Complex Numbers & the Argand Plane — Maths for Quantum Chemistry

Numbers that point in two dimensions

Why quantum chemistry can't manage without √−1

There is no real number whose square is negative, so mathematicians defined one: $i = - 1$ , with $i^{2} = - 1$ . A complex number $z = a + bi$ then has a real part $a = Re (z)$ and an imaginary part $b = Im (z)$ , and lives as a point on the Argand plane — a two-dimensional coordinate system where the x-axis is real and the y-axis is imaginary.

Arithmetic is straightforward: add by adding components, multiply by expanding the bracket and using $i^{2} = - 1$ . The complex conjugate $z^{*} = a - bi$ is the mirror image across the real axis; the product $z z^{*} = a^{2} + b^{2} = ∣ z ∣^{2}$ is always real and non-negative — a trick exploited constantly in quantum mechanics to build real probability densities from complex wavefunctions.

Writing $z$ in polar form $z = r e^{i θ}$ (where $r = ∣ z ∣$ is the modulus and $θ$ is the argument) makes multiplication transparent: $z_{1} z_{2} = r_{1} r_{2} e^{i (θ_{1} + θ_{2})}$ — moduli multiply and arguments add. Multiplying by $e^{i ϕ}$ simply rotates a number through angle $ϕ$ without changing its magnitude. This is why complex exponentials are the natural language for anything that oscillates.

Core identities at a glance

Rectangular form

z = a + bi, i^{2} = - 1

Modulus & conjugate

∣ z ∣ = a^{2} + b^{2}, z^{*} = a - bi, z z^{*} = ∣ z ∣^{2}

Euler's formula

e^{i θ} = cos θ + i sin θ

Polar form

z = r e^{i θ}, r = ∣ z ∣, θ = arctan (b / a)

Why chemists need it

Wavefunctions are complex-valued; their time part is

e^{- i Et /ℏ}

. Conjugation builds the real probability density

∣ ψ ∣^{2} = ψ^{*} ψ

. AC impedance in electrochemistry, the Fourier mathematics behind FT-NMR and FT-IR, and the phase information in NMR free-induction decays are all expressed in complex arithmetic. Open the Argand plane to get a feel for

z

, then Euler's formula to see how phase rotation becomes a wave.

De Moivre's theorem follows immediately from Euler's formula: $(e^{i θ})^{n} = e^{in θ}$ , which in trigonometric form reads

(cos θ + i sin θ)^{n} = cos n θ + i sin n θ

This is the fastest route to multiple-angle identities, the $n$ th roots of unity, and the time-evolution of a superposition wavefunction. The $n$ roots of unity are the complex numbers $e^{2 πik / n}$ for $k = 0, 1, \dots, n - 1$ — they sit equally spaced on the unit circle and are extensively used in quantum chemistry (molecular orbital phases), signal processing (DFT roots), and symmetry operations.

De Moivre

(r e^{i θ})^{n} = r^{n} e^{in θ}

. Raise to a power by multiplying the modulus exponent and adding

n θ

to the argument.

nth roots of unity

ω_{k} = e^{2 πik / n}, k = 0, \dots, n - 1

. Equally spaced on the unit circle; sum to zero for

n \geq 2

Multiplying in polar form

z_{1} z_{2} = r_{1} r_{2} e^{i (θ_{1} + θ_{2})}

. Dividing:

z_{1} / z_{2} = (r_{1} / r_{2}) e^{i (θ_{1} - θ_{2})}

Probability density

∣ ψ ∣^{2} = ψ^{*} ψ

. The conjugate flips

i \to - i

; the product is always real and non-negative.

Common pitfalls

Argument in the wrong quadrant. $θ = arctan (b / a)$ only gives the correct angle when $a > 0$ . For $a < 0$ add $π$ (or 180°). Always sketch the point in the Argand plane before computing $θ$ .
Forgetting i² = −1. When expanding $(a + bi) (c + d i)$ the mixed term $(bi) (d i) = b d i^{2} = - b d$ contributes to the real part. Overlooking the sign change turns a correct calculation into one that is wrong by $2 b d$ .
Writing |z|² = z² (wrong). The modulus squared is $∣ z ∣^{2} = z z^{*} = a^{2} + b^{2}$ , which is real. The square $z^{2} = (a + bi)^{2} = a^{2} - b^{2} + 2 abi$ is complex. Using the wrong one in a wavefunction integral gives a complex probability — physically nonsensical.
Phase cancellation in superpositions. Adding amplitudes and then squaring is not the same as squaring and adding. Interference terms arise from the cross-product $ψ_{1}^{*} ψ_{2} + ψ_{1} ψ_{2}^{*}$ and can be constructive or destructive depending on the relative phase.

Going deeper

The three cube roots of unity ( $1, ω, ω^{2}$ with $ω = e^{2 πi /3}$ ) appear in the symmetry-adapted linear combinations of molecular orbitals of a triangular molecule like the cyclopropenyl cation. The three p-orbitals on each carbon combine with phase factors that are exactly these roots — and they must sum to zero on an atom not participating in the bonding, mirroring the fact that $1 + ω + ω^{2} = 0$ . This connection between roots of unity and orbital symmetry is one of the most elegant bridges between pure mathematics and physical chemistry.

