Worked example: Apply the momentum operator to ψ(x) = A·e^{ikx} and to a Gaussian packet φ(x) = B·e^{−x²/2σ²}. What momenta do they carry? Step 1: p̂ψ = −iℏ(d/dx)[A·e^{ikx}] = −iℏ·(ik)·A·e^{ikx} = ℏk·ψ. Eigenvalue ℏk — definite momentum, as expected for a plane wave. Step 2: p̂φ = −iℏ(d/dx)[B·e^{−x²/2σ²}] = −iℏ·(−x/σ²)·φ = (iℏx/σ²)·φ. This is NOT a multiple of φ — φ is not a momentum eigenfunction. Step 3: Interpretation: the Gaussian has fairly definite position (width σ) but no single momentum; expanding it needs many plane waves — position–momentum trade-off in action. Conclusion: eigenfunctions of p̂ are plane waves; localised packets are superpositions — operators make the uncertainty trade-off explicit.