Fourier series, Fourier transforms, and conjugate variables

On [0, L] the operator −i d/dx is only symmetric once the boundary term f*g at the ends vanishes, and periodic boundary conditions f(0) = f(L) are one choice that makes it genuinely self-adjoint — each twisted condition f(L) = e^(iθ) f(0) is another, and shifts every eigenvalue by θ/L. Its eigenvalue problem −i e′ = k e has solutions e^(ikx) for every real k, but the boundary condition keeps only kₙ = 2πn/L: the discreteness of a Fourier series is not a property of the exponentials, it is the boundary condition thinning a continuum. Orthonormality is a two-line check — ⟨eₘ|eₙ⟩ = (1/L)∫₀ᴸ e^{(i(kₙ−kₘ)x)} dx integrates a whole number of turns to zero unless m = n — and completeness is the spectral theorem's promise for this operator. Expansion is then the standard move of topic 1.4: each coefficient is a projection, cₙ = ⟨eₙ|f⟩, and the discrete spectrum is why this eigenbasis is countable, as the separability of L² requires.

The coefficient list is more than a representation; its decay rate is a diagnosis. Integrate cₙ = ⟨eₙ|f⟩ by parts once and the boundary terms cancel by periodicity, leaving the coefficients of f′ equal to ikₙcₙ: every derivative the function owns in L² buys one extra power of 1/n of decay. A unit square wave, with a jump, has odd-harmonic sine amplitudes 4/(πn) and nothing else, so |cₙ| falls only as 1/n: the series converges pointwise but not uniformly, and a partial sum overshoots each jump by about 9 per cent of its height however many terms are kept — Gibbs's overshoot narrows with N but never shrinks. A C∞ periodic function's coefficients fall faster than any power of 1/n. So before computing anything you can say how many terms a truncation needs, and afterwards you can read a measured spectrum's slope as the smoothness of the thing that produced it.

Let the interval grow. The allowed wavenumbers kₙ = 2πn/L pack together with spacing Δk = 2π/L, so a sum over n is (L/2π) Σ Δk — primed to become an integral. The coefficients themselves shrink like 1/√L, so define f̃(kₙ) = √(L/2π) cₙ = (1/√(2π)) ∫ e^(−ikₙx) f(x) dx over the interval, which stays finite; as L → ∞ the series pair turns into the symmetric transform pair, with (1/√(2π)) e^(ikx) playing the basis role. But the basis has paid a price: ∫|e^(ikx)|² dx diverges, so no plane wave is a vector in L². What replaces ⟨eₘ|eₙ⟩ = δₘₙ is the distributional statement (1/2π)∫e^{(i(k−k′)x)} dx = δ(k − k′): continuum basis elements are normalised to a δ, labels for a continuous spectrum rather than members of the space. The genuine vectors are wave packets — integrals of e^(ikx) against a square-integrable f̃(k) — and this is exactly the status |p⟩ and |x⟩ will hold from Unit 3 onwards.

Plancherel's theorem upgrades the transform from an integral formula to a unitary operator: ⟨f|g⟩ = ⟨f̃|g̃⟩ for every pair, so in particular ∫|f|² dx = ∫|f̃|² dk. Nothing about the vector changed — only the basis it is written in, the same insert-a-resolution-of-the-identity move as topic 1.5, executed with the |k⟩ family. Unitarity is what the transform buys operationally. Differentiation becomes multiplication: −i d/dx acting on f appears as k f̃(k), and d²/dx² as −k² f̃(k), so any constant-coefficient differential equation collapses to algebra in the k basis — the transform diagonalising every operator that commutes with translation. Numerically it is why one free-particle Schrödinger step is an FFT, a multiplication by e^{(−iħk²t/2m)}, and an inverse FFT.

Define the widths honestly: Δx is the rms spread of |f|² about its mean, Δk the same for |f̃|². Cauchy–Schwarz — the inequality of topic 1.3 — applied to the pair xf and f′ gives Δx Δk ≥ 1/2, with equality exactly when f′ ∝ xf, i.e. for Gaussians, which transform into Gaussians with widths σ and 1/(2σ). The theorem owns every transform pair, not only quantum ones: a Gaussian light pulse of rms duration Δt = 10 ns has bandwidth Δω ≥ 1/(2 × 10⁻⁸ s) = 5 × 10⁷ rad s⁻¹, about 8 MHz, and laser physicists call the equality case transform-limited. Nothing here was measured and no ħ appeared. Unit 3 will multiply k by ħ and call the result momentum; the inequality comes along unchanged, and the quantum content is that matter is described by an amplitude at all.

numpy.fft.fft does not compute the transform of your function; it diagonalises the cyclic shift on N samples, which is the Fourier series of a surrogate that repeats with period NΔx. Its k grid has spacing Δk = 2π/(NΔx) and stops at the Nyquist value π/Δx. Two artefacts follow from the basis, not the signal. Aliasing: on the sample grid e^(ikx) and e^{(i(k−2π/Δx)x)} agree at every sample, so a component above Nyquist lands folded — 900 Hz sampled at 1000 samples per second is indistinguishable from 100 Hz, and since the two are identical at the sampler no later processing can separate them; anti-alias filters therefore act in hardware, before digitisation. Leakage: a frequency that does not complete a whole number of cycles in the window is orthogonal to no grid exponential, so its energy smears into sinc-shaped sidelobes across the bins; windows such as Hann trade sidelobe height for main-lobe width. Reading an FFT is reading a projection onto a basis you chose.