Bra–ket notation

Template:Quantum mechanics

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".

Quantum mechanics

In quantum mechanics and quantum computing, bra–ket notation is used ubiquitously to denote quantum states. The notation uses angle brackets, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle } and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rangle } , and a vertical bar $|$ , to construct "bras" and "kets".

A ket is of the form Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |v\rangle } . Mathematically it denotes a vector, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {v}}} , in an abstract (complex) vector space $V$ , and physically it represents a state of some quantum system.

A bra is of the form Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle f|} . Mathematically it denotes a linear form Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f:V\to \mathbb {C} } , i.e. a linear map that maps each vector in $V$ to a number in the complex plane $\mathbb {C}$ . Letting the linear functional Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle f|} act on a vector Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |v\rangle } is written as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle f|v\rangle \in \mathbb {C} } .

Assume that on $V$ there exists an inner product Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (\cdot ,\cdot )} with antilinear first argument, which makes $V$ an inner product space. Then with this inner product each vector Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {\phi }}\equiv |\phi \rangle } can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ({\boldsymbol {\phi }},\cdot )\equiv \langle \phi |} . The correspondence between these notations is then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ({\boldsymbol {\phi }},{\boldsymbol {\psi }})\equiv \langle \phi |\psi \rangle } . The linear form $\langle \phi |$ is a covector to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\phi \rangle } , and the set of all covectors forms a subspace of the dual vector space Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V^{\vee }} , to the initial vector space $V$ . The purpose of this linear form $\langle \phi |$ can now be understood in terms of making projections onto the state Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {\phi }},} to find how linearly dependent two states are, etc.

For the vector space Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {C} ^{n}} , kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted using matrix multiplication. If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {C} ^{n}} has the standard Hermitian inner product Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ({\boldsymbol {v}},{\boldsymbol {w}})=v^{\dagger }w} , under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the Hermitian conjugate (denoted Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {\sigma }}_{z}} on a two-dimensional space $\Delta$ of spinors has eigenvalues Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\textstyle \pm {\frac {1}{2}}} with eigenspinors ${\boldsymbol {\psi }}_{+},{\boldsymbol {\psi }}_{-}\in \Delta$ . In bra–ket notation, this is typically denoted as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {\psi }}_{+}=|+\rangle } , and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {\psi }}_{-}=|-\rangle } . As above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular, when also identified with row and column vectors, kets and bras with the same label are identified with Hermitian conjugate column and row vectors.

Bra–ket notation was effectively established in 1939 by Paul Dirac;^[1]^[2] it is thus also known as Dirac notation, despite the notation having a precursor in Hermann Grassmann's use of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [\phi {\mid }\psi ]} for inner products nearly 100 years earlier.^[3]^[4]

Vector spaces

Vectors vs kets

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 ( ${\vec {r}}$ ), boldface (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbf {p} } ) or indices (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 $\phi$ of an abstract complex vector space as a ket Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\phi \rangle } , to refer to it as a "ket" rather than as a vector, and to pronounce it "ket- $\phi$ " or "ket-A" for $| A ⟩$ .

Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\ \rangle } making clear that the label indicates a vector in vector space. 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. For example, $|1⟩ + |2⟩$ is not necessarily equal to $|3⟩$ . 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. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {x}}} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {p}}} , ${\hat {L}}_{z}$ , etc.

Notation

Since kets are just vectors in a Hermitian vector space, they can be manipulated using the usual rules of linear algebra. For example:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}|A\rangle &=|B\rangle +|C\rangle \\|C\rangle &=(-1+2i)|D\rangle \\|D\rangle &=\int _{-\infty }^{\infty }e^{-x^{2}}|x\rangle \,\mathrm {d} x\,.\end{aligned}}}

Note how the last line above involves infinitely many different kets, one for each real number $x$ .

Since the ket is an element of a vector space, a bra Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle A|} is an element of its dual space, i.e. a bra is a linear functional which is a linear map from the vector space to the complex numbers. Thus, it is useful to think of kets and bras as being elements of different vector spaces (see below however) with both being different useful concepts.

