Bra-ket notation

In quantum mechanics, Bra-ket notation is a standard notation for describing quantum states, composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics. It is so called because the inner product (or dot product) of two states is denoted by a <bra|c|ket>;

$\langle\phi|\psi\rangle$,

consisting of a left part, $\langle\phi|$, called the bra ( /brɑː/), and a right part, $|\psi\rangle$, called the ket ( /kɛt/). The notation was introduced in 1930 by Paul Dirac, and is also known as Dirac notation.

Bra-ket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantum mechanics—including a large portion of modern physics — is usually explained with the help of bra-ket notation. The expression $\langle\phi|\psi\rangle$ is typically interpreted as the probability amplitude for the state ψ to collapse into the state ϕ.

Vector representations

Vectors in Euclidean spaces

In physics, basis vectors allow any vector to be represented geometrically using angles and lengths, in different directions, i.e. in terms of the spatial orientations. It is simpler to see the notational equivalences between ordinary notation and bra-ket notation, so for now; consider a vector A as an element of 3-d Euclidean space using the field of real numbers, symbolically stated as $\bold{A}\in\mathbb{R}^3\,\!$.

The vector A can be written using any set of basis vectors and corresponding coordinate system. Informally basis vectors are like "building blocks of a vector", they are added together to make a vector, and the coordinates are the number of basis vectors in each direction. Two useful representations of a vector are simply a linear combination of basis vectors, and column matricies. Using the familiar cartesian basis, a vector A is written;

Illustration of cartesian vectors, bases, coordinates and components. The coordinates of the vector are equal to the projections of the vector (yellow) onto the x-component basis vector (green) - using the dot product (a special case of an inner product, see below).

$\bold{A} = A_x \bold{e}_x + A_y \bold{e}_y + A_z \bold{e}_z = \begin{pmatrix} A_x \\ A_y \\ A_z \\ \end{pmatrix}$

respectively, where e_x, e_y, e_z denotes the cartesian basis vectors (all are orthogonal unit vectors) and A_x, A_y, A_z are the corresponding coordinates, in the x, y, z directions. Natural alternatives to Cartesian are spherical and cylindrical systems. In general for any basis in 3d space we write;

$\bold{A} = A_1 \bold{e}_1 + A_2 \bold{e}_2 + A_3 \bold{e}_3 = \begin{pmatrix} A_1 \\ A_2 \\ A_3 \\ \end{pmatrix}$

Generalizing further, consider a vector A in an N dimensional vector space over the field of complex numbers $\mathbb{C}$, symbolically stated as $\bold{A} \in \mathbb{C}^N$. The vector A is still conventionally represented by a linear combination of basis vectors or a column matrix:

$\bold{A} = \sum_{n=1}^N A_n \bold{e}_n = \begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_N \\ \end{pmatrix}$

though the coordinates and vectors are now all complex-valued.

Bras and kets in Hilbert spaces

Rather than boldtype, over/under-arrows, underscores etc conventionally used elsewhere; $\bold{A},\,\underline{A}, \, \vec{A}$, Dirac's notation for a vector uses vertical bars and angular brackets; $| A\rangle$. This applies to all vectors, the resultant vector and the basis. The previous vectors are now written

Illustration of cartesian vectors, bases, coordinates and components. The coordinates of the vector are equal to the projections of the vector (yellow) onto the x-component basis vector (green) - using the inner product (see below).

