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Template:Braket - Wikipedia

From Wikipedia, the free encyclopedia

"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}}.

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.

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:

iħ⁠d/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⟩ ≡ |ξ⟩ ⊗ |ψ⟩ ≡ |ξ, ψ⟩,

that is

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

{{Angle bracket}}: surrounding angle brackets
{{Langle}}, {{Rangle}}: single angle brackets

Math templates

Functions
Numeral systems

Functions

Numeral systems

Conversions

convert many units (see: list) cvt abbreviated {{convert}} convinfobox {{convert}} for infoboxes bbl to t barrels of oil to tonnes long ton long hundredweights, quarters and pounds to kilograms;
long tons and hundredweights to pounds and metric tons miles-chains miles and chains to kilometres linking "chains" decdeg degrees, minutes, and seconds to decimal degrees deg2dms decimal degrees to degrees, minutes, and seconds deg2hms decimal degrees to hour angle (in hours, minutes, and seconds) hms2deg hour angle (in hours, minutes, and seconds) to decimal degrees inflation calculate inflation of Consumer Price Index-related prices pop density population density in an area track gauge railway track gauges

Notation and

formatting bigmath bigger font to match TeX \displaystyle (standalone formulas only)

bra–ket notation ceil, floor calculations :mw:Help:#expr; formatting indicators ⌈3.14⌉, ⌊3.14⌋ (no calculation performed) fraction slant fractions 3⁄5 (not for maths/science articles; use standing or upright fractions {{sfrac}} instead) intmath integral symbols

multiline delimiters (2–5 lines inclusive) abs absolute values (paired vertical lines) math short text-based formulas mathcal [mathematical] calligraphic font; alternative to LaTeX \mathcal{...} mvar individual italicized maths variables in normal text

overline

underline

a line set above/below a sequence of characters overarc an arc set above a sequence of characters

overset

underset

arbitrary characters/diacritics set above/below one another pars parentheses that can be resized (∑) sfrac "standing" or upright fractions ⁠3/5⁠ (use in maths/science articles instead of{{fraction}})

sub

sup

subscripts and superscripts tmath Wrap TeX in <math> tags tombstone symbol indicating the end of a proof val measurement values, uncertainties and units vec various overarrows, underarrows, etc.

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