wavefunction (ψ)
The wavefunction, written with the Greek letter psi (ψ), is the central object of quantum mechanics: a mathematical function that carries everything the theory can tell you about a particle or system. For a single particle in one dimension it is written ψ(x, t) — a number assigned to every place x and every moment t. Crucially that number is complex, meaning it has both a size and a phase, like an arrow with a length and a direction.
By itself ψ is not something you can directly see or measure. What you can get from it is probability: the square of its magnitude, |ψ|², tells you how likely you are to find the particle at each location. Where |ψ|² is large, the particle is likely to turn up; where it is zero, the particle is never found. The wavefunction does not say where the particle is, only the odds of finding it there.
It is tempting to picture ψ as a real ripple spreading through space, and for one particle that picture is sometimes helpful. But the honest statement is subtler: for two or more particles, ψ lives not in ordinary three-dimensional space but in a higher-dimensional configuration space, and its values are complex. The wavefunction is best thought of as a bookkeeping device for probabilities and phases, extraordinarily successful at prediction, rather than as a tangible wave you could touch.
The wavefunction is complex-valued; only its magnitude squared yields a measurable probability.
A common misconception is that ψ is a physical wave in ordinary space. For a single particle that image roughly works, but for many particles ψ is a complex function on a high-dimensional configuration space — a tool for computing probabilities, not a substance.