$\begin{align} |A \rangle & = A_x|e_x \rangle + A_y|e_y \rangle + A_z|e_z \rangle = \begin{pmatrix} A_x \\ A_y \\ A_z \\ \end{pmatrix}, \\ |A \rangle & = A_1|e_1 \rangle + A_2|e_2 \rangle + A_3|e_3 \rangle = \begin{pmatrix} A_1 \\ A_2 \\ A_3 \\ \end{pmatrix}, \end{align}$

The last one may be written for short by

$|A \rangle = A_1|1 \rangle + A_2|2 \rangle + A_3|3 \rangle$

More generally, a vector $|A\rangle$ is an element of a Hilbert space $\mathcal{H}$, meaning $|A\rangle \in \mathcal{H}$, and represented by:

$|A\rangle= \sum_{n=1}^N A_n |e_n\rangle = \begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_N \\ \end{pmatrix}$

These vectors are kets, read as "ket-A". For reasons of efficient manipulation, and particularly due to Heisenberg's matrix mechanics, the matrix representation remains. In principle Dirac's notation can be applied universally, though its strict application is to abstract vector spaces - most frequently a projective Hilbert space, denoted $\mathcal{H}$. This space uses the field of complex numbers, again meaning the vector basis and coordinates are complex-valued.

An extra feature not shown above is a dual ket, given by:

$\langle A | = \sum_{n=1}^N A_n^* \langle e_n | = \begin{pmatrix} A_1 \\ A_2\\ \vdots \\ A_N\\ \end{pmatrix}^{*T} = \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix}$

These are bra vectors, read as "bra-A".

The bra is simply the conjugate transpose and matrix transpose (taken together the Hermitian conjugate) of the ket and vice versa. The above cases are finite-dimensional Hilbert spaces, i.e. column/row vectors with a finite number of elements. In infinite-dimensional spaces there are infinitely many coordinates and the ket may be written in complex function notation, by prepending it with a bra (see below).

Technically; bras are continuous linear functionals from $\mathcal H$ to the complex numbers $\mathbb{C}$, defined by:

$\langle\psi| : \mathcal H \to \mathbb{C} \,\!$

in which the functional takes a ket, and returns a complex number;

$\langle \psi | \left( \, |\rho\rangle \, \right) = \operatorname{IP}\left( |\psi\rangle \; , \; |\rho\rangle \right),$

for all kets in the Hilbert space (symbolically $\forall\,|\rho\rangle\in\mathcal{H}$), where IP( , ) denotes the inner product defined on the Hilbert space. Here the origin 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 $\langle\psi|\rho\rangle$, 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.

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. More precisely, if

$J: \mathcal H \rightarrow \mathcal H^*$

is the Riesz isomorphism between $\mathcal H$ and its dual space, then

$\forall \phi \in \mathcal H: \; \langle\phi| = J(|\phi\rangle).$

Note that this only applies to states that are actually vectors in the Hilbert space. Non-normalizable states, such as those whose wavefunctions are Dirac delta functions or infinite plane waves, do not technically belong to the Hilbert space. So if such a state is written as a ket, it will not have a corresponding bra according to the above definition. This problem can be dealt with in either of two ways. First, since all physical quantum states are normalizable, one can carefully avoid non-normalizable states. Alternatively, the underlying theory can be modified and generalized to accommodate such states, as in the Gelfand–Naimark–Segal construction or rigged Hilbert spaces. In fact, physicists routinely use bra-ket notation for non-normalizable states, taking the second approach either implicitly or explicitly.

Inner products

In Euclidean space of any finite dimension, the dot product can be defined for vectors using orthonormal bases:

$\bold{A}\cdot\bold{B} = \sum_{n=1}^N A_nB_n$

for the same vector, the dot product of a vector with itself is the square of its norm (magnitude)

$\bold{A}\cdot\bold{A} = \sum_{n=1}^N A_n^2 = \|A\|^2$

Orthonormality means if two vectors are perpendicular, their dot product is zero, for any two orthonormal basis vectors e_i and e_j this reads,

$\bold{e}_i\cdot\bold{e}_j = \delta_{ij}$

where δ_ij is the Kronecker delta.

This operation has the interpretation as a projection of one magnitude of a vector onto the other. Using this fact, the coordinates with respect to the chosen basis are projections of the vector itself to the basis vectors. For the cartesian coordinates they are:

$A_x = \bold{e}_x\cdot\bold{A} \quad A_y = \bold{e}_y \cdot\bold{A} \quad A_z = \bold{e}_z \cdot\bold{A}$

For space of finite dimension N, the coordinates are:

$A_n = \bold{e}_n\cdot\bold{A}$

for n = 1, 2, ... N.

