(0 < x < 1/a) = ∫₀^{1/a} 2a·e^{−2ax}dx = 1 − e^{−2} ≈ 0.865. Conclusion: the particle is found within one decay length 86.5% of the time — normalization turned an arbitrary A into quantitative predictions.