Time-Independent Schrödinger Equation — David J. Griffiths
Physics — Quantum Mechanics · fourier_transform
Problem
What is the Fourier transform of δ(x)? Using Plancherel's theorem, show that
δ(x)=2π1∫−∞+∞eikxdk.
Comment: This formula gives any respectable mathematician apoplexy. Although the integral is clearly infinite when x=0, it doesn't converge (to zero or anything else) when x=0, since the integrand oscillates forever. There are ways to patch it up (for instance, you can integrate from −L to +L, and interpret Equation 2.147 to mean the average value of the finite integral, as L→∞). The source of the problem is that the delta function doesn't meet the requirement (square-integrability) for Plancherel's theorem (see footnote 42). In spite of this, Equation 2.147 can be extremely useful, if handled with care.