Complex Numbers and Phases for Quantum Mechanics

Classical probability theory assigns real numbers between 0 and 1 to events. Quantum mechanics assigns a complex probability amplitude to each possible outcome, and the observed probability is the squared modulus of that amplitude. This single change forces us to work with complex arithmetic from the very first calculation.

Complex numbers are not just a convenience; they are required to describe interference. When two paths lead to the same outcome, the total amplitude is a sum of complex numbers. Depending on their relative phase, the probabilities can add constructively, destructively, or anywhere in between.

Wave functions, state vectors, and matrix elements in quantum mechanics are all complex-valued. The time-dependent Schrödinger equation, $i\hbar \partial_t \psi = \hat{H}\psi$, explicitly contains $i = \sqrt{-1}$. Without complex numbers the equation would not be first order in time and unitary evolution would be impossible.

The modulus of an amplitude gives a probability, while the argument gives a phase. Both pieces of information are physically meaningful in combinations, even though an overall phase is not observable by itself. This distinction is central to the logic of quantum superposition.

A complex number is an ordered pair $(x,y)$ written as $z = x + iy$, where $x$ and $y$ are real and $i^2 = -1$. The set of all such numbers forms the field $\mathbb{C}$, in which addition, subtraction, multiplication, and division by nonzero elements are all well defined.

The real part is $\Re(z) = x$ and the imaginary part is $\Im(z) = y$. The complex conjugate of $z$ is $z^* = x - iy$, obtained by reflecting across the real axis in the complex plane. Conjugation turns $i$ into $-i$ and is an involution: $(z^*)^* = z$.

The modulus, or absolute value, of $z$ is $|z| = \sqrt{x^2 + y^2}$. It equals the distance from the origin to the point representing $z$. A crucial identity is $|z|^2 = z^* z = x^2 + y^2$, which is always a nonnegative real number.

Multiplication and division are easiest in polar form, but the Cartesian form makes addition transparent. Every nonzero complex number has a multiplicative inverse $z^{-1} = z^*/|z|^2$. These algebraic properties make $\mathbb{C}$ the natural arena for quantum amplitudes.

Any complex number can be written in polar form as $z = r e^{i\theta}$, where $r = |z|$ is the modulus and $\theta = \arg(z)$ is the argument. Euler's formula states that $e^{i\theta} = \cos\theta + i\sin\theta$, connecting the exponential function with trigonometry.

Euler's formula is not merely a definition; it follows from the Taylor series of the exponential, sine, and cosine functions. Because the exponential obeys $e^{z+w} = e^z e^w$, multiplication of complex numbers corresponds to multiplying moduli and adding arguments: $z_1 z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)}$.

The reciprocal is $1/z = (1/r) e^{-i\theta}$, and the quotient is $z_1/z_2 = (r_1/r_2)e^{i(\theta_1-\theta_2)}$. These rules make complex phases behave additively, which is why they appear so naturally in wave mechanics and quantum evolution.

Unit-modulus complex numbers lie on the unit circle and are written $e^{i\theta}$. They are the building blocks of quantum phases. A general normalized quantum state in a two-level system can be written $e^{i\alpha}(\cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle)$, where the overall phase $e^{i\alpha}$ is physically irrelevant.

A phase is the argument of a complex amplitude. When two amplitudes add, their relative phase determines whether they reinforce or cancel. If $\psi_1 = r_1 e^{i\theta_1}$ and $\psi_2 = r_2 e^{i\theta_2}$, then $\psi_1 + \psi_2 = e^{i\theta_1}(r_1 + r_2 e^{i(\theta_2-\theta_1)})$. The relative angle $\delta = \theta_2 - \theta_1$ controls the interference pattern.

The probability of the combined outcome is $|\psi_1 + \psi_2|^2 = r_1^2 + r_2^2 + 2 r_1 r_2 \cos\delta$. The cross term $2 r_1 r_2 \cos\delta$ is the interference term. It can be positive, negative, or zero, giving probabilities that differ from the sum of the individual probabilities.

This interference is the origin of many quantum phenomena, from electron diffraction to the stability of atoms. In a double-slit experiment, the phase difference accumulated along each path determines the positions of bright and dark fringes on the detection screen.

Because probabilities are obtained from amplitudes, the relative phase is just as important as the magnitude. Two states with the same component magnitudes but different relative phases can have completely different measurement statistics.

Multiplying an entire state vector by a unit-modulus complex number $e^{i\chi}$ produces a state that is physically indistinguishable from the original. Every probability $P_n = |\langle n|\psi\rangle|^2$ is unchanged because the global phase factor cancels between the amplitude and its conjugate.

A relative phase, by contrast, appears inside a superposition. For example, $(|0\rangle + e^{i\phi}|1\rangle)/\sqrt{2}$ and $(|0\rangle + |1\rangle)/\sqrt{2}$ differ only by the relative phase $\phi$, yet they are orthogonal when $\phi = \pi$. Relative phases are directly observable through interference experiments.

The distinction matters whenever a state is expanded in more than one basis component. A global phase redefinition of the form $|\psi\rangle \mapsto e^{i\chi}|\psi\rangle$ does not change inner products with other states: $\langle\phi|e^{i\chi}\psi\rangle = e^{i\chi}\langle\phi|\psi\rangle$, whose modulus is unchanged.

However, relative phases between different components of the same state do change inner products with fixed basis states. They also affect the action of gates and the outcomes of measurements in rotated bases. Learning to track which phase is global and which is relative is a key skill in quantum mechanics.

Consider a particle that can reach a detector by two paths. The amplitude for path A is $\psi_A = \frac{1}{\sqrt{5}} e^{i\pi/3}$ and the amplitude for path B is $\psi_B = \frac{2}{\sqrt{5}} e^{-i\pi/6}$. Find the total detection probability.

Step 1 -- Add the amplitudes. The total amplitude is $\psi = \psi_A + \psi_B = \frac{1}{\sqrt{5}} e^{i\pi/3} + \frac{2}{\sqrt{5}} e^{-i\pi/6}$. The relative phase is $\delta = \pi/3 - (-\pi/6) = \pi/2$.

Step 2 -- Compute the probability. Using $|\psi|^2 = |\psi_A|^2 + |\psi_B|^2 + 2|\psi_A||\psi_B|\cos\delta$, we obtain $P = 1/5 + 4/5 + 2(1/\sqrt{5})(2/\sqrt{5})\cos(\pi/2) = 1 + 0 = 1$. The two paths interfere destructively in the cross term because their relative phase is $\pi/2$.

Step 3 -- Interpret the result. Even though the magnitudes do not change, the phase relationship makes the total probability exactly 1, consistent with a normalized two-path process. A different relative phase would yield a different probability and illustrate constructive or partial interference.