# bra-ket notation

**Bra-ket notation**is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract vectors and linear functionals in pure mathematics. It is so called because the inner product (or dot product) of two states is denoted by a

**bra**c

**ket**, , consisting of a left part, , called the

**bra**, and a right part, , called the

**ket**. The notation was invented by Paul Dirac, and is also known as

**Dirac notation**.

## Bras and kets

### Most common use: Quantum mechanics

In quantum mechanics, the state of a physical system is identified with a unit ray in a complex separable Hilbert space, , or, equivalently, by a point in the projective Hilbert space of the system. Each vector in the ray is called a "**ket**" and written as , which would be read as "

*psi ket*". The ket can be viewed as a column vector and (given a basis for the Hilbert space) written out in components,

**bra**(see below). For example,

Every ket has a dual

**bra**, written as . For example, the bra corresponding to the ket above would be the row vector

This is a continuous linear functional from to the complex numbers , defined by:

- for all kets

where denotes the inner product defined on the Hilbert space. Here an advantage of the bra-ket notation becomes clear: when we drop the parentheses (as is common with linear functionals) and meld the bars together we get , which is common notation for an inner product in a Hilbert space. This combination of a bra with a ket to form a complex number is called a

*bra-ket*or

*bracket*.

In quantum mechanics the expression (mathematically: the coefficient for the projection of onto ) is typically interpreted as the probability amplitude for the state to collapse into the state

### More general uses

The bra is simply the conjugate transpose (also called the Hermitian conjugate) of the ket and vice versa. The notation is justified by the Riesz representation theorem, which states that a Hilbert space and its dual space are isometrically conjugate isomorphic. Thus, each bra corresponds to exactly one ket, and vice versa. However, this is not always the case; on page 111 of Quantum Mechanics by Cohen-Tannoudji*et al.*it is clarified that there is such a relationship between bras and kets, so long as the defining functions used are square integrable. This does not hinder quantum mechanics because all physically realistic wave functions are square integrable and thus elements of the Hilbert space L2, whereas sine and cosine are not in L2 and the Dirac delta function is not even a function, but a measure.

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 kets and the continuous linear functionals by bras. Over any vector space without topology, we may also notate the vectors by kets and the 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.

## Properties

Because each ket is a vector in a complex Hilbert space and each bra-ket is an inner product, it follows directly that bras and kets can be manipulated in the following ways:- Given any bra , kets and , and complex numbers
*c*_{1}and*c*_{2}, then, since bras are*linear*functionals,

- :

- Given any ket , bras and , and complex numbers
*c*_{1}and*c*_{2}, then, by the definition of addition and scalar multiplication of linear functionals,

- :

- Given any kets and , and complex numbers
*c*_{1}and*c*_{2}, from the properties of the inner product (with c* denoting the complex conjugate of c),

- : is dual to

- Given any bra and ket , 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 to obtain the ket . Linear operators are ubiquitous in the theory of quantum mechanics. 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.

Operators can also be viewed as acting on bras

*from the right hand side*. Composing the bra with the operator

*A*results in the bra , defined as a linear functional on

*H*by the rule

- .

This expression is commonly written as

If the same state vector appears on both bra and ket side, 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 , written as

A convenient way to define linear operators on

*H*is given by the outer product: if is a bra and is a ket, the outer product

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

Just as kets and bras can be transformed into each other (making into ) the element from the dual space corresponding with is where

*A*denotes the Hermitian conjugate of the operator

^{†}*A*.

It is usually taken as a postulate or axiom of quantum mechanics, that any operator corresponding to an observable quantity (shortly called

*observable*) is self-adjoint, that is, it satisfies

*A*. Then the identity

^{†}= A## Composite bras and kets

Two Hilbert spaces*V*and

*W*may form a third space by a tensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described in

*V*and

*W*respectively, then the Hilbert space of the entire system is the tensor product of the two spaces. (The exception to this is if the subsystems are actually identical particles. In that case, the situation is a little more complicated.)

If is a ket in V and is a ket in W, the direct product of the two kets is a ket in . This is written variously as

- or or or

## Representations in terms of bras and kets

In quantum mechanics, it is often 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 SchrÃ¶dinger equation). This process is very similar 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 , where the label

**x**extends over the set of position vectors. Starting from any ket 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

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 differentiating wavefunctions once the expression is projected into the position basis:

*For further details, see rigged Hilbert space.*

## The unit operator

Consider a complete orthonormal system (*basis*), , for a Hilbert space

*H*, with respect to the norm from an inner product . From basic functional analysis we know that any ket can be written as

In quantum mechanics it often occurs that little or no information about the inner product of two arbitrary (state) kets is present, while it is possible to say something about the expansion coefficients and of those vectors with respect to a chosen (orthonormalized) basis. In this case it is particularly useful to insert the unit operator into the bracket one time or more.

## Notation used by mathematicians

The object physicists are considering when using the "bra-ket" notation is a Hilbert space (a complete inner product space).Let be a Hilbert space and . What physicists would denote as is the vector itself. That is

- :.