Dot products are special cases of the general inner product. In bra-ket notation this is:

$\langle A | B \rangle = \left(\sum_{n=1}^N A_n^* \langle e_n |\right)\left(\sum_{n=1}^N B_n | e_n \rangle \right) = \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \begin{pmatrix} B_1 \\ B_2 \\ \vdots \\ B_N \\ \end{pmatrix}$

For the case of the same vector,

$\langle A | A \rangle = \left(\sum_{n=1}^N A_n^* \langle e_n | \right)\left(\sum_{n=1}^N A_n | e_n \rangle \right) = \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_N \\ \end{pmatrix} = \sum_{n=1}^N |A_n|^2 = \| A \|^2$

so analogous to the dot product yeilding the square of the magnitude of a vector, the inner product of a bra and ket is the square of the vector norm. Again, orthonormality reads,

$\langle e_i|e_j\rangle = \delta_{ij}$

Using the inner product on the above euclidean vectors, written in bra-ket notation the cartesian coordinates are

$A_x = \langle e_x | A \rangle, \quad A_y = \langle e_y | A \rangle ,\quad A_z = \langle e_z | A \rangle$

Usage in quantum mechanics

In quantum mechanics, since wavefunctions can be added in linear combinations just like vectors,

Ψ =	∑	cnϕn
	n

Dirac introduced a notation to:

1. Extend the idea of multiple spatial basis sets (such as above) to incorperate the state of the system into the a basis set. Momentum components, energy levels, quantum numbers, spins, can be thought of as basis vectors for a wavefunction in the Hilbert space (in this instance a simple, real-valued Euclidean vector space is insufficient). Hence the basis vectors indicate the state of the system, and the state of a physical system is identified with a ray in a complex separable Hilbert space, $\mathcal{H}$, or, equivalently, by a point in the projective Hilbert space of the system. Each vector in the ray a "ket" written as $|\psi\rangle$.
2. Use any useful set of basis vectors to construct the overall quantum state: the wavefunction as a vector in the vector space, rather than a mathematical function. Since any basis can be used, the wavefunction is basis-independent.

State vector

The advantage of the bra-ket notation over explicit wave function algebra is the possibility of expressing operations on quantum states independent of a basis. For example the time-independent Schrödinger equation is simply expressed as

$\hat H|\Psi\rangle=E |\Psi\rangle.$

in which the above vector is re-written

$|\Psi\rangle = \sum_n c_n|\phi_n\rangle$

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 Schrödinger equation). This process is very similar to the use of coordinate vectors in linear algebra.

For instance, the Hilbert space of a spin-0 point particle is spanned by a position basis $\{|\mathbf{r}\rangle\}$, where the label r extends over the set of position vectors. Starting from any ket $|\Psi\rangle$ in this Hilbert space, we can define a complex scalar function of r, known as a wavefunction:

$\Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|\Psi\rang.$

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

$A \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|A|\Psi\rang.$

For instance, the momentum operator p has the following form:

$\mathbf{p} \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r} |\mathbf{p}|\Psi\rang = - i \hbar \nabla \Psi(\mathbf{r}).$

In quantum mechanics the expression $\langle\phi|\psi\rangle$ is typically interpreted as the probability amplitude for the state $\psi\!$ to collapse into the state $\phi\!$. Mathematically, this means the coefficient for the projection of $\psi\!$ onto $\phi\!$.

