Complex amplitudes and the phase that survives

A real amplitude added to another can only reinforce it or oppose it. A complex amplitude z = x + iy is a point in a plane, addition is head-to-tail vector addition there, and two amplitudes can therefore partially cancel at any angle — exactly the freedom interference patterns demand. Cartesian form suits addition; polar form z = r e^(iφ) suits multiplication. Converting between them has one trap: the argument is φ = atan2(y, x), not arctan(y/x). The point z = −1 − i sits at φ = −3π/4, while blind arctan of y/x = 1 reports +π/4; that sign error later flips a cross term from constructive to destructive. Conjugation z* = x − iy = r e^(−iφ) reflects across the real axis, and the product z*z = r² is real and non-negative — unlike z², which for z = e^(iπ/4) equals i: still complex, still unit modulus. Quantum mechanics squares moduli, never the numbers themselves.

Define f(θ) = e^(iθ) as the solution of f′ = i f with f(0) = 1. Then (f*f)′ = −i f*f + i f*f = 0, so |f| stays 1: the solution runs around the unit circle at unit speed, starting at 1 and turning anticlockwise — which is the point (cos θ, sin θ). That is Euler's formula, and it makes phase arithmetic trivial: e^(iφ₁) e^(iφ₂) = e^{(i(φ₁+φ₂))} turns composing rotations into adding angles, and de Moivre's theorem (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ is one line. Read backwards it dissolves trigonometry: cos θ = (e^(iθ) + e^(−iθ))/2 and sin θ = (e^(iθ) − e^(−iθ))/(2i), which is how every trig integral in this course becomes an exponential integral. Numerically e^(iπ/3) = 0.500 + 0.866i, e^(iπ) = −1, e^(2πi) = 1: the phase is periodic, which is why arguments are only ever defined modulo 2π.

Work on the circle: square-integrable functions with the boundary condition f(θ + 2π) = f(θ). Apply d/dθ to cos θ and you get −sin θ — a different function, so cos θ is no eigenfunction; the derivative keeps knocking sines and cosines into each other. Apply it to e^(ikθ) and the function returns as ik e^(ikθ): an eigenfunction with eigenvalue ik, and the boundary condition admits only integer k. Integrating by parts around the closed circle kills the boundary terms and shows d/dθ is anti-Hermitian, so its eigenvalues are pure imaginary; multiply by −i and L̂ = −i d/dθ is Hermitian, with real eigenvalues k ∈ ℤ and the orthonormal eigenbasis uₖ(θ) = e^(ikθ)/√(2π), ⟨uⱼ|uₖ⟩ = δⱼₖ. This is the template the whole course stamps out: name the space, the boundary condition and the operator, then read the eigenvalue equation as the list of measurable values. Attach ħ and it is already physics: L̂_z = −iħ d/dθ on the ring, p̂ = −iħ d/dx on the line.

The exit to experiment is |z|² = z*z, and it is worth being exact about what that step forgets. Multiply an entire state by e^(iα): every amplitude ⟨φ|ψ⟩ acquires the same factor, and |e^(iα)⟨φ|ψ⟩|² = |⟨φ|ψ⟩|² for every |φ⟩; every expectation value ⟨ψ|A|ψ⟩ is equally untouched, since e^(−iα) e^(iα) = 1. No experiment, in any basis, distinguishes e^(iα)|ψ⟩ from |ψ⟩ — the physical state is the whole ray of kets e^(iα)|ψ⟩, and normalisation ⟨ψ|ψ⟩ = 1 fixes the overall modulus while leaving the overall phase forever free. But that is the only phase the squaring erases. Rephase one branch of a superposition, c₂ → e^(iδ)c₂, and no common factor pulls out: (|0⟩ + |1⟩)/√2 and (|0⟩ − |1⟩)/√2 have identical moduli in this basis — probabilities ½, ½ for both — yet their inner product is zero. They are orthogonal: as different as two states can be, on the strength of one relative sign.

A relative phase hides in the basis that carries it and appears in any rotated one. Prepare |ψ_δ⟩ = (|0⟩ + e^(iδ)|1⟩)/√2 and measure in |±⟩ = (|0⟩ ± |1⟩)/√2. The amplitude for + is ⟨+|ψ_δ⟩ = (1 + e^(iδ))/2, so P(+) = (1 + cos δ)/2 and P(−) = (1 − cos δ)/2: certainty of + at δ = 0, an even split at δ = π/2, certainty of − at δ = π. The pattern generalises through one identity: whenever two amplitudes reach the same outcome they add before the square, and |z₁ + z₂|² = r₁² + r₂² + 2r₁r₂ cos(φ₂ − φ₁). The first two terms are the incoherent sum a classical mixture would give; the cross term, bounded by ±2r₁r₂, is interference, and only the phase difference enters it. Interference is not a wave phenomenon bolted onto the formalism — it is complex addition followed by a single squaring.

You have used complex exponentials before, in AC circuits, and the habit learned there must be unlearned here. A phasor computes with V₀ e^(iωt), but the physical voltage is Re[V₀ e^(iωt)]: the imaginary part is scaffolding, removable at the price of clumsier trigonometry. The quantum amplitude is not scaffolding. Take the real part of the momentum eigenfunction e^(ikx) and you get cos kx = (e^(ikx) + e^(−ikx))/2 — no longer an eigenfunction of −iħ d/dx but an equal superposition of momenta +ħk and −ħk: Re destroys the very property the exponential was chosen for. The time factor tells the same story. A stationary state evolves by e^(−iEt/ħ), whose unit modulus keeps |Ψ|² frozen; a real cos(Et/ħ) in its place would make every atom's charge density pulse at angular frequency 2E/ħ, which stable atoms do not do. The i in iħ ∂Ψ/∂t = ĤΨ is load-bearing. No measurement ever returns Re z, Im z, or φ itself; what is measurable is |z|², and phase differences through the cross terms they feed.