Argand plane

Drag the point to any position — read off the rectangular form, polar form, modulus, argument, and conjugate.

Drag anywhere to move the complex number z = a + bi

z (rectangular)3 + 2i

modulus |z| = r3.61

argument θ34°

conjugate z*3 − 2i

z = r (cos h e t a + i sin h e t a) = r e^{i h e t a}

The same number has two faces: rectangular (a + bi) for adding, and polar (r, θ) for multiplying and for waves. Multiplying by z* gives r², a real number — the trick behind every probability density |ψ|².

Worked example 1Arithmetic with complex numbers: probability amplitude

Two quantum pathways contribute amplitudes $ψ_{1} = 3 + 4 i$ and $ψ_{2} = 1 - 2 i$ (in arbitrary units). Find (a) the total amplitude $ψ = ψ_{1} + ψ_{2}$ and (b) the probability density $∣ ψ ∣^{2} = ψ^{*} ψ$ .

Worked example 2Polar form and the real probability density |ψ|²

A stationary-state wavefunction evaluated at a point is $ψ = 3 - 4 i$ . Find (a) its modulus, (b) its argument, and (c) confirm that $∣ ψ ∣^{2} = ψ^{*} ψ$ gives a real, non-negative number.

Euler's formula

Sweep θ on the unit circle — the real part traces the cosine wave that is the backbone of all periodic chemistry.

Sweep the angle θ — e^{iθ} walks the unit circle, and its real part traces a cosine wave

e^{i θ} = cos θ + i sin θ

angle θ50° = 0.87 rad

real part cos θ0.643

imag part sin θ0.766

|e^{iθ}|1.00

The magnitude is always 1, so e^{iθ} is a pure phase. A quantum wavefunction's time evolution e^{−iEt/ħ} spins exactly like this — the probability |ψ|² stays constant while the phase rotates.

Worked example 3Time-dependent wavefunction: probability density is constant

The time-dependent part of a quantum state is $Φ (t) = e^{- i Et /ℏ}$ , where $E$ is the energy eigenvalue and $ℏ$ the reduced Planck constant. Show that the probability density $∣Φ ∣^{2} = Φ^{*} Φ$ is independent of time.

Worked example 4AC impedance of a capacitor in electrochemistry

In AC electrochemistry an angular frequency $ω$ is applied. The impedance of a capacitor with capacitance $C$ is $Z_{C} = \frac{1}{iω C}$ . Rewrite $Z_{C}$ in rectangular form and identify its phase angle.

Worked examples

Complex numbers across quantum chemistry, NMR signal processing, and molecular spectroscopy.

Worked example 5Euler's identity and verifying e^{iπ} + 1 = 0

Use Euler's formula $e^{i θ} = cos θ + i sin θ$ to evaluate $e^{iπ}$ and hence verify Euler's identity $e^{iπ} + 1 = 0$ .

Worked example 6FT-NMR: the free-induction decay as a complex exponential

The free-induction decay (FID) signal of a single ¹H resonance in FT-NMR is modelled as $s (t) = e^{- t / T_{2}} e^{i ω_{0} t}$ , where $T_{2}$ is the transverse relaxation time and $ω_{0}$ is the Larmor frequency. Show that the signal envelope decays as $e^{- t / T_{2}}$ and find $∣ s (t) ∣$ .

Worked example 7Complex conjugate and normalisation of a wavefunction

A trial wavefunction (un-normalised) is $ψ (x) = N (1 + 2 i) f (x)$ where $N$ is a real normalisation constant and $\int ∣ f ∣^{2} d x = 1$ . Find $N$ such that $\int ∣ ψ ∣^{2} d x = 1$ .

Worked example 8De Moivre's theorem: derive cos 3θ and sin 3θ

Use de Moivre's theorem $(cos θ + i sin θ)^{3} = cos 3 θ + i sin 3 θ$ to derive expressions for $cos 3 θ$ and $sin 3 θ$ in terms of $cos θ$ and $sin θ$ . (These identities appear in the analysis of vibrational overtones of C₃v molecules.)

Worked example 9Dividing complex impedances: find the phase of a RC circuit

In an electrochemical RC circuit the total impedance is $Z = R + \frac{1}{iω C}$ where $R = 500 Ω$ , $C = 10 μ F$ , and $ω = 1000 rad s^{- 1}$ . Express $Z$ in polar form and find the phase angle by which the voltage leads the current.

ChallengeChallenge — normalise a superposition wavefunction and confirm |ψ|² is real

A particle is in the superposition state $ψ = N (ψ_{1} + e^{i δ} ψ_{2})$ where $ψ_{1}, ψ_{2}$ are normalised, real, orthogonal basis states ( $\int ψ_{1} ψ_{2} d x = 0$ , $\int ψ_{1, 2}^{2} d x = 1$ ) and $δ$ is a real phase difference. (a) Find $N$ . (b) Show that the probability density $∣ ψ ∣^{2} = ψ^{*} ψ$ is real. (c) Evaluate for $δ = π /2$ .

Check yourself

Five questions on complex arithmetic, the Argand plane, and Euler's formula in chemistry.

Question 1 of 5 · Score 0

What is i²?

Choose an answer.