Richard Feynman

Why does quantum mechanics need complex numbers? Can't we just use real numbers like in classical physics?

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?" The answer isn't "because the math works out prettier." 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. First, let's get clear on what complex numbers do. 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: e^(iθ) = cos(θ) + i·sin(θ). This transforms growth into rotation. The magnitude stays at 1 while the angle changes. Why does this matter for physics? Because quantum states need to do exactly this: maintain constant total probability while their phases rotate in time.

Can math predict everything, or does quantum mechanics say nature is fundamentally random?

Here's something that keeps physicists up at night: we've got these beautiful mathematical equations that predict everything perfectly—from planets orbiting the sun to neural networks learning to recognize your face. Give me the starting conditions, and I'll tell you exactly where the planet will be a thousand years from now. That's determinism at its finest. But then there's quantum mechanics, which says: "Not so fast, Feynman. Nature rolls dice." Let me be straight with you—both views are right. And that's not a cop-out. It's actually more interesting than if one side simply won. Large systems—planets, baseballs, buildings—behave deterministically. Not because quantum mechanics stops applying, but because with 10^23 particles, the randomness averages out. Quantum fluctuations become negligible. The classical equations emerge as limiting cases of quantum mechanics, like how Newton's gravity emerges from Einstein's relativity at low speeds. But for individual particles, individual measurements on individual quantum systems? Fundamentally random. You cannot predict which way a specific electron will spin. The wavefunction gives you probabilities, and those probabilities evolve deterministically via the Schrödinger equation—but when measurement happens, outcome selection is random.

What did the Stern-Gerlach experiment reveal about quantum mechanics?

In 1922, Stern and Gerlach shot silver atoms through a magnetic field gradient. Classical physics predicted a smooth spread—atoms with magnetic moments pointing all different directions should deflect continuously. Instead, the beam split into exactly two lines. Half up, half down. Nothing in between. This isn't measurement error. The atoms genuinely don't have a definite spin direction until you check. Before measurement, an electron exists in superposition—simultaneously spin-up AND spin-down in a way that's not just ignorance on our part. It's not that we don't know which it is; it genuinely IS both until the measurement collapses it onto one outcome. Here's where it gets weird. Spin isn't even physical rotation. We initially thought electrons were tiny spinning spheres, but the math says they'd need surface velocities exceeding light speed. Impossible. Instead, spin is an intrinsic quantum property with no classical analog. It's angular momentum that doesn't come from anything rotating.

Why do you say that Feynman diagrams are "just keeping score"? What do they actually do?

When you try to understand how things work—really understand them—you find out pretty quickly that words are slippery. They're not good enough. If I tell you "the electron interacts with the photon," you might picture a billiard ball hitting a marble. But that's wrong. It's not a collision; it's a dance. It's a probability. It's a mess! So, when I was trying to figure out Quantum Electrodynamics, the math was getting out of hand. Pages and pages of integrals. You couldn't see what was happening. You were lost in the algebra. So I started drawing pictures. Not to make it pretty, but to keep track of the score. People look at my diagrams now—these little arrows and squiggles—and they think, "Ah, that's what an electron looks like!" No! It's not a photograph. It's a map of what could happen. Here's the trick: you draw a line. That's a particle, say, an electron, moving through time. Then you draw a wavy line. That's a photon, a piece of light. Where they meet—bloop!—that's a vertex. That's an interaction.

What are virtual particles and why do we need them?

If you have an electron and a positron (that's an anti-electron, going backwards in time—don't panic, it's just a direction!), and they smash together, they might annihilate and turn into a photon. Then that photon might turn back into a new pair. You draw the lines: arrow in, arrow out, squiggle in the middle. You can see the "virtual particles" in the middle—those are the lines that start and end inside the diagram. We call them "mathematical intermediaries." They're like the money changing hands in a bank transfer; you don't see the cash, you just see the balance change. They are the ghosts in the machine that make the force happen. Think about electromagnetic repulsion. Two electrons push each other away. How? They exchange virtual photons. These photons exist only inside the calculation—they never appear as real, detectable particles. They're the accounting system for how forces work.