Editing Bra–ket notation (section)

===Linear operators acting on kets===

A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have [[linear operator|certain properties]].) In other words, if <math>\hat A</math> is a linear operator and <math>|\psi\rangle</math> is a ket-vector, then <math>\hat A |\psi\rangle</math> is another ket-vector.

In an <math>N</math>-dimensional Hilbert space, we can impose a basis on the space and represent <math>|\psi\rangle</math> in terms of its coordinates as a <math>N \times 1</math> [[column vector]]. Using the same basis for <math>\hat A</math>, it is represented by an <math>N \times N</math> complex matrix. The ket-vector <math>\hat A |\psi\rangle</math> can now be computed by matrix multiplication.

Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by [[self-adjoint operator]]s, such as [[energy]] or [[momentum]], whereas transformative processes are represented by [[unitary operator|unitary]] linear operators such as rotation or the progression of time.