Let be the dual space of . This is the space of linear functionals on . The isomorphism is defined by where for all we have

- :,

- :

*any*inner product space). Notational confusion arises when identifying and with and respectively. This is because of literal symbolic substitutions. Let and . This gives

- :

One ignores the parentheses and removes the double bars. Some properties of this notation are convenient since we are dealing with linear operators and composition acts like a ring multiplication.

## Further reading

- Feynman, Leighton and Sands (1965).
*The Feynman Lectures on Physics Vol. III*. Addison-Wesley. ISBN 0-201-02115-3.

## External links

- Richard Fitzpatrick, "Quantum Mechanics: A graduate level course", The University of Texas at Austin.
- 1. Ket space
- 2. Bra space
- 3. Operators
- 4. The outer product
- 5. Eigenvalues and eigenvectors

**.....**Click the link for more information.

**quantum mechanics**is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.

**.....**Click the link for more information.

In mathematics, a

**vector space**(or**linear space**) is a collection of objects (called*vectors*) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are**.....**Click the link for more information.**.....**Click the link for more information.

**Mathematics**(colloquially,

**maths**or

**math**) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".

**.....**Click the link for more information.

**inner product space**is a vector space of arbitrary (possibly infinite) dimension with additional structure, which, among other things, enables generalization of concepts from two or three-dimensional Euclidean geometry.

**.....**Click the link for more information.

**dot product**, also known as the

**scalar product**, is an operation which takes two vectors over the real numbers

**R**and returns a real-valued scalar quantity. It is the standard

**inner product**of the Euclidean space.

**.....**Click the link for more information.

**Paul Dirac**

Paul Adrien Maurice Dirac

Born July 8 1902

Bristol, England

Died September 20 1984 (aged 82)

**.....**Click the link for more information.

**quantum mechanics**is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.

**.....**Click the link for more information.

**Physics**is the science of

*matter*

^{[1]}

*and its motion*

^{[2]}

^{[3]}, as well as

*space and time*

^{[4]}

^{[5]}—the science that deals with concepts such as force, energy, mass, and charge.

**.....**Click the link for more information.

In mathematics, a

where

**complex number**is a number of the formwhere

*a*and*b*are real numbers, and*i*is the imaginary unit, with the property*i*Â² = −1.**.....**Click the link for more information. In topology and related areas of mathematics a topological space is called

**separable**if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space.**.....**Click the link for more information.**Hilbert space**, named after the David Hilbert, generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.

**.....**Click the link for more information.

In mathematics, a

**vector space**(or**linear space**) is a collection of objects (called*vectors*) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are**.....**Click the link for more information. In mathematics, any vector space

*V*has a corresponding**dual vector space**(or just dual space for short) consisting of all linear functionals on*V*. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in**.....**Click the link for more information.**.....**Click the link for more information.

**inner product space**is a vector space of arbitrary (possibly infinite) dimension with additional structure, which, among other things, enables generalization of concepts from two or three-dimensional Euclidean geometry.

**.....**Click the link for more information.

**Hilbert space**, named after the David Hilbert, generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.

**.....**Click the link for more information.

In quantum mechanics, a

**probability amplitude**is a complex-valued function that describes an uncertain or unknown quantity. For example, each particle has a probability amplitude describing its position. This amplitude is the wave function, expressed as a function of position.**.....**Click the link for more information. In certain interpretations of quantum mechanics,

**wave function collapse**is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. It is also called*collapse of the state vector*or*reduction of the wave packet*.**.....**Click the link for more information.**conjugate transpose**,

**Hermitian transpose**, or

**adjoint matrix**of an

*m*-by-

*n*matrix

*A*with complex entries is the

*n*-by-

*m*matrix

*A*

^{*}obtained from

*A*

**.....**Click the link for more information.

**adjoint operator**. Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a Hilbert space as "generalized complex numbers", then the adjoint of an operator plays the role of the complex

**.....**Click the link for more information.

There are several well-known theorems in functional analysis known as the

**Riesz representation theorem**. They are named in honour of Frigyes Riesz.## The Hilbert space representation theorem

**.....**Click the link for more information. In mathematics, any vector space

*V*has a corresponding**dual vector space**(or just dual space for short) consisting of all linear functionals on*V*. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in**.....**Click the link for more information. In mathematics, an

**integrable function**is a function whose integral exists. Unless specifically stated, the integral in question is usually the Lebesgue integral. Otherwise, one can say that the function is "Riemann-integrable" (i.e.**.....**Click the link for more information.**Dirac delta**or

**Dirac's delta**, often referred to as the

**unit impulse function**and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(

*x*) that has the value of infinity for

*x*

**.....**Click the link for more information.

**Hilbert space**, named after the David Hilbert, generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.

**.....**Click the link for more information.

In mathematics,

**Banach spaces**(pronounced ['banaɣ]), named after Stefan Banach, are one of the central objects of study in functional analysis.**.....**Click the link for more information.**.....**Click the link for more information.

In mathematics, a

**vector space**(or**linear space**) is a collection of objects (called*vectors*) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are**.....**Click the link for more information.This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.