= Position representation
{wiki}
A way to write the wavefunction $\psi(x)$ such that the <position operator> is:
$$
x
$$
i.e., a function that takes the wavefunction as input, and outputs another function:
$$
x \psi(x)
$$
If you believe that <mathematicians> took care of <continuous spectrum (functional analysis)> for us and that everything just works, the most concrete and direct thing that this representation tells us is that:
\Q[the probability of finding a particle between $x_0$ and $x_1$ at time $t$]
equals:
$$
\int_{x_0}^{x_1}x \psi{x, t} dx
$$