position representation

The position representation is the way of writing a quantum state as a function of position — the familiar ψ(x). It amounts to choosing position as the 'axes' along which to lay out the abstract state vector, so that ψ(x) is the amplitude for finding the particle at each point x. This is the representation most beginners meet first, because |ψ(x)|² maps so directly onto the question 'where is the particle likely to be?'.

Choosing this representation fixes how the basic observables look. Position itself becomes simple — to apply it you just multiply ψ(x) by x — while momentum turns into a derivative with respect to x, an instruction to read off how steeply the phase of the wavefunction varies through space. The Schrödinger equation, in this dress, becomes a differential equation for ψ(x, t), which is why so much of practical quantum mechanics is the art of solving such equations.

It is worth holding lightly to the idea that ψ(x) is 'the' wavefunction. It is one representation among several of a single underlying state; the momentum representation describes the very same physics in terms of momentum instead, and the two are linked by a Fourier transform. The position representation is privileged only by familiarity and by how naturally it answers questions about location, not by any deeper claim to be more real.