imaginary unit — from formula anatomyThe imaginary unit, denoted i, is defined by the property i² = −1. It extends the real number system to the complex numbers, allowing every polynomial equation to have solutions. In physics, the imaginary unit is not just a mathematical convenience but a fundamental ingredient in the equations of quantum mechanics, where wave functions are complex-valued, and in the description of oscillations and waves across all branches of physics.
The imaginary unit was first introduced by Gerolamo Cardano in the 16th century for solving cubic equations, and later formalized by Rafael Bombelli. Its geometric interpretation as rotation came from Caspar Wessel and Jean-Robert Argand in the late 18th century.
Complex numbers are written as a + bi, where a and b are real numbers. The imaginary unit enables compact representation of rotations and periodic phenomena: Euler's formula e^{iθ} = cos θ + i sin θ links exponentials to trigonometry. In quantum mechanics, the time evolution of a state is governed by the Schrödinger equation iħ ∂ψ/∂t = Hψ, where the factor i ensures that probabilities remain conserved and the evolution is unitary.
The imaginary unit is used in alternating current circuit analysis, signal processing (Fourier transforms), and quantum computing, where qubit states are represented as complex vectors.
The imaginary unit is the only number whose reciprocal equals its negative: 1/i = −i, a property that follows directly from i² = −1.
| ρ | variable | density matrix of the open quantum system — density matrix |
| t | variable | time — time second |
| H | variable | Hamiltonian of the system — hamiltonian joule |
| L_n | variable | Lindblad operators (jump operators) for each dissipation channel — lindblad operator |
| i | constant | imaginary unit, square root of -1 — imaginary unit = 1 |
| ħ | constant | Planck constant divided by 2π — reduced planck constant = 1.054571817e-34 joule second |
| d/dt | operator | derivative with respect to time — time derivative |
| [H, ρ] | operator | commutator of H and ρ — commutator |
| {L_n† L_n, ρ} | operator | anticommutator of L_n† L_n and ρ — anticommutator |
| ∑_n | operator | sum over all dissipation channels — sum over channels |
| Ψ | variable | Spinor wave function — wave function |
| t | variable | Time — time second |
| i | constant | Square root of -1 — imaginary unit = 1 |
| ħ | constant | Fundamental quantum of action — reduced planck constant = 1.054571817e-34 joule second |
| c | constant | Universal speed limit — speed of light = 299792458 metre per second |
| m | constant | Intrinsic mass of the particle — mass = variable kilogram |
| ∂_t | operator | Differentiates the wave function with respect to time — partial derivative |
| ∇ | operator | Spatial derivative vector — nabla operator |
| A | variable | Polarization vector — polarization vector |
| Δ | variable | Amplitude of the field — amplitude volt per metre |
| e_x | variable | Unit vector along x-axis — unit vector |
| e_y | variable | Unit vector along y-axis — unit vector |
| i | operator | Complex number with square -1 — imaginary unit |
| z | variable | Analytic signal — analytic signal |
| x | variable | Real signal — signal |
| a | variable | Instantaneous amplitude — instantaneous amplitude |
| φ | variable | Instantaneous phase — instantaneous phase radian |
| t | variable | Time variable — time second |
| H | operator | Hilbert transform, shifts phase by -90° — hilbert transform operator |
| i | operator | Imaginary unit, √(-1) — imaginary unit |
