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Template:Langle - Wikipedia

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⟨

This is the left-handed angular bracket used for writing averages or bra–ket notation, with other applications primarily in mathematics and physics, for use when inline html rendering is desired rather than TeX rendering.

This is used in the {{braket}} template. When creating bra or ket vectors, or inner products, use {{Braket}} to save the trouble of typing | (for the pipe symbol), {{langle}}, or {{rangle}} every time.

Bras

The superposition of states can be written ⟨p| + ⟨q| + ⟨χ| + ⟨ψ|, which is inline with the text.

Another superposition of states: ⟨P| + ⟨Q| + ⟨Φ| + ⟨Ψ|, again inline.

The superposition of states can be written {{langle}}p| + {{langle}}q| + {{langle}}χ| + {{langle}}ψ|, which is inline with the text.

Another superposition of states: {{langle}}P| + {{langle}}Q| + {{langle}}Φ| + {{langle}}Ψ|, again inline.

Tables (also hidden boxes)

Due to the vertical bar | used in template coding, the html code | must be used when bra–ket notation is used in tables, else some parts will not show up because of code interference.

The correct way:

Left bracket alone Bra ⟨Φ + ⟨Ψ ⟨Φ| + ⟨Ψ|

and the wrong way:

Left bracket alone Bra ⟨Φ + ⟨Ψ + ⟨Ψ|

The correct way: 

{| class="wikitable"
! Left bracket alone 
! Bra 
|-
| {{langle}}Φ + {{langle}}Ψ
| {{langle}}Φ&amp;#124; + {{langle}}Ψ&amp;#124;
|}

and the wrong way:

{| class="wikitable"
! Left bracket alone 
! Bra 
|-
| {{langle}}Φ + {{langle}}Ψ
| {{langle}}Φ| + {{langle}}Ψ|
|}

In conjunction with {{rangle}}

One sum of inner products is ⟨p|q⟩ + ⟨χ|ψ⟩, a real number.

A sum of average values could be ⟨P|E|Q⟩ + ⟨Φ|p|Ψ⟩, another real number.

One sum of inner products is {{langle}}p|q{{rangle}} + {{langle}}χ|ψ{{rangle}}, a real number.

A sum of average values could be {{langle}}P|''E''|Q{{rangle}} + {{langle}}Φ|''p''|Ψ{{rangle}}, another real number.

The average of a quantity q may be written ⟨q⟩. The root mean square is then √⟨q²⟩, i.e. square every value, then average, then take the root.

The average of a quantity ''q'' may be written {{langle}}''q''{{rangle}}. The root mean square is 
then √{{langle}}''q''<sup>2</sup>{{rangle}}, i.e. square every value, then average, then take the root.

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.

Boxes