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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bra–ket notation | 3/8 | https://en.wikipedia.org/wiki/Bra–ket_notation | reference | science, encyclopedia | 2026-05-05T14:40:03.882193+00:00 | kb-cron |
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{\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:
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{\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:
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{\displaystyle \langle A|^{\dagger }=|A\rangle ,\quad |A\rangle ^{\dagger }=\langle A|}
because if one starts with the bra
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{\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
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{\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). 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 B, 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 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 ===
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:
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{\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,
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{\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,
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{\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
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{\displaystyle {\hat {A}}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {A}}|\Psi \rangle \,.}