A bra $\langle \phi |$ and a ket Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\psi \rangle } (i.e. a functional and a vector), can be combined to an operator Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\psi \rangle \langle \phi |} of rank one with outer product

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\psi \rangle \langle \phi |\colon |\xi \rangle \mapsto |\psi \rangle \langle \phi |\xi \rangle ~.}

Inner product and bra–ket identification on Hilbert space

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). The inner product on Hilbert space Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (\ ,\ )} (with the first argument anti linear as preferred by physicists) is fully equivalent to an (anti-linear) identification between the space of kets and that of bras in the bra–ket notation: for a vector ket Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \psi =|\psi \rangle } define a functional (i.e. bra) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{\phi }=\langle \phi |} by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (\phi ,\psi )=(|\phi \rangle ,|\psi \rangle )=:f_{\phi }(\psi )=\langle \phi |\,{\bigl (}|\psi \rangle {\bigr )}=:\langle \phi {\mid }\psi \rangle }

Bras and kets as row and column vectors

In the simple case where we consider the vector space Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {C} ^{n}} , a ket can be identified with a column vector, and a bra as a row vector. If, moreover, we use the standard Hermitian inner product on $\mathbb {C} ^{n}$ , the bra corresponding to a ket, in particular a bra $⟨ m |$ and a ket $| m ⟩$ with the same label are conjugate transpose. Moreover, conventions are set up in such a way that writing bras, kets, and linear operators next to each other simply imply matrix multiplication.^[6] In particular the outer product Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\psi \rangle \langle \phi |} of a column and a row vector ket and bra can be identified with matrix multiplication (column vector times row vector equals matrix).

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: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle A|B\rangle \doteq A_{1}^{*}B_{1}+A_{2}^{*}B_{2}+\cdots +A_{N}^{*}B_{N}={\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}{\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}} Based on this, the bras and kets can be defined as: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\langle A|&\doteq {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\\|B\rangle &\doteq {\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}\end{aligned}}} 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: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle A|^{\dagger }=|A\rangle ,\quad |A\rangle ^{\dagger }=\langle A|} because if one starts with the bra Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\,,} then performs a complex conjugation, and then a matrix transpose, one ends up with the ket Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{pmatrix}A_{1}\\A_{2}\\\vdots \\A_{N}\end{pmatrix}}}

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 " $| m ⟩$ " 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 " $| + ⟩$ ".

Non-normalizable states and non-Hilbert spaces

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 more general context.

Banach spaces are a different generalization of Hilbert spaces. In a Banach space $Template:Mathcal$ , the vectors may be notated by kets and the continuous linear functionals by bras. Over any vector space without a given topology, we may still 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.

Usage in quantum mechanics

The mathematical structure of quantum mechanics is based in large part on linear algebra:

Wave functions and other quantum states can be represented as vectors in a separable complex Hilbert space. (The exact structure of this Hilbert space depends on the situation.) In bra–ket notation, for example, an electron might be in the "state" $| ψ ⟩$ . (Technically, the quantum states are rays of vectors in the Hilbert space, as $c | ψ ⟩$ corresponds to the same state for any nonzero complex number $c$ .)
Quantum superpositions can be described as vector sums of the constituent states. For example, an electron in the state $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/√2|1⟩ + i/√2|2⟩$ is in a quantum superposition of the states $|1⟩$ and $|2⟩$ .
Measurements are associated with linear operators (called observables) on the Hilbert space of quantum states.
Dynamics are also described by linear operators on the Hilbert space. For example, in the Schrödinger picture, there is a linear time evolution operator $U$ with the property that if an electron is in state $| ψ ⟩$ right now, at a later time it will be in the state $U | ψ ⟩$ , the same $U$ for every possible $| ψ ⟩$ .
Wave function normalization is scaling a wave function so that its norm is 1.

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:

Spinless position–space wave function

Error creating thumbnail:

Discrete components

A k

of a complex vector

| A ⟩ = Σ k A k | e k ⟩

.

Error creating thumbnail:

Continuous components

ψ (x)

of a complex vector

| ψ ⟩ = \int d x ψ (x) | x ⟩

.

Components of complex vectors plotted against index number; discrete

k

and continuous

x

. Two particular components out of infinitely many are highlighted.

