Complex Numbers

A complex number combines two real components into one algebraic object:

$$z=x+iy, \qquad x,y\in\mathbb R, \qquad i^2=-1.$$

Complex numbers are not optional notation in quantum mechanics. States, amplitudes, inner products, and unitary time evolution use their magnitude and phase as independent structures. Probabilities discard an overall phase, but interference detects relative phase.

Cartesian form and equality

The real and imaginary parts are

$$\operatorname{Re}z=x, \qquad \operatorname{Im}z=y.$$

Two complex numbers are equal exactly when their real and imaginary parts are equal:

$$\begin{aligned} x+iy=u+iv &\quad\Longleftrightarrow\quad x=u,\\ &\quad\text{and}\quad y=v. \end{aligned}$$

The complex plane identifies $z=x+iy$ with the point $(x,y)$. The real axis is the horizontal coordinate and the imaginary axis is the vertical coordinate. Addition is therefore ordinary vector addition in this plane.

Arithmetic

For $z=x+iy$ and $w=u+iv$,

$$z+w = (x+u)+i(y+v),$$

and

$$\begin{aligned} zw &= (x+iy)(u+iv)\\ &= (xu-yv)+i(xv+yu). \end{aligned}$$

Subtraction is componentwise. Division uses the conjugate of the denominator. If $w\ne0$,

$$\begin{aligned} \frac{z}{w} &= \frac{(x+iy)(u-iv)} {u^2+v^2}\\ &= \frac{xu+yv}{u^2+v^2} +i\frac{yu-xv}{u^2+v^2}. \end{aligned}$$

Multiplying numerator and denominator by $u-iv$ converts the denominator into a positive real number.

The usual field laws hold: addition and multiplication are associative and commutative, multiplication distributes over addition, and every nonzero complex number has a multiplicative inverse.

Complex conjugation

The conjugate reflects a point across the real axis:

$$z^* = x-iy.$$

Conjugation obeys

$$\begin{aligned} (z+w)^*&=z^*+w^*,\\ (zw)^*&=z^*w^*,\\ \left(\frac{z}{w}\right)^* &= \frac{z^*}{w^*}, \qquad w\ne0,\\ (z^*)^*&=z. \end{aligned}$$

It extracts the Cartesian parts:

$$\operatorname{Re}z = \frac{z+z^*}{2}, \qquad \operatorname{Im}z = \frac{z-z^*}{2i}.$$

Conjugation reverses phase. This reversal is why a complex inner product must conjugate one argument and why probabilities involve an amplitude times its conjugate.

Modulus and distance

The modulus is

$$\lvert z\rvert = \sqrt{x^2+y^2}.$$

It is the Euclidean distance from the origin. The distance between two complex numbers is $\lvert z-w\rvert$.

The squared modulus is

$$\lvert z\rvert^2 = z^*z.$$

Important properties are

$$\begin{aligned} \lvert z\rvert&\ge0,\\ \lvert z\rvert=0 &\Longleftrightarrow z=0,\\ \lvert zw\rvert &= \lvert z\rvert\lvert w\rvert,\\ \left\lvert\frac{z}{w}\right\rvert &= \frac{\lvert z\rvert}{\lvert w\rvert}. \end{aligned}$$

The triangle inequality and its reverse form are

$$\lvert z+w\rvert \le \lvert z\rvert+\lvert w\rvert$$

and

$$\bigl\lvert \lvert z\rvert-\lvert w\rvert \bigr\rvert \le \lvert z-w\rvert.$$

These inequalities become norm bounds for complex vectors and wavefunctions.

Polar form and argument

Every nonzero complex number can be written

$$z = re^{i\theta}, \qquad r=\lvert z\rvert>0.$$

The angle $\theta$ is an argument of $z$. It is not unique:

$$\theta \sim \theta+2\pi n, \qquad n\in\mathbb Z.$$

A chosen single-valued representative is the principal argument $\operatorname{Arg}z$. A common convention is

$$-\pi < \operatorname{Arg}z \le \pi.$$

That convention introduces a jump across the negative real axis. Other branch choices are possible. The number $z=0$ has no defined argument because it has no direction from the origin.

Cartesian and polar coordinates are related by

$$x=r\cos\theta, \qquad y=r\sin\theta.$$

In numerical work, the phase should be computed with a two-argument arctangent so the quadrant is retained.

Euler formula

Euler’s formula is

$$e^{i\theta} = \cos\theta+i\sin\theta.$$

It immediately gives

$$\lvert e^{i\theta}\rvert=1$$

and

$$\left(e^{i\theta}\right)^* = e^{-i\theta}.$$

The polar form follows:

$$z = r\left( \cos\theta+i\sin\theta \right).$$

The analytic definition, derivatives, and oscillatory applications are developed in Complex Exponentials.

Multiplication as scaling and rotation

Write

$$z=r e^{i\theta}, \qquad w=s e^{i\varphi}.$$

Then

$$zw = rs e^{i(\theta+\varphi)}.$$

Multiplication multiplies magnitudes and adds phases. In the complex plane, multiplication by $e^{i\varphi}$ is a rotation through angle $\varphi$, while multiplication by $s>0$ is a radial scaling.

