Introduction to Quantum Mechanics · Problem 2.19

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · plancherel_theorem_proof

Problem

This problem is designed to guide you through a "proof" of Plancherel's theorem, by starting with the theory of ordinary Fourier series on a *finite* interval, and allowing that interval to expand to infinity. (a) Dirichlet's theorem says that "any" function f(x)f(x) on the interval [−a,+a][-a, +a] can be expanded as a Fourier series: f(x)=∑n=0∞[ansin⁡(nπxa)+bncos⁡(nπxa)].f(x) = \sum_{n=0}^{\infty} \left[a_n \sin\left(\frac{n\pi x}{a}\right) + b_n \cos\left(\frac{n\pi x}{a}\right)\right]. Show that this can be written equivalently as f(x)=∑n=−∞∞cneinπx/a.f(x) = \sum_{n=-\infty}^{\infty} c_n e^{in\pi x/a}. What is cnc_n, in terms of ana_n and bnb_n? (b) Show (by appropriate modification of Fourier's trick) that cn=12a∫−a+af(x) e−inπx/a dx.c_n = \frac{1}{2a} \int_{-a}^{+a} f(x)\, e^{-in\pi x/a}\, dx. (c) Eliminate nn and cnc_n in favor of the new variables k=(nπ/a)k = (n\pi/a) and F(k)=2/π a cnF(k) = \sqrt{2/\pi}\, a\, c_n. Show that (a) and (b) now become f(x)=12π∑n=−∞∞F(k) eikxΔk;F(k)=12π∫−a+af(x) e−ikx dx,f(x) = \frac{1}{\sqrt{2\pi}} \sum_{n=-\infty}^{\infty} F(k)\, e^{ikx} \Delta k; \quad F(k) = \frac{1}{\sqrt{2\pi}} \int_{-a}^{+a} f(x)\, e^{-ikx}\, dx, where Δk\Delta k is the increment in kk from one nn to the next. (d) Take the limit a→∞a \to \infty to obtain Plancherel's theorem. *Comment:* In view of their quite different origins, it is surprising (and delightful) that the two formulas—one for F(k)F(k) in terms of f(x)f(x), the other for f(x)f(x) in terms of F(k)F(k)—have such a similar structure in the limit a→∞a \to \infty.

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