Bra-ket notation Guide, Meaning , Facts, Information and Description

Bra-ket notation is the standard notation used for describing quantum mechanical states; it can also be used as a notation for vectors and linear functionals in pure mathematics. It was invented by Paul Dirac, and is also known as Dirac notation. It is so called because the inner product of two states is denoted by a bracket, ‹φ|ψ›, consisting of a left part, ‹φ|, called the bra, and a right part, |ψ›, called the ket.

Table of contents

1 Bras and kets 2 Duals 3 Properties 4 Linear operators 5 Composite bras and kets 6 Representations in terms of bras and kets

Bras and kets

In quantum mechanics, the state of a physical system is identified with a vector in a complex Hilbert space, H. Each vector is called a "ket", and written as

where ψ denotes the particular ket. Each element of the continuous dual of H (i.e. each continuous linear functional from H to the complex numbers C) is known as a "bra", and written as

where φ names the bra in question. Applying the bra ‹φ| to the ket |ψ› results in a complex number, called a "bra-ket" or "bracket", which is written as

Duals

Every ket |ψ› has a dual bra, written as ‹ψ|, a continuous linear function on H defined as follows:

for all kets

where ( , ) denotes the inner product defined on the Hilbert space. The notation is justified by the Riesz representation theorem, which states that a Hilbert space and its dual space are isometrically isomorphic. Thus, each bra corresponds to exactly one ket, and vice versa.

Incidentally, bra-ket notation can be used even if the vector space is not a Hilbert space. In any Banach space B, the vectors may be notated by bras and the continuous linear functionals by kets. Over any vector space without topology, we may also notate the vectors by bras and the linear functionals by kets. In these more general contexts, the bracket does not the meaning of an inner product, because the Riesz representation theorem does not hold.

Properties

Bras and kets can be manipulated in the following ways:

• Given any bra ‹φ|, kets |ψ₁› and |ψ₂›, and complex numbers c₁ and c₂, then, since bras are linear functionals,

• Given any ket |ψ›, bras ‹φ₁| and ‹φ₂|, and complex numbers c₁ and c₂, then, by the definition of addition and scalar multiplication of linear functionals,

• Given any kets |ψ₁› and |ψ₂›, and complex numbers c₁ and c₂, from the properties of the inner product (with c* denoting the complex conjugate of c),

is dual to

• Given any bra ‹φ| and ket |ψ›, an axiomatic property of the inner product gives

Linear operators

If A : H → H is a linear operator, we can apply A to the ket |ψ› to obtain the ket (A|ψ›). The operator also acts on bras: applying the operator A to the bra ‹φ| results in the bra (‹φ|A), defined as a linear functional on H by the rule

This expression is commonly written as

A convenient way to define linear operators on H is given by the outer product: if ‹φ| is a bra and |ψ› is a ket, the outer product

denotes the operator that maps the ket |ρ› to the ket |φ›‹ψ|ρ› (where ‹ψ|ρ› is a scalar multiplying the vector |φ›). One of the uses of the outer product is to construct projection operators. Given a ket |ψ› of norm 1, the orthogonal projection onto the subspace spanned by |ψ› is

Composite bras and kets

Two Hilbert spaces V and W may form a third space by a tensor product. If |ψ› is a ket in V and |φ› is a ket in W, the tensor product of the two kets is a ket in . This is written variously as

or or .

Representations in terms of bras and kets

It is sometimes more convenient to work with the projections of state vectors onto a particular basis, rather than the vectors themselves. The reason is that the former are simply complex numbers, and can be formulated in terms of partial differential equations (see, for example, the derivation of the position-basis Schrödinger equation.) This process is very similiar to the use of coordinate vectors in linear algebra.

For instance, the Hilbert space of a zero-spin point particle is spanned by a position basis {|x>}, where the label x extends over the set of position vectors. Starting from any ket |ψ> in this Hilbert space, we can define a complex scalar function of x, known as a wavefunction:

It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by

Although the operator A on the left hand side of this equation is, by convention, labelled in the same way as the operator on the right hand side, it should be borne in mind that the two are conceptually different entities: the first acts on wavefunctions, and the second acts on kets. For instance, the momentum operator p has the following form:

One occasionally encounters an expression like

This is something of an abuse of notation, though a fairly common one. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differenting wavefunctions once the expression is projected into the position basis. For further details, see rigged Hilbert space.

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