One occasionally encounters an expression like

$- i \hbar \nabla |\Psi\rang.$

though is something of a (fairly common) abuse of notation. 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:

$- i \hbar \nabla \lang\mathbf{r}|\Psi\rang.$

By analogy with the above, the wave functions momentum space or reciprocal space (i.e. using the momentum p or the wave-vector k instead of position r) can be represented using bra-kets in the following way:

$\Psi_\bold{p}=\langle \bold{p} | \Psi \rangle, \Psi_\bold{k}=\langle \bold{k} | \Psi \rangle$

and all basis conversions can be performed via relations such as

$\langle \bold{k}|\bold{r} \rangle = e^{-i\bold{k}\cdot\bold{r}}, \langle \bold{p}|\bold{r} \rangle = e^{-i\bold{p}\cdot\bold{r}/\hbar}$

For further details, see rigged Hilbert space.

States and basis vectors

Examples of bases where state properties are incorperated into a basis follow. The components of the momentum of a particle can be written as:

$| p \rangle = p_x|p_x \rangle + p_y|p_y \rangle + p_z|p_z \rangle = \begin{pmatrix} p_x \\ p_y \\ p_z \\ \end{pmatrix},$

Superpositions of energy eigenstates can be written as:

$| E \rangle = c_1|E_1 \rangle + c_3|E_3 \rangle = \begin{pmatrix} c_1 \\ c_3 \\ \end{pmatrix},$

where E₁ and E₃ are energies corresponding to energy levels 1 and 3 in (say) a particle in a box. The state of electrons in multi-electron atoms use the principal n, orbital l, total angular momentum j, and assocaited magnetic m_j quantum numbers;

$| n, \ell, j, m_j \rangle$

Overlaps of two 1s atomic orbitals (energy eigenstates) in Hydrogen-like atoms, which form σ bonding and antibonding, can be written respectively as

$\begin{align} & | \sigma \rangle = c_A|1s_A \rangle + c_B|1s_B \rangle = \begin{pmatrix} c_A \\ c_B \\ \end{pmatrix}, \\ & | \sigma^* \rangle = c_A|1s_A \rangle - c_B|1s_B \rangle = \begin{pmatrix} c_A \\ c_B \\ \end{pmatrix}, \end{align}$

where A denotes one atom, B denotes the other, and * (asterisk) denotes an antibonded state. Eigenfunctions for two-electron systems (non-interacting) can be written as

$| \psi \rangle = \frac{|\uparrow\downarrow\rangle - |\downarrow\uparrow \rangle }{\sqrt{2}}$

where ↑ indicates "spin-up (+1/2)", ↓ "spin-down (–1/2)", and is antisymmetric becuase electrons are fermions, which obey the Pauli principle.

Notice how any symbols, letters, numbers, or even words — whatever serves as a convenient label, can be used for bras and kets.

Linear operators

Given the linear operator

$A \, : \, \mathcal{H} \rightarrow \mathcal{H}$

we can apply A to the ket $|\psi\rangle$ to obtain the ket $(A|\psi\rangle)$. 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 $\langle\phi|$ with the operator A results in the bra $\bigg(\langle\phi|A\bigg)$, defined as a linear functional on H by the rule

$\bigg(\langle\phi|A\bigg) \; |\psi\rangle = \langle\phi| \; \bigg(A|\psi\rangle\bigg)$.

This expression is commonly written as (cf. energy inner product)

$\langle\phi|A|\psi\rangle.$

Note that the second symbol | is completely optional, i.e. $\langle\phi|A|\psi\rangle=\langle\phi|A\psi\rangle$, since $A|\psi\rangle$ is in itself a ket and may be written $|A\psi\rangle$.

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 $|\psi\rangle$, written as

$\langle\psi|A|\psi\rangle.$

A convenient way to define linear operators on H is given by the outer product: if $\langle\phi|$ is a bra and $|\psi\rangle$ is a ket, the outer product

$|\phi\rang \lang \psi|$

denotes the rank-one operator that maps the ket $|\rho\rangle$ to the ket $|\phi\rangle\langle\psi|\rho\rangle$ (where $\langle\psi|\rho\rangle$ is a scalar multiplying the vector $|\phi\rangle$):