The Hilbert space of a spin-0 point particle can be represented in terms of a "position basis" ${| r ⟩}$ , where the label $r$ extends over the set of all points in position space. These states satisfy the eigenvalue equation for the position operator: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {\mathbf {r} }}|\mathbf {r} \rangle =\mathbf {r} |\mathbf {r} \rangle .} The position states are "generalized eigenvectors", not elements of the Hilbert space itself, and do not form a countable orthonormal basis. However, as the Hilbert space is separable, it does admit a countable dense subset within the domain of definition of its wavefunctions. That is, starting from any ket $|Ψ⟩$ in this Hilbert space, one may define a complex scalar function of $r$ , known as a wavefunction, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |\Psi \rangle \,.}

On the left-hand side, $Ψ(r)$ is a function mapping any point in space to a complex number; on the right-hand side, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left|\Psi \right\rangle =\int d^{3}\mathbf {r} \,\Psi (\mathbf {r} )\left|\mathbf {r} \right\rangle } 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {A}}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {A}}|\Psi \rangle \,.}

For instance, the momentum operator Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {\mathbf {p} }}} has the following coordinate representation, ${\hat {\mathbf {p} }}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {\mathbf {p} }}|\Psi \rangle =-i\hbar \nabla \Psi (\mathbf {r} )\,.$

One occasionally even encounters an expression such as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \nabla |\Psi \rangle } , 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, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \nabla \langle \mathbf {r} |\Psi \rangle \,,} even though, in the momentum basis, this operator amounts to a mere multiplication operator (by $iħ p$ ). That is, to say, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle \mathbf {r} |{\hat {\mathbf {p} }}=-i\hbar \nabla \langle \mathbf {r} |~,} or Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {\mathbf {p} }}=\int d^{3}\mathbf {r} ~|\mathbf {r} \rangle (-i\hbar \nabla )\langle \mathbf {r} |~.}

Overlap of states

In quantum mechanics the expression $⟨ φ | ψ ⟩$ is typically interpreted as the probability amplitude for the state $ψ$ to collapse into the state $φ$ . Mathematically, this means the coefficient for the projection of $ψ$ onto $φ$ . It is also described as the projection of state $ψ$ onto state $φ$ .

Changing basis for a spin-1/2 particle

A stationary [[spin-1/2|spin-Template:1/2]] particle has a two-dimensional Hilbert space. One orthonormal basis is: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |{\uparrow }_{z}\rangle \,,\;|{\downarrow }_{z}\rangle } where $|↑ z ⟩$ is the state with a definite value of the spin operator $S z$ equal to +Template:1/2 and $|↓ z ⟩$ is the state with a definite value of the spin operator $S z$ equal to −Template:1/2.

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: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\psi \rangle =a_{\psi }|{\uparrow }_{z}\rangle +b_{\psi }|{\downarrow }_{z}\rangle } where $a ψ$ and $b ψ$ are complex numbers.

A different basis for the same Hilbert space is: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |{\uparrow }_{x}\rangle \,,\;|{\downarrow }_{x}\rangle } defined in terms of $S x$ rather than $S z$ .

Again, any state of the particle can be expressed as a linear combination of these two: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\psi \rangle =c_{\psi }|{\uparrow }_{x}\rangle +d_{\psi }|{\downarrow }_{x}\rangle }

In vector form, you might write Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\psi \rangle \doteq {\begin{pmatrix}a_{\psi }\\b_{\psi }\end{pmatrix}}\quad {\text{or}}\quad |\psi \rangle \doteq {\begin{pmatrix}c_{\psi }\\d_{\psi }\end{pmatrix}}} depending 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{\psi }} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b_{\psi }} , $c_{\psi }$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d_{\psi }} ; see change of basis.

Pitfalls and ambiguous uses

There are some conventions and uses of notation that may be confusing or ambiguous for the non-initiated or early student.

Separation of inner product and vectors

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {\psi }}} , and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (\cdot ,\cdot )} for the inner product. Consider the following dual space bra-vector in the basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{|e_n\rangle\}} , where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{\psi _{n}\}} are the complex number coefficients of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \psi | } : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi| = \sum_n \langle e_n| \psi_n}

It has to be determined by convention if the complex numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\psi_n\}} are inside or outside of the inner product, and each convention gives different results.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n \psi_n, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n^*}

Reuse of symbols

It is common to use the same symbol for labels and constants. For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat \alpha |\alpha\rangle = \alpha |\alpha \rangle} , where the symbol Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is used simultaneously as the name of the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat \alpha} , its eigenvector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle} and the associated eigenvalue Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} . Sometimes the hat is also dropped for operators, and one can see notation such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A |a\rangle = a |a \rangle} .^[7]

Hermitian conjugate of kets

It is common to see the usage Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle^\dagger = \langle\psi|} , where the dagger (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dagger} ) corresponds to the Hermitian conjugate. This is however not correct in a technical sense, since the ket, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} , represents a vector in a complex Hilbert-space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , and the bra, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi|} , is a linear functional on vectors in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} . In other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} is just a vector, while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi|} is the combination of a vector and an inner product.

