Complex phase: why a wave can be a spiral

One number, two parts

A complex number can be written in polar form as a magnitude and an angle:

z = A · e^(iφ) = A cos φ + i · A sin φ

A = |z|  ≥ 0     magnitude (how far from the origin)
φ = arg z        phase (which direction, in radians)

Phase is defined only when A > 0; at z = 0 there is no direction. Directions repeat every 2π. The explorer tracks accumulated phase from 0 to 4π, so it remembers two turns rather than just the direction modulo 2π. With growth enabled, the radius can differ on the second turn.

The identity on the first line is Euler's formula. It is a definition-level fact about the complex exponential, not a physical hypothesis; the NIST Digital Library of Mathematical Functions states it in its section on the exponential function.

Amplitude says how big. Phase says where in the cycle. They are independent: doubling A stretches the picture without moving the point around the circle, and adding to φ rotates the point without changing its distance from the origin.

Real and imaginary projections

Read the two parts of z separately and each one is an ordinary wave:

Re z = A cos φ      (shadow on the horizontal axis)
Im z = A sin φ      (shadow on the vertical axis)

With constant A, as φ increases steadily, the point moves round the circle, and each shadow slides back and forth. The two waves are identical in shape, offset by a quarter cycle (π/2). Neither one alone tells you which way the point is turning; together they do. This is the same reason the θ reference encodes an angle as a (sin θ, cos θ) pair rather than a single number.

For example, just before and after midnight, a daily angle wraps from nearly 2π back to 0. As raw numbers those angles look far apart; their (sin θ, cos θ) pairs stay close together. This gives a model a continuous representation of position within a cycle. The pair alone does not identify the date or count completed cycles.

Circle, spiral, helix

Quantum phase: snapshots and stationary states

For a time-independent Hamiltonian, an energy eigenstate satisfies Ĥψ = Eψ. Substituting that into the Schrödinger equation gives ∂ψ/∂t = −iEψ/ħ, whose solution is a rotating phase factor:

ψ(x, t) = ψ(x) · e^(−iEt/ħ)

Every point of the wavefunction rotates at the same rate. Because measurable probabilities come from |ψ|², and |e^(−iEt/ħ)| = 1, this global phase does not by itself change any observable. That is why such states are called stationary. Phase becomes physically visible only through relative phase, for example between two superposed energy states, where it produces interference. MIT OpenCourseWare's 8.05 notes on wave mechanics cover this carefully.

For the deeper treatment, including the U(1) symmetry and how it relates to the Temporal Spiral encoding as an analogy, see Time as a Phase-Space Trajectory.