One of the uses of the outer product is to construct projection operators. Given a ket $|\psi\rangle$ of norm 1, the orthogonal projection onto the subspace spanned by $|\psi\rangle$ is

$|\psi\rangle\langle\psi|.$

Just as kets and bras can be transformed into each other (making $|\psi\rangle$ into $\langle\psi|$) the element from the dual space corresponding with $A|\psi\rangle$ is $\langle \psi | A^\dagger$ where A^† denotes the Hermitian conjugate (or adjoint) 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^† = A. Then the identity

$\langle \psi | A | \psi \rangle^\star = \langle \psi |A^\dagger |\psi \rangle = \langle \psi | A | \psi \rangle$

holds (for the first equality, use the scalar product's conjugate symmetry and the conversion rule from the preceding paragraph). This implies that expectation values of observables are real.

Quantum operators

Quantum operators themselves can be conveniently expressed in different bases (see next section for the operations used here - the action of a linear operator and outer product of a ket and a bra):

$\hat p=\int \frac{\mathrm{d} p}{2\pi} |p\rangle p \langle p| = \int \mathrm{d} x |x\rangle (- i \hbar \part_x) \langle x |$

$\hat x=\int \mathrm{d} x |x\rangle x \langle x| = \int \frac{\mathrm{d} p}{2\pi} |p\rangle \left(+i \hbar \part_p\right) \langle p |.$

For a rigorous treatment of the Dirac inner product of non-normalizable states see the definition given by D. Carfì in and . For a rigorous definition of basis with a continuous set of indices and consequently for a rigorous definition of position and momentum basis see . For a rigorous statement of the expansion of an S-diagonalizable operator - observable - in its eigenbasis or in another basis see .

Properties

Bra-ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions. Some of the properties that allow this manipulation are listed herein. In what follows, c₁ and c₂ denote arbitrary complex numbers, c* denotes the complex conjugate of c, A and B denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets.

Linearity

• Since bras are linear functionals,

$\langle\phi| \; \bigg( c_1|\psi_1\rangle + c_2|\psi_2\rangle \bigg) = c_1\langle\phi|\psi_1\rangle + c_2\langle\phi|\psi_2\rangle.$

• By the definition of addition and scalar multiplication of linear functionals in the dual space,

$\bigg(c_1 \langle\phi_1| + c_2 \langle\phi_2|\bigg) \; |\psi\rangle = c_1 \langle\phi_1|\psi\rangle + c_2 \langle\phi_2|\psi\rangle.$

Associativity

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). For example:

$\lang \psi| (A |\phi\rang) = (\lang \psi|A)|\phi\rang$

$(A|\psi\rang)\lang \phi| = A(|\psi\rang \lang \phi|)$

and so forth. The expressions can thus be written, unambiguously, with no parentheses whatsoever. Note that the associative property does not hold for expressions that include non-linear operators, such as the antilinear time reversal operator in physics.

Hermitian conjugation

Bra-ket notation makes it particularly easy to compute the Hermitian conjugate (also called dagger, and denoted †) of expressions. The formal rules are:

• The Hermitian conjugate of a bra is the corresponding ket, and vice-versa.
• The Hermitian conjugate of a complex number is its complex conjugate.
• The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is itself—i.e.,

$(x^\dagger)^\dagger=x$.

• Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as follows:

• Kets:

$\left(c_1|\psi_1\rangle + c_2|\psi_2\rangle\right)^\dagger = c_1^* \langle\psi_1| + c_2^* \langle\psi_2|.$

• Inner products:

$\lang \phi | \psi \rang^* = \lang \psi|\phi\rang$

• Matrix elements:

$\lang \phi| A | \psi \rang^* = \lang \psi | A^\dagger |\phi \rang$

$\lang \phi| A^\dagger B^\dagger | \psi \rang^* = \lang \psi | BA |\phi \rang$

• Outer products:

$\left((c_1|\phi_1\rang\lang \psi_1|) + (c_2|\phi_2\rang\lang\psi_2|)\right)^\dagger = (c_1^* |\psi_1\rang\lang \phi_1|) + (c_2^*|\psi_2\rang\lang\phi_2|)$