Operations inside bras and kets

This is done for a fast notation of scaling vectors. For instance, if the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha \rangle} is scaled by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/\sqrt{2}} , it may be denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha/\sqrt{2} \rangle} . This can be ambiguous since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is simply a label for a state, and not a mathematical object on which operations can be performed. This usage is more common when denoting vectors as tensor products, where part of the labels are moved outside the designed slot, e.g. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha \rangle = |\alpha/\sqrt{2} \rangle_1 \otimes |\alpha/\sqrt{2} \rangle_2} .

Linear operators

Linear operators acting on kets

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat A} is a linear operator and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} is a ket-vector, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat A |\psi\rangle} is another ket-vector.

In an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} -dimensional Hilbert space, we can impose a basis on the space and represent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} in terms of its coordinates as a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \times 1} column vector. Using the same basis for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat A} , it is represented by an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \times N} complex matrix. The ket-vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat A |\psi\rangle} can now be computed by matrix multiplication.

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.

Linear operators acting on bras

Operators can also be viewed as acting on bras from the right hand side. Specifically, if $A$ is a linear operator and $⟨ φ |$ is a bra, then $⟨ φ | A$ is another bra defined by the rule Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl(\langle\phi|\boldsymbol{A}\bigr) |\psi\rangle = \langle\phi| \bigl(\boldsymbol{A}|\psi\rangle\bigr) \,,} (in other words, a function composition). This expression is commonly written as (cf. energy inner product) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\phi| \boldsymbol{A} |\psi\rangle \,.}

In an $N$ -dimensional Hilbert space, $⟨ φ |$ can be written as a $1 \times N$ row vector, and $A$ (as in the previous section) is an $N \times N$ matrix. Then the bra $⟨ φ | A$ can be computed by normal matrix multiplication.

If the same state vector appears on both bra and ket side, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi|\boldsymbol{A}|\psi\rangle \,,} 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 $| ψ ⟩$ .

Outer products

A convenient way to define linear operators on a Hilbert space $Template:Mathcal$ is given by the outer product: if $⟨ ϕ |$ is a bra and $| ψ ⟩$ is a ket, the outer product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rang \, \lang \psi| } denotes the rank-one operator with the rule Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl(|\phi\rang \lang \psi|\bigr)(x) = \lang \psi | x \rang |\phi \rang.}

For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi \rangle \, \langle \psi | \doteq \begin{pmatrix} \phi_1 \\ \phi_2 \\ \vdots \\ \phi_N \end{pmatrix} \begin{pmatrix} \psi_1^* & \psi_2^* & \cdots & \psi_N^* \end{pmatrix} = \begin{pmatrix} \phi_1 \psi_1^* & \phi_1 \psi_2^* & \cdots & \phi_1 \psi_N^* \\ \phi_2 \psi_1^* & \phi_2 \psi_2^* & \cdots & \phi_2 \psi_N^* \\ \vdots & \vdots & \ddots & \vdots \\ \phi_N \psi_1^* & \phi_N \psi_2^* & \cdots & \phi_N \psi_N^* \end{pmatrix} } The outer product is an $N \times N$ matrix, as expected for a linear operator.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle \, \langle\psi| \,.} This is an idempotent in the algebra of observables that acts on the Hilbert space.

Hermitian conjugate operator

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 $A$ . In other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle = A |\psi\rangle \quad \text{if and only if} \quad \langle\phi| = \langle \psi | A^\dagger \,.}

If $A$ is expressed as an $N \times N$ matrix, then $A †$ is its conjugate transpose.

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, $c 1$ and $c 2$ 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\phi| \bigl( c_1|\psi_1\rangle + c_2|\psi_2\rangle \bigr) = 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,^[8] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl(c_1 \langle\phi_1| + c_2 \langle\phi_2|\bigr) |\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \lang \psi| \bigl(A |\phi\rang\bigr) = \bigl(\lang \psi|A\bigr)|\phi\rang \, &\stackrel{\text{def}}{=} \, \lang \psi | A | \phi \rang \\ \bigl(A|\psi\rang\bigr)\lang \phi| = A\bigl(|\psi\rang \lang \phi|\bigr) \, &\stackrel{\text{def}}{=} \, A | \psi \rang \lang \phi | \end{align}}

