Why Complex Numbers Make Quantum Mechanics Work

Here’s a question that bothered me for years: Why does quantum mechanics need complex numbers? I mean really need them, not just find them convenient.

You might think, “Well, real numbers describe everything in classical physics just fine—positions, velocities, forces. So why can’t we describe quantum states with just real numbers? Why do we need this imaginary unit $i = \sqrt{-1}$ cluttering up the equations?”

The answer isn’t “because the math works out prettier” or “because Schrödinger wrote it that way.” Complex numbers are baked into the fabric of quantum reality. Nature actually uses them. Let me show you why it has to be this way.

What Complex Numbers Actually Do

First, let’s get clear on what complex numbers are. On the complex plane, any number $a + bi$ is a point with a real part $a$ (horizontal) and an imaginary part $b$ (vertical). You can also think of it in polar form: a magnitude (distance from origin) and a phase (angle from the positive real axis).

Here’s where it gets interesting. When you raise $e$ to an imaginary power, something remarkable happens. Real exponents cause exponential growth—$e^{2}$ is bigger than $e^{1}$, marching off to infinity. But imaginary exponents? They rotate around the unit circle. Euler’s formula tells us:

This isn’t just mathematical poetry—it’s showing us that imaginary exponents transform growth into rotation. The magnitude stays at 1 while the angle changes. Exponentials with imaginary arguments trace circular paths, oscillating forever without growing or shrinking.

Why does this matter for physics? Because quantum states need to do exactly this: maintain constant total probability while their phases rotate in time.

Quantum States Need Phases

Now let’s talk about quantum states. A quantum system isn’t described by just probabilities—it’s described by probability amplitudes. These are complex numbers whose squared magnitudes give probabilities.

For an $n$-qubit system, the quantum state lives in $2^n$-dimensional Hilbert space. Each basis state (like $|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle$ for two qubits) gets assigned a complex amplitude. The unit normalization requirement ensures the probabilities sum to one, but the individual amplitudes can point anywhere in the complex plane.

Here’s the crucial bit: these amplitudes aren’t just numbers sitting there. They evolve through time by rotating their phases. An electron sitting in an energy eigenstate has an amplitude that rotates as:

The magnitude stays constant (probability doesn’t spontaneously change), but the phase rotates at a rate proportional to the energy. This is how quantum states evolve when you’re not looking at them—they spin in complex phase space.

Could you do this with just real numbers? No! Real numbers can’t rotate. They can only change magnitude. You need complex numbers to have phases that can rotate while keeping magnitude fixed.

Interference Requires Imaginary Components

But it gets better. The real reason nature uses complex numbers is interference.

Quantum mechanics is fundamentally about superposition. When an electron goes through a double slit, it doesn’t go through one slit OR the other—it goes through both simultaneously. The amplitude for the electron to arrive at a detector screen is the sum of amplitudes from both paths.

Here’s where we need complex numbers: these amplitudes can interfere constructively or destructively. If both amplitudes are $+1$, they add to $2$. But if one is $+1$ and the other is $-1$, they cancel to $0$. That’s how you get the dark fringes in an interference pattern—destructive interference.

With just positive real numbers (like classical probabilities), you can’t get destructive interference. Probabilities always add: $P(\text{total}) = P(\text{path 1}) + P(\text{path 2})$. But amplitudes can subtract. And to subtract, you need to be able to have opposite phases—angles that differ by $180°$ on the complex plane.

Actually, it’s even more general. Two amplitudes with a phase difference of $\phi$ combine as:

When $\phi = 0$ (same phase), you get maximum constructive interference. When $\phi = \pi$ (opposite phase), you get destructive interference. For values in between, you get partial interference. This continuous range of phase relationships enables the quantum interference patterns that fundamentally distinguish quantum from classical behavior.

Quantum algorithms exploit this ruthlessly. In Grover’s search algorithm or any quantum computation, you’re not just exploring multiple possibilities in parallel—you’re carefully orchestrating interference patterns. You design operations that amplify the amplitudes of correct answers through constructive interference while canceling wrong answers through destructive interference. This amplitude manipulation is quantum computing’s actual power source, and it’s completely impossible without complex phases.

The Physical Meaning of Phase

Now you might ask: “Okay, but what IS this phase physically?”

Great question. Electric charge, for example, is literally the rate at which a particle’s complex phase rotates through time. Electrons are described by complex-valued fields, and as they propagate forward in time, their phases spin. That rotation rate defines their electric charge. Photons, built from real-valued vectors, have no phase rotation and therefore no electric charge.

When quantum field theory calculates the probability for some process (say, two electrons repelling), it computes amplitudes for every possible interaction scenario—every Feynman diagram. Each scenario contributes a complex amplitude. These amplitudes superpose, interfering constructively and destructively, and the squared magnitude of the total amplitude gives the observable probability.

Nature explores all possibilities simultaneously through quantum superposition. Each possibility has its amplitude with its own magnitude and phase. The phases determine how possibilities interfere. Without complex numbers—without phases—you lose interference, and quantum mechanics collapses into something that doesn’t match what nature actually does.

You Can’t Fake It With Real Numbers

Some people have tried to reformulate quantum mechanics using only real numbers. You can do it—sort of—but you end up needing twice as many variables. Instead of one complex number, you need two real numbers (the real and imaginary parts). The equations become more cumbersome, and you lose the geometric insight that complex numbers provide.

More importantly, the phase relationships become obscured. Euler’s formula $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ elegantly connects exponentials to rotation. Without complex numbers, you’re writing separate equations for sine and cosine components, losing the conceptual unity.

The universe seems to be built on a complex plane, not a real line. Quantum states naturally live in complex vector spaces. Time evolution is naturally described by unitary operators that rotate phases. Observables correspond to Hermitian operators with real eigenvalues (because measurements give real numbers), but the states themselves must be complex to enable the phase relationships that create interference.

What This Tells Us About Nature

The necessity of complex numbers in quantum mechanics tells us something profound: nature doesn’t just track “how much” of something exists at each point—it tracks both magnitude and phase. The universe computes with complex-valued fields where phase relationships matter as much as magnitudes.

This isn’t mathematical abstraction. It’s physical reality. When you measure electron spin and get “up” or “down,” the 50-50 probability comes from complex amplitudes whose squared magnitudes equal $1/2$. The phases of those amplitudes determine what happens in subsequent measurements.

The imaginary unit $i$ isn’t imaginary at all—it’s how nature rotates quantum phases. Without it, you can’t describe superposition, you can’t describe interference, you can’t describe how quantum states evolve. You lose quantum mechanics entirely.

So why do we need complex numbers in quantum theory? Because nature needs them. The universe is solving complex-valued differential equations at every moment. We’re just along for the ride, writing down what it does.