Composite bras and kets

Two Hilbert spaces V and W may form a third space $V \otimes W$ 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 $|\psi\rangle$ is a ket in V and $|\phi\rangle$ is a ket in W, the direct product of the two kets is a ket in $V \otimes W$. This is written variously as

$|\psi\rangle|\phi\rangle$ or $|\psi\rangle \otimes |\phi\rangle$ or $|\psi \phi\rangle$ or $|\psi ,\phi\rangle.$

The unit operator

Consider a complete orthonormal system (basis), $\{ e_i \ | \ i \in \mathbb{N} \}$, for a Hilbert space H, with respect to the norm from an inner product $\langle\cdot,\cdot\rangle$. From basic functional analysis we know that any ket $|\psi\rangle$ can be written as

$|\psi\rangle = \sum_{i \in \mathbb{N}} \langle e_i | \psi \rangle | e_i \rangle,$

with $\langle\cdot|\cdot\rangle$ the inner product on the Hilbert space. From the commutativity of kets with (complex) scalars now follows that

$\sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | = \hat{1}$

must be the unit operator, which sends each vector to itself. This can be inserted in any expression without affecting its value, for example

$\langle v | w \rangle = \langle v | \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | w \rangle = \langle v | \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | \sum_{j \in \mathbb{N}} | e_j \rangle \langle e_j | w \rangle = \langle v | e_i \rangle \langle e_i | e_j \rangle \langle e_j | w \rangle$

where in the last identity Einstein summation convention has been used.

In quantum mechanics it often occurs that little or no information about the inner product $\langle\psi|\phi\rangle$ of two arbitrary (state) kets is present, while it is possible to say something about the expansion coefficients $\langle\psi|e_i\rangle = \langle e_i|\psi\rangle^*$ and $\langle e_i|\phi\rangle$ 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 (for more information see Resolution of the identity).

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 $\mathcal{H}$ be a Hilbert space and $h\in\mathcal{H}$ is a vector in $\mathcal{H}$. What physicists would denote as $|h\rangle$ is the vector itself. That is

$(|h\rangle)\in \mathcal{H}$.

Let $\mathcal{H}^*$ be the dual space of $\mathcal{H}$. This is the space of linear functionals on $\mathcal{H}$. The isomorphism $\Phi:\mathcal{H}\to\mathcal{H}^*$ is defined by Φ(h) = ϕ_h where for all $g\in\mathcal{H}$ we have

$\phi_h(g) = \mbox{IP}(h,g) = (h,g) = \langle h,g \rangle = \langle h|g \rangle$,

Where

$\mbox{IP}(\,,\,), (\,,\,),\langle \,,\, \rangle, \langle \, | \, \rangle$

are just different notations for expressing an inner product between two elements in a Hilbert space (or for the first three, in any inner product space). Notational confusion arises when identifying ϕ_h and g with $\langle h |$ and $|g \rangle$ respectively. This is because of literal symbolic substitutions. Let $\phi_h = H = \langle h|$ and let $g=G=|g\rangle$. This gives

$\phi_h(g) = H(g) = H(G)=\langle h|(G) = \langle h|( |g\rangle).$

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.

Moreover (and more embarrasingly, although this is esssentially trivial) the mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they don't use the *-symbol, but an overline (which the physicists reserve to averages) to denote conjugate-complex numbers, i.e. for scalar products mathematicians usually write

$(\phi ,\psi )=\int {\rm d}x\,\phi (x)\cdot \overline{\psi(x)}\,,$

whereas physisists would write for the same quantity

$\langle\psi |\phi \rangle=\int {\rm d}x\,\psi^*(x)\cdot\phi(x)\,.$

More general uses

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.

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
• Robert Littlejohn, Lecture notes on "The Mathematical Formalism of Quantum mechanics", including bra-ket notation.