Division subtracts phase:

$$\frac{z}{w} = \frac{r}{s} e^{i(\theta-\varphi)}.$$

This geometry makes phase evolution and interference easier to visualize than Cartesian component arithmetic alone.

Powers and roots

De Moivre’s formula gives

$$\left( r e^{i\theta} \right)^n = r^n e^{in\theta}$$

for integer $n$. Equivalently,

$$\left( \cos\theta+i\sin\theta \right)^n = \cos(n\theta)+i\sin(n\theta).$$

If $z=re^{i\theta}\ne0$, its $n$ distinct $n$th roots are

$$w_k = r^{1/n} \exp\left[ \frac{i(\theta+2\pi k)}{n} \right],$$

where

$$k=0,1,\ldots,n-1.$$

The roots lie equally spaced on a circle. Choosing only the principal argument gives one principal root, not all algebraic roots.

Exponential and logarithm

For $z=x+iy$,

$$e^z = e^x \left( \cos y+i\sin y \right).$$

The complex exponential is periodic in the imaginary direction:

$$e^{z+2\pi i n}=e^z.$$

Consequently, the complex logarithm is multivalued:

$$\log z = \ln\lvert z\rvert +i\left( \arg z+2\pi n \right).$$

A single-valued logarithm requires a branch choice and a branch cut. Fractional powers inherit that choice. Contours, analytic continuation, and branch cuts belong in Complex Analysis Essentials.

Complex amplitudes

If a quantum process has amplitude $\mathcal A$, its probability is

$$P = \lvert\mathcal A\rvert^2 = \mathcal A^*\mathcal A.$$

Squaring the amplitude without conjugation generally gives a complex number and is not a probability.

If two indistinguishable alternatives have amplitudes $\mathcal A_1$ and $\mathcal A_2$, the total probability is

$$\begin{aligned} P &= \lvert\mathcal A_1+\mathcal A_2\rvert^2\\ &= \lvert\mathcal A_1\rvert^2 +\lvert\mathcal A_2\rvert^2\\ &\quad+ 2\operatorname{Re} \left( \mathcal A_1^*\mathcal A_2 \right). \end{aligned}$$

The final term is interference. If

$$\mathcal A_j = r_j e^{i\theta_j},$$

then

$$P = r_1^2+r_2^2 +2r_1r_2 \cos(\theta_2-\theta_1).$$

Only the relative phase enters.

Global and relative phase

Multiplying a normalized state by a common phase,

$$\lvert\psi\rangle \longmapsto e^{i\chi}\lvert\psi\rangle,$$

does not change expectation values:

$$\begin{aligned} \langle\psi\vert e^{-i\chi}A e^{i\chi} \lvert\psi\rangle &= \langle\psi\vert A\lvert\psi\rangle. \end{aligned}$$

Equivalently, the density operator is unchanged:

$$e^{i\chi} \lvert\psi\rangle\langle\psi\rvert e^{-i\chi} = \lvert\psi\rangle\langle\psi\rvert.$$

Relative phase is observable through interference. Consider

$$\lvert\psi\rangle = \frac{1}{\sqrt2} \left( \lvert0\rangle +e^{i\varphi}\lvert1\rangle \right)$$

and the basis

$$\lvert\mathord\pm\rangle = \frac{1}{\sqrt2} \left( \lvert0\rangle \mathord\pm\lvert1\rangle \right).$$

The probabilities are

$$\begin{aligned} P(+) &= \left\lvert \frac{1+e^{i\varphi}}{2} \right\rvert^2 = \cos^2\left(\frac{\varphi}{2}\right),\\ P(-) &= \sin^2\left(\frac{\varphi}{2}\right). \end{aligned}$$

The phase is invisible in the original $\{\lvert0\rangle,\lvert1\rangle\}$ probabilities but visible after changing basis. See Superposition and Relative Phase.

Complex inner products

For vectors $u,v\in\mathbb C^n$, the standard inner product is

$$\langle u,v\rangle = \sum_{j=1}^{n}u_j^*v_j.$$

It is conjugate-linear in the first argument and linear in the second:

$$\begin{aligned} \langle au+bv,w\rangle &= a^*\langle u,w\rangle +b^*\langle v,w\rangle,\\ \langle u,av+bw\rangle &= a\langle u,v\rangle +b\langle u,w\rangle. \end{aligned}$$

It also obeys conjugate symmetry and positivity:

$$\langle u,v\rangle = \langle v,u\rangle^*, \qquad \langle u,u\rangle\ge0.$$

Without conjugation, the proposed squared norm of $(1,i)$ would be $1+i^2=0$ despite the vector being nonzero. Conjugation produces $1+\lvert i\rvert^2=2$ instead.

The vector-space and bra–ket conventions are developed in Complex Vector Spaces.

