Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread.
Bra–ket notation was created by Paul Dirac in his 1939 publication A New Notation for Quantum Mechanics. The notation was introduced as an easier way to write quantum mechanical expressions.[1] The name comes from the English word "bracket".
In quantum mechanics, bra–ket notation is used ubiquitously to denote quantum states. The notation uses angle brackets, and, and a vertical bar, to construct "bras" and "kets".
A ket is of the form
|v\rangle
\boldsymbolv
V
A bra is of the form
\langlef|
f:V\to\Complex
V
\Complex
\langlef|
|v\rangle
\langlef|v\rangle\in\Complex
Assume that on
V
( ⋅ , ⋅ )
V
\boldsymbol\phi\equiv|\phi\rangle
(\boldsymbol\phi, ⋅ )\equiv\langle\phi|
(\boldsymbol\phi,\boldsymbol\psi)\equiv\langle\phi|\psi\rangle
\langle\phi|
|\phi\rangle
V\vee
V
\langle\phi|
\boldsymbol\phi,
For the vector space
\Complexn
\Complexn
(\boldsymbolv,\boldsymbolw)=v\daggerw
\dagger
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. For example, the spin operator
\hat\sigmaz
\Delta
\boldsymbol\psi+,\boldsymbol\psi-\in\Delta
\boldsymbol\psi+=|+\rangle
\boldsymbol\psi-=|-\rangle
Bra–ket notation was effectively established in 1939 by Paul Dirac; it is thus also known as Dirac notation, despite the notation having a precursor in Hermann Grassmann's use of
[\phi{\mid}\psi]
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 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 (
\vecr
p
v\mu
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 wavefunctions (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
\phi
|\phi\rangle
\phi
Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the
| \rangle
\hatx
\hatp
\hatLz
Since kets are just vectors in a Hermitian vector space, they can be manipulated using the usual rules of linear algebra. For example:
\begin{align} |A\rangle&=|B\rangle+|C\rangle\\ |C\rangle&=(-1+2i)|D\rangle\\ |D\rangle&=
infty | |
\int | |
-infty |
-x2 | |
e |
|x\rangledx. \end{align}
Note how the last line above involves infinitely many different kets, one for each real number .
Since the ket is an element of a vector space, a bra
\langleA|
A bra
\langle\phi|
|\psi\rangle
|\psi\rangle\langle\phi|
|\psi\rangle\langle\phi|\colon|\xi\rangle\mapsto|\psi\rangle\langle\phi|\xi\rangle~.
See main article: Inner product.
The bra–ket notation is particularly useful in Hilbert spaces which have an inner product[3] 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). The inner product on Hilbert space
( , )
\phi=|\phi\rangle
f\phi=\langle\phi|
(\phi,\psi)=(|\phi\rangle,|\psi\rangle)=:f\phi(\psi)=\langle\phi|l(|\psi\rangler)=:\langle\phi{\mid}\psi\rangle
In the simple case where we consider the vector space
\Complexn
\Complexn
|\psi\rangle\langle\phi|
For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector:Based on this, the bras and kets can be defined as:and then it is understood that a bra next to a ket implies matrix multiplication.
The conjugate transpose (also called Hermitian conjugate) of a bra is the corresponding ket and vice versa:because if one starts with the brathen performs a complex conjugation, and then a matrix transpose, one ends up with the ket
Writing elements of a finite dimensional (or mutatis mutandis, countably infinite) vector space as a column vector of numbers requires picking a basis. Picking a basis is not always helpful because quantum mechanics calculations involve frequently switching between different bases (e.g. position basis, momentum basis, energy eigenbasis), and one can write something like "" without committing to any particular basis. In situations involving two different important basis vectors, the basis vectors can be taken in the notation explicitly and here will be referred simply as "" and "".
Bra–ket notation can be used even if the vector space is not a Hilbert space.
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. These do not, technically, belong to the Hilbert space itself. However, the definition of "Hilbert space" can be broadened to accommodate these states (see the Gelfand–Naimark–Segal construction or rigged Hilbert spaces). The bra–ket notation continues to work in an analogous way in this broader context.
Banach spaces are a different generalization of Hilbert spaces. In a Banach space, 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.
The mathematical structure of quantum mechanics is based in large part on linear algebra:
Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and often does involve, bra–ket notation. A few examples follow:
The Hilbert space of a spin-0 point particle is spanned by a "position basis", where the label extends over the set of all points in position space. This label is the eigenvalue of the position operator acting on such a basis state,
\hat{r
Starting from any ket in this Hilbert space, one may define a complex scalar function of, known as a wavefunction,
On the left-hand side, is a function mapping any point in space to a complex number; on the right-hand side, is a ket consisting of a superposition of kets with relative coefficients specified by that function.
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
\hatp
One occasionally even encounters an expression such as , though this is something of an 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 onto the position basis,
\nabla\langr|\Psi\rang,
In quantum mechanics the expression is typically interpreted as the probability amplitude for the state to collapse into the state . Mathematically, this means the coefficient for the projection of onto . It is also described as the projection of state onto state .
A stationary spin- particle has a two-dimensional Hilbert space. One orthonormal basis is:where is the state with a definite value of the spin operator equal to + and is the state with a definite value of the spin operator equal to −.
Since these are a basis, any quantum state of the particle can be expressed as a linear combination (i.e., quantum superposition) of these two states:where and are complex numbers.
A different basis for the same Hilbert space is:defined in terms of rather than .
