Editing Bra–ket notation (section)

===Vectors vs kets===

In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" tends to refer almost exclusively to quantities like [[displacement (vector)|displacement]] or [[velocity]], which have components that relate directly to the three dimensions of [[space]], or relativistically, to the four of [[spacetime]]. Such vectors are typically denoted with over arrows (<math>\vec r</math>), boldface (<math>\mathbf{p}</math>) or indices (<math>v^\mu</math>).

In quantum mechanics, a quantum state is typically represented as an element of a complex [[Hilbert space]], for example, the infinite-dimensional vector space of all possible [[wavefunction]]s (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically. To distinguish this type of vector from those described above, it is common and useful in physics to denote an element <math>\phi</math> of an abstract complex vector space as a ket <math>|\phi\rangle</math>, to refer to it as a "ket" rather than as a vector, and to pronounce it "ket-<math>\phi</math>" or "ket-A" for {{math|{{ket|''A''}}}}.

Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the <math>|\ \rangle</math> making clear that the label indicates a vector in vector space. In other words, the symbol "{{math|{{ket|''A''}}}}" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "{{math|''A''}}" by itself does not. For example, {{math|{{ket|1}} + {{ket|2}}}} is not necessarily equal to {{math|{{ket|3}}}}. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling [[stationary state|energy eigenkets]] in quantum mechanics through a listing of their [[quantum number]]s. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as <math>\hat x</math>, <math>\hat p</math>, <math>\hat L_z</math>, etc.