Adding waves together
Waves have a friendly property: you can add them. Drop two pebbles in a pond and the ripples pass through each other, their heights simply summing where they meet. Wavefunctions behave the same way. If ψ_A is a perfectly good wavefunction for one situation — say, an electron over *here* — and ψ_B is a perfectly good wavefunction for another — the electron over *there* — then their sum, ψ_A + ψ_B, is *also* a perfectly good wavefunction. And it describes something genuinely new: an electron that is, in a precise sense, both here and there at the same time. This combined state is called a [[qm-superposition|superposition]].
Not a blurry average, but both crisp options
It is tempting to picture a superposition as a vague compromise — the electron sort of half-here, half-there, like a faded photograph. Resist that. If you measure the position of an electron in the "here + there" superposition, you never find it stretched across the middle. You find it sharply *here*, or sharply *there*, each with a probability set by the Born rule. The superposition is not a smeared third option; it is the two crisp options held together, with the answer chosen only at the instant you look. Between looks, both possibilities are fully present and able to influence each other.
The proof that both possibilities are really present — not just one we are ignorant of — is interference. Return to the double-slit experiment. A particle passing the slits enters a superposition of "went through the left slit" and "went through the right slit." Those two pieces of its wavefunction then overlap on the far screen and add up, building the bright-and-dark interference pattern. Crucially, if you sneak a detector onto the slits to learn which one each particle truly used, the superposition is destroyed and the fringes vanish. The pattern only appears when both routes genuinely coexist — visible proof that superposition is real.
Weights and the freedom to combine
You are not limited to equal mixes. You can build a superposition that leans heavily one way, like 90% "here" and 10% "there," by adding the two wavefunctions with different weights: a big helping of ψ_A plus a small splash of ψ_B. The weights are themselves amplitudes, and squaring them (Born rule again) gives the probability of each outcome. This freedom to combine any valid states, in any weighted blend, into another valid state is one of the load-bearing pillars of all quantum theory.
- Start with two (or more) valid wavefunctions for distinct possibilities — for example, "spinning up" and "spinning down."
- Add them with chosen weights (amplitudes) to form one combined wavefunction — the superposition.
- Square each weight to read off how likely each individual outcome is when you finally measure.
- Re-normalize if needed so the squared weights total 1 — the probabilities must still add to 100%.
Superposition is also where quantum strangeness scales up alarmingly. If a tiny particle can be in two states at once, what about a system *built* from particles? Push the logic far enough and you reach Schrödinger's cat — a thought experiment in which a cat is, on paper, placed in a superposition of alive and dead, hostage to a single quantum event. Schrödinger meant it as a provocation: surely a cat is never literally both. Resolving why we never *see* such monstrous superpositions, only crisp single outcomes, is the cliff-edge the final guide steps up to.