Time-evolution phases

An energy eigenstate evolves as

$$\lvert E,t\rangle = e^{-iEt/\hbar} \lvert E,0\rangle.$$

For one energy eigenstate this is a global phase. For a superposition,

$$\lvert\psi(t)\rangle = \sum_n c_n e^{-iE_nt/\hbar} \lvert E_n\rangle,$$

relative phases change at rates set by energy differences:

$$\frac{e^{-iE_nt/\hbar}} {e^{-iE_mt/\hbar}} = e^{-i(E_n-E_m)t/\hbar}.$$

Those relative phases generate observable oscillations. The imaginary unit makes the exponential a unit-modulus rotation, which is the scalar prototype of unitary time evolution.

Complex matrices

For a complex matrix $A$, the adjoint is

$$A^\dagger = (A^*)^{\mathsf T}.$$

Hermitian matrices satisfy $A^\dagger=A$ and represent finite-dimensional observables. Unitary matrices satisfy

$$U^\dagger U=I$$

and preserve complex inner products. Entrywise conjugation, transposition, and adjunction are different operations and should not be interchanged.

Numerical practice

Complex computations have a few recurring traps:

• compare complex numbers by magnitude of their difference, not by exact floating-point equality;
• use a two-argument arctangent for phase;
• unwrap phases only when continuity along a chosen path is physically intended;
• compute probabilities with $\lvert z\rvert^2$, not $z^2$;
• distinguish transpose from conjugate transpose;
• avoid subtracting nearly equal phases when a ratio or inner product gives a more stable relative phase.

At a zero of an amplitude, phase is undefined and numerical phase plots can jump arbitrarily. This is geometry, not necessarily a numerical defect.

Common mistakes

• Writing $i^2=1$ or $1/i=i$ instead of $1/i=-i$.
• Dividing complex numbers without conjugating the denominator.
• Confusing $z^2$ with $\lvert z\rvert^2$.
• Treating $\arg z$ as single-valued without a branch convention.
• Assigning a phase to $z=0$.
• Keeping only one algebraic root of a complex number.
• Assuming $\log(zw)=\log z+\log w$ globally without branch qualifications.
• Omitting conjugation from a complex inner product.
• Confusing a global state phase with a relative component phase.
• Treating phase velocity or time-dependent phase as a directly observable probability.
• Using $A^{\mathsf T}$ where $A^\dagger$ is required.

Exercises

1. Compute $(2+i)/(1-2i)$ in Cartesian and polar form.

Solution

Multiply by the conjugate of the denominator:

$$\begin{aligned} \frac{2+i}{1-2i} &= \frac{(2+i)(1+2i)} {(1-2i)(1+2i)}\\ &= \frac{5i}{5}\\ &= i. \end{aligned}$$

Thus the Cartesian form is $0+i$, the modulus is one, and a polar form is

$$i=e^{i\pi/2}.$$

2. Find all cube roots of $-8$.

Solution

Write

$$-8 = 8e^{i(\pi+2\pi n)}.$$

The roots have modulus $2$ and arguments

$$\theta_k = \frac{\pi+2\pi k}{3}, \qquad k=0,1,2.$$

Therefore,

$$\begin{aligned} z_0&=2e^{i\pi/3}=1+i\sqrt3,\\ z_1&=2e^{i\pi}=-2,\\ z_2&=2e^{i5\pi/3}=1-i\sqrt3. \end{aligned}$$

Cubing any of these gives $-8$.

3. Two alternatives have amplitudes

   $$\mathcal A_1=r, \qquad \mathcal A_2=r e^{i\varphi}.$$

   Find the total probability and identify complete constructive and destructive interference.

Solution

The total amplitude is

$$\mathcal A = r\left(1+e^{i\varphi}\right).$$

Hence

$$\begin{aligned} P &= r^2 \left\lvert 1+e^{i\varphi} \right\rvert^2\\ &= 2r^2(1+\cos\varphi)\\ &= 4r^2 \cos^2\left(\frac{\varphi}{2}\right). \end{aligned}$$

Constructive interference occurs at $\varphi=2\pi n$, giving $P=4r^2$. Destructive interference occurs at $\varphi=(2n+1)\pi$, giving $P=0$.

4. Show that multiplying a state by a global phase leaves every transition probability unchanged.

Solution

Let the transformed state be

$$\lvert\psi'\rangle = e^{i\chi}\lvert\psi\rangle.$$

For any normalized target state $\lvert\phi\rangle$,

$$\langle\phi\vert\psi'\rangle = e^{i\chi} \langle\phi\vert\psi\rangle.$$

Taking the squared modulus gives

$$\begin{aligned} \left\lvert \langle\phi\vert\psi'\rangle \right\rvert^2 &= \left\lvert e^{i\chi}\right\rvert^2 \left\lvert \langle\phi\vert\psi\rangle \right\rvert^2\\ &= \left\lvert \langle\phi\vert\psi\rangle \right\rvert^2. \end{aligned}$$

The common phase cancels. A phase attached to only one component would generally change interference with the other components.

References

• T. Needham, Visual Complex Analysis, Oxford University Press, 1997.
• J. W. Brown and R. V. Churchill, Complex Variables and Applications, 9th ed., McGraw–Hill, 2014.
• M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2005.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific, 1998.