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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bra–ket notation | 6/8 | https://en.wikipedia.org/wiki/Bra–ket_notation | reference | science, encyclopedia | 2026-05-05T14:40:03.882193+00:00 | kb-cron |
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,
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if and only if
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{\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 × 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, 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,
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{\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,
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{\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:
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def
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{\displaystyle {\begin{aligned}\langle \psi |{\bigl (}A|\phi \rangle {\bigr )}={\bigl (}\langle \psi |A{\bigr )}|\phi \rangle \,&{\stackrel {\text{def}}{=}}\,\langle \psi |A|\phi \rangle \\{\bigl (}A|\psi \rangle {\bigr )}\langle \phi |=A{\bigl (}|\psi \rangle \langle \phi |{\bigr )}\,&{\stackrel {\text{def}}{=}}\,A|\psi \rangle \langle \phi |\end{aligned}}}
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.,
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{\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:
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{\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:
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{\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.,
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{\displaystyle {\bigl (}\langle \phi |\psi \rangle {\bigr )}^{\dagger }=\langle \phi |\psi \rangle ^{*}}
Matrix elements:
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{\displaystyle {\begin{aligned}\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{aligned}}}
Outer products:
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{\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 ⊗ 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 |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 ⊗ W. This is written in various notations: