= Wave function
{title2=$\psi(x,t)$}
{wiki}
= Wavefunction
{synonym}
A <wavefunction> is the position representation of a <pure state>. In the one-dimensional model used here, $\psi(x,t)$ takes values in the <complex numbers>, and the <Born rule> gives
$$
\rho(x,t)=|\psi(x,t)|^2,\qquad \Pr(a<x<b)=\int_a^b\rho(x,t)\,dx.
$$
The <probability density> $\rho$ has units of inverse length; the <wavefunction> has units of inverse square root of length. A value of $\rho$ at a point is not itself a probability: probabilities come from an <integral> over a region. For the <infinite square well> <ground state>, $\rho(L/2)=2/L$, whereas the probability of the exact single point $L/2$ is zero. The interesting question is how much probability a finite interval contains.
Two <wavefunctions> differing only by a <global phase> represent the same <pure state>. Relative <quantum phases> between components of a <quantum superposition> affect <matter-wave interference>, so retaining the full <complex number> amplitude is essential. This scalar description covers a particle without active <spin> degrees of freedom; a general <Electron> <wavefunction> can have two <spin> components, with the position <probability density> obtained by adding the squared <modulus> of each component.
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