Again, any state of the particle can be expressed as a linear combination of these two:
In vector form, you might writedepending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used.
There is a mathematical relationship between
a\psi
b\psi
c\psi
d\psi
There are some conventions and uses of notation that may be confusing or ambiguous for the non-initiated or early student.
A cause for confusion is that the notation does not separate the inner-product operation from the notation for a (bra) vector. If a (dual space) bra-vector is constructed as a linear combination of other bra-vectors (for instance when expressing it in some basis) the notation creates some ambiguity and hides mathematical details. We can compare bra–ket notation to using bold for vectors, such as
\boldsymbol\psi
( ⋅ , ⋅ )
\{|en\rangle\}
It has to be determined by convention if the complex numbers
\{\psin\}
It is common to use the same symbol for labels and constants. For example,
\hat\alpha|\alpha\rangle=\alpha|\alpha\rangle
\alpha
\hat\alpha
|\alpha\rangle
\alpha
A|a\rangle=a|a\rangle
It is common to see the usage
|\psi\rangle\dagger=\langle\psi|
\dagger
|\psi\rangle
l{H}
\langle\psi|
l{H}
|\psi\rangle
\langle\psi|
This is done for a fast notation of scaling vectors. For instance, if the vector
|\alpha\rangle
1/\sqrt{2}
|\alpha/\sqrt{2}\rangle
\alpha
|\alpha\rangle=|\alpha/\sqrt{2}\rangle1 ⊗ |\alpha/\sqrt{2}\rangle2
See also: Linear operator.
A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have certain properties.) In other words, if
\hatA
|\psi\rangle
\hatA|\psi\rangle
In an
N
|\psi\rangle
N x 1
\hatA
N x N
\hatA|\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. Specifically, if is a linear operator and is a bra, then is another bra defined by the rule(in other words, a function composition). This expression is commonly written as (cf. energy inner product)
In an -dimensional Hilbert space, can be written as a row vector, and (as in the previous section) is an matrix. Then the bra can be computed by normal matrix multiplication.
If the same state vector appears on both bra and ket side,then this expression gives the expectation value, or mean or average value, of the observable represented by operator for the physical system in the state .
A convenient way to define linear operators on a Hilbert space is given by the outer product: if is a bra and is a ket, the outer productdenotes the rank-one operator with the rule
For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:The outer product is an matrix, as expected for a linear operator.
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 isThis is an idempotent in the algebra of observables that acts on the Hilbert space.
See main article: Hermitian conjugate. Just as kets and bras can be transformed into each other (making into), the element from the dual space corresponding to is, where denotes the Hermitian conjugate (or adjoint) of the operator . In other words,
If is expressed as an matrix, then is its conjugate transpose.
Self-adjoint operators, where, play an important role in quantum mechanics; for example, an observable is always described by a self-adjoint operator. If is a self-adjoint operator, then is always a real number (not complex). This implies that expectation values of observables are real.
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, and denote arbitrary complex numbers, denotes the complex conjugate of, and denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets.
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:
\begin{align} \lang\psi|l(A|\phi\rangr)=l(\lang\psi|Ar)|\phi\rang&\stackrel{def
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:
\bigl(c_1|\psi_1\rangle + c_2|\psi_2\rangle\bigr)^\dagger = c_1^* \langle\psi_1| + c_2^* \langle\psi_2| \,.
\langle \phi| A | \psi \rangle^\dagger &= \left\langle \psi \left| A^\dagger \right|\phi \right\rangle \\\left\langle \phi\left| A^\dagger B^\dagger \right| \psi \right\rangle^\dagger &= \langle \psi | BA |\phi \rangle \,.\end
Two Hilbert spaces and 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 and 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 and is a ket in, the tensor product of the two kets is a ket in . This is written in various notations:
|\psi\rangle|\phi\rangle, |\psi\rangle ⊗ |\phi\rangle, |\psi\phi\rangle, |\psi,\phi\rangle.
See quantum entanglement and the EPR paradox for applications of this product.
Consider a complete orthonormal system (basis),for a Hilbert space, with respect to the norm from an inner product .
From basic functional analysis, it is known that any ket
|\psi\rangle
From the commutativity of kets with (complex) scalars, it follows thatmust be the identity operator, which sends each vector to itself.
This, then, can be inserted in any expression without affecting its value; for examplewhere, in the last line, the Einstein summation convention has been used to avoid clutter.
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 still possible to say something about the expansion coefficients and of those vectors with respect to a specific (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,[8] where
Since, plane waves follow,
In his book (1958), Ch. III.20, Dirac defines the standard ket which, up to a normalization, is the translationally invariant momentum eigenstate in the momentum representation, i.e.,
\hat{p}|\varpi\rangle=0
\langlex|\varpi\rangle\sqrt{2\pi\hbar}=1
Typically, when all matrix elements of an operator such as are available, this resolution serves to reconstitute the full operator,
The object physicists are considering when using bra–ket notation is a Hilbert space (a complete inner product space).
Let
(lH,\langle ⋅ , ⋅ \rangle)
\Phi:lH\hookrightarrowlH*
\Phi(h)=\varphih
\varphih:lH\toC
\varphih(g)=\langleh,g\rangle=\langleh\midg\rangle
\varphih=H=\langleh\mid
One ignores the parentheses and removes the double bars.
Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they usually use not an asterisk but an overline (which the physicists reserve for averages and the Dirac spinor adjoint) to denote complex conjugate numbers; i.e., for scalar products mathematicians usually writewhereas physicists would write for the same quantity