and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguously because of the equalities on the left. Note that the associative property does not hold for expressions that include nonlinear 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., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(x^\dagger\right)^\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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl(c_1|\psi_1\rangle + c_2|\psi_2\rangle\bigr)^\dagger = c_1^* \langle\psi_1| + c_2^* \langle\psi_2| \,. }
Inner products: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \phi | \psi \rangle^* = \langle \psi|\phi\rangle \,.} Note that $⟨ φ | ψ ⟩$ is a scalar, so the Hermitian conjugate is just the complex conjugate, i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl(\langle \phi | \psi \rangle\bigr)^\dagger = \langle \phi | \psi \rangle^*}
Matrix elements: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}
Outer products: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big(\bigl(c_1|\phi_1\rangle\langle \psi_1|\bigr) + \bigl(c_2|\phi_2\rangle\langle\psi_2|\bigr)\Big)^\dagger = \bigl(c_1^* |\psi_1\rangle\langle \phi_1|\bigr) + \bigl(c_2^*|\psi_2\rangle\langle\phi_2|\bigr) \,.}

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.)^{[citation needed]}

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 $V \otimes W$ . This is written in various notations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle|\phi\rangle \,,\quad |\psi\rangle \otimes |\phi\rangle\,,\quad|\psi \phi\rangle\,,\quad|\psi ,\phi\rangle\,.}

See quantum entanglement and the EPR paradox for applications of this product.

The unit operator

Consider a complete orthonormal system (basis), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ e_i \ | \ i \in \mathbb{N} \} \,,} for a Hilbert space $H$ , with respect to the norm from an inner product $⟨\cdot,\cdot⟩$ .

From basic functional analysis, it is known that any ket Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle } can also be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle = \sum_{i \in \mathbb{N}} \langle e_i | \psi \rangle | e_i \rangle,} with $⟨\cdot|\cdot⟩$ the inner product on the Hilbert space.

From the commutativity of kets with (complex) scalars, it follows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | = \mathbb{I}} must be the identity operator, which sends each vector to itself.

This, then, can be inserted in any expression without affecting its value; for example Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \langle v | w \rangle &= \langle v | \left( \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i| \right) | w \rangle \\ &= \langle v | \left( \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i| \right) \left( \sum_{j \in \mathbb{N}} | e_j \rangle \langle e_j |\right)| w \rangle \\ &= \langle v | e_i \rangle \langle e_i | e_j \rangle \langle e_j | w \rangle \,, \end{align}} where, 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 $⟨ ψ | φ ⟩$ of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients $⟨ ψ | e i ⟩ = ⟨ e i | ψ ⟩ *$ and $⟨ e i | φ ⟩$ 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,^[9] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb I} = \int\! dx~ | x \rangle \langle x |= \int\! dp ~| p \rangle \langle p |,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |p\rangle = \int dx \frac{e^{ixp / \hbar} |x\rangle}{\sqrt{2\pi\hbar}}.}

Since $⟨ x Template:Prime | x ⟩ = δ (x - x Template:Prime)$ , plane waves follow, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x | p \rangle = \frac{e^{ixp / \hbar}}{\sqrt{2\pi\hbar}}.}

In his book (1958), Ch. III.20, Dirac defines the standard ket which, up to a normalization, is the translationally invariant momentum eigenstate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |\varpi\rangle=\lim_{p\to 0} |p\rangle} in the momentum representation, i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{p}|\varpi\rangle=0} . Consequently, the corresponding wavefunction is a constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x|\varpi\rangle \sqrt{2\pi \hbar}=1} , and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |x\rangle =\delta ({\hat {x}}-x)|\varpi \rangle {\sqrt {2\pi \hbar }},} as well as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |p\rangle =\exp(ip{\hat {x}}/\hbar )|\varpi \rangle .}

Typically, when all matrix elements of an operator such as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle x|A|y\rangle } are available, this resolution serves to reconstitute the full operator, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int dx\,dy\,|x\rangle \langle x|A|y\rangle \langle y|=A\,.}

Notation used by mathematicians

The object physicists are considering when using bra–ket notation is a Hilbert space (a complete inner product space).

Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle )} be a Hilbert space and $h \in Template:Mathcal$ a vector in $Template:Mathcal$ . What physicists would denote by $| h ⟩$ is the vector itself. That is, $|h\rangle \in {\mathcal {H}}.$

Let $Template:Mathcal *$ be the dual space of $Template:Mathcal$ . This is the space of linear functionals on $Template:Mathcal$ . The embedding Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Phi :{\mathcal {H}}\hookrightarrow {\mathcal {H}}^{*}} is defined by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Phi (h)=\varphi _{h}} , where for every $h \in Template:Mathcal$ the linear functional Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varphi _{h}:{\mathcal {H}}\to \mathbb {C} } satisfies for every $g \in Template:Mathcal$ the functional equation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varphi _{h}(g)=\langle h,g\rangle =\langle h\mid g\rangle } . Notational confusion arises when identifying $φ h$ and $g$ with $⟨ h |$ and $| g ⟩$ respectively. This is because of literal symbolic substitutions. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varphi _{h}=H=\langle h\mid } and let $g = G = | g ⟩$ . This gives Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varphi _{h}(g)=H(g)=H(G)=\langle h|(G)=\langle h|{\bigl (}|g\rangle {\bigr )}\,.}

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 write Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle \phi ,\psi \rangle =\int \phi (x){\overline {\psi (x)}}\,dx\,,} whereas physicists would write for the same quantity Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle \psi |\phi \rangle =\int dx\,\psi ^{*}(x)\phi (x)~.}

Notes

↑ ^1.0 ^1.1 Dirac 1939
↑ Shankar 1994, Chapter 1
↑ Grassmann 1862
↑ Lecture 2 | Quantum Entanglements, Part 1 (Stanford), Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.
↑ Lecture 2 | Quantum Entanglements, Part 1 (Stanford), Leonard Susskind on inner product, 2006-10-02.
↑ "Gidney, Craig (2017). Bra–Ket Notation Trivializes Matrix Multiplication".
↑ Sakurai & Napolitano 2021 Sec 1.2, 1.3
↑ Lecture notes by Robert Littlejohn Archived 2012-06-17 at the Wayback Machine, eqns 12 and 13
↑ Sakurai & Napolitano 2021 Sec 1.2, 1.3

References

Dirac, P. A. M. (1939). "A new notation for quantum mechanics". Mathematical Proceedings of the Cambridge Philosophical Society. 35 (3): 416–418. Bibcode:1939PCPS...35..416D. doi:10.1017/S0305004100021162. S2CID 121466183. Also see his standard text, The Principles of Quantum Mechanics, IV edition, Clarendon Press (1958), ISBN 978-0198520115
Grassmann, H. (1862). Extension Theory. History of Mathematics Sources. 2000 translation by Lloyd C. Kannenberg. American Mathematical Society, London Mathematical Society.
Cajori, Florian (1929). A History Of Mathematical Notations Volume II. Open Court Publishing. p. 134. ISBN 978-0-486-67766-8.
Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). ISBN 0-306-44790-8.
Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1965). The Feynman Lectures on Physics. III. Reading, MA: Addison-Wesley. ISBN 0-201-02118-8.
Sakurai, J J; Napolitano, J (2021). Modern Quantum Mechanics (3rd ed.). Cambridge University Press. ISBN 978-1-108-42241-3.

External links

Richard Fitzpatrick, "Quantum Mechanics: A graduate level course", The University of Texas at Austin. Includes:
- 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. University of California, Berkeley.
Gieres, F. (2000). "Mathematical surprises and Dirac's formalism in quantum mechanics". Rep. Prog. Phys. 63 (12): 1893–1931. arXiv:quant-ph/9907069. Bibcode:2000RPPh...63.1893G. doi:10.1088/0034-4885/63/12/201. S2CID 10854218.

Template:Quantum mechanics topics

[Dirac-1] 1.0 ^1.1 Dirac 1939

[2] Shankar 1994, Chapter 1

[Grassmann-3] Grassmann 1862

[4] Lecture 2 | Quantum Entanglements, Part 1 (Stanford), Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.

[5] Lecture 2 | Quantum Entanglements, Part 1 (Stanford), Leonard Susskind on inner product, 2006-10-02.

[bra–ket_Notation_Trivializes_Matrix_Multiplication-6] "Gidney, Craig (2017). Bra–Ket Notation Trivializes Matrix Multiplication".

[7] Sakurai & Napolitano 2021 Sec 1.2, 1.3

[8] Lecture notes by Robert Littlejohn Archived 2012-06-17 at the Wayback Machine, eqns 12 and 13

[9] Sakurai & Napolitano 2021 Sec 1.2, 1.3

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Bra–ket notation

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Quantum mechanics