It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics.
can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product:
, and the set of all covectors forms a subspace of the dual vector space
Combinations of bras, kets, and linear operators are interpreted using matrix multiplication.
It is common to suppress the vector or linear form from the bra–ket notation and only use a label inside the typography for the bra or ket.
In physics, however, the term "vector" tends to refer almost exclusively to quantities like displacement or velocity, which have components that relate directly to the three dimensions of space, or relativistically, to the four of spacetime.
To distinguish this type of vector from those described above, it is common and useful in physics to denote an element
Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the
In other words, the symbol "|A⟩" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "A" by itself does not.
Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling energy eigenkets in quantum mechanics through a listing of their quantum numbers.
For example: Note how the last line above involves infinitely many different kets, one for each real number x.
of rank one with outer product The bra–ket notation is particularly useful in Hilbert spaces which have an inner product[5] that allows Hermitian conjugation and identifying a vector with a continuous linear functional, i.e. a ket with a bra, and vice versa (see Riesz representation theorem).
Moreover, conventions are set up in such a way that writing bras, kets, and linear operators next to each other simply imply matrix multiplication.
In quantum mechanics, it is common practice to write down kets which have infinite norm, i.e. non-normalizable wavefunctions.
Examples include states whose wavefunctions are Dirac delta functions or infinite plane waves.
In a Banach space B, the vectors may be notated by kets and the continuous linear functionals by bras.
In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply.
Starting from any ket |Ψ⟩ in this Hilbert space, one may define a complex scalar function of r, known as a wavefunction,[clarification needed]
This usage is more common when denoting vectors as tensor products, where part of the labels are moved outside the designed slot, e.g.
For example, observable physical quantities are represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time.
In an N-dimensional Hilbert space, ⟨φ| can be written as a 1 × N row vector, and A (as in the previous section) is an N × N matrix.
then this expression gives the expectation value, or mean or average value, of the observable represented by operator A for the physical system in the state |ψ⟩.
For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:
Just as kets and bras can be transformed into each other (making |ψ⟩ into ⟨ψ|), the element from the dual space corresponding to A|ψ⟩ is ⟨ψ|A†, where A† denotes the Hermitian conjugate (or adjoint) of the operator A.
Given any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators (but not addition), written in bra–ket notation, the parenthetical groupings do not matter (i.e., the associative property holds).
The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguously because of the equalities on the left.
Bra–ket notation makes it particularly easy to compute the Hermitian conjugate (also called dagger, and denoted †) of expressions.
The formal rules are: These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as follows: Two Hilbert spaces V and W may form a third space V ⊗ W by a tensor product.
In quantum mechanics, it often occurs that little or no information about the inner product ⟨ψ|φ⟩ of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients ⟨ψ|ei⟩ = ⟨ei|ψ⟩* and ⟨ei|φ⟩ of those vectors with respect to a specific (orthonormalized) basis.
III.20, Dirac defines the standard ket which, up to a normalization, is the translationally invariant momentum eigenstate