Template:Braket

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"Template:Dirac notation" redirects here.

This is for producing templates {{bra}}, {{ket}}, and {{bra-ket}}. It can also produce quantum state vectors in bra–ket notation, using wikicode, ideally with {{math}}, as an alternative to LaTeX in <math> mode, but using this template ( {{braket}} ) is more clumsy than the simpler and more directly applicable {{bra}}, {{ket}}, and {{bra-ket}}.

Application

There are three parameters, use as many as needed in this order:

Brackets: choose one of:
- bra (for a bra vector),
- ket (for a ket vector),
- bra-ket (for the inner product), or
Symbol 1:
- if 1 is set to bra or ket: enter the first symbol for the bra or ket,
- if 1 is set to bra-ket: enter the symbol for the bra part of the inner product
Symbol 2:
- if 1 is set to bra or ket: this parameter is not needed.
- if 1 is set to bra-ket: enter the symbol for the ket part of the inner product

If 1 is set to bra-ket, the symbols are entered in the order they are read, left to right. The symbols can of course be bold, italic, underlined, any unicode symbol, etc.

Examples

Ket

A ket can be written: |ψ⟩, that is {{braket|ket|ψ}}.

Using {{math}}, a ket can be written: $|ψ⟩$ , that is {{math|{{braket|ket|ψ}}}}.

Bra

A bra can be written: ⟨ψ| = |ψ⟩^†, that is {{braket|bra|ψ}} = {{braket|ket|ψ}}†.

Using {{math}}, a bra can be written: $⟨ψ| = |ψ⟩ †$ , that is {{math|{{braket|bra|ψ}} {{=}} {{braket|ket|ψ}}†}}.

Bra-ket

The inner product of the kets |ξ⟩ and |ψ⟩ can be written: ⟨ψ|ξ⟩ = ⟨ξ|ψ⟩^†, that is {{braket|bra-ket|ψ|ξ}} = {{braket|bra-ket|ξ|ψ}}†.

Using {{math}}, the inner product of the kets $|ξ⟩$ and $|ψ⟩$ can be written: $⟨ψ|ξ⟩ = ⟨ξ|ψ⟩ †$ , that is {{math|{{braket|bra-ket|ψ|ξ}} {{=}} {{braket|bra-ket|ξ|ψ}}†}}.

Outer products

The outer product of the kets |ξ⟩ and |ψ⟩ can be written: |ψ⟩⟨ξ| = [|ξ⟩⟨ψ|]^†, that is {{braket|ket|ψ}}{{braket|bra|ξ}} = [{{braket|ket|ξ}}{{braket|bra|ψ}}]†.

Using {{math}}, the outer product of the kets $|ξ⟩$ and $|ψ⟩$ can be written: $|ψ⟩ ⟨ξ| = [|ξ⟩ ⟨ψ|] †$ , that is {{braket|ket|ψ}}{{braket|bra|ξ}} {{=}} [{{braket|ket|ξ}}{{braket|bra|ψ}}]†.

Inner products including operators

The inner product of the kets |ξ⟩ and Ĥ|ψ⟩ is written using a bra and ket separately between the operator (there is no third parameter for the operator symbol):

⟨ψ|Ĥ|ξ⟩ = ⟨ξ|Ĥ^†|ψ⟩,

that is

{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} = {{braket|bra|ξ}}''Ĥ''†{{braket|ket|ψ}}.

Using {{math}}, the inner product of the kets $|ξ⟩$ and $Ĥ |ψ⟩$ is written using a bra and ket separately between the operator:

⟨ψ| Ĥ |ξ⟩ = ⟨ξ| Ĥ † |ψ⟩

,

that is

{{math|{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} {{=}} {{braket|bra|ξ}}''Ĥ''†{{braket|ket|ψ}}}}.

Schrödinger equation

In wiki-markup rather than LaTeX:

iħ.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{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠d/dt⁠|Ψ(t)⟩ = Ĥ|Ψ(t)⟩ ↔ −iħ⟨Ψ(t)|⁠d/dt⁠ = ⟨Ψ(t)|Ĥ†

that is,

{{math|''iħ''{{sfrac|''d''|''dt''}}{{braket|ket|Ψ(''t'')}} {{=}} ''Ĥ''{{braket|ket|Ψ(''t'')}} ↔ −''iħ''{{braket|bra|Ψ(''t'')}}{{sfrac|''d''|''dt''}} {{=}} {{braket|bra|Ψ(''t'')}}''Ĥ''<sup>†</sup>}}

Tensor products

The tensor product of the kets |ξ⟩ and |ψ⟩ is written using the ket mode only (there is no parameter for tensor products):

|ξ⟩|ψ⟩ ≡ |ξ⟩ ⊗ |ψ⟩ ≡ |ξ, ψ⟩,

that is

{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}} &otimes; {{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}}.

Using {{math}}, the tensor product of the kets $|ξ⟩$ and $|ψ⟩$ is written using the ket mode only:

|ξ⟩ |ψ⟩ \equiv |ξ⟩ \otimes |ψ⟩ \equiv |ξ, ψ⟩

,

that is

{{math|{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}} &otimes; {{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}}}}.

Application

Examples

See also

From Wikipedia, the free encyclopedia · View on Wikipedia