Template:Langle/doc

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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 $T e X$ 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.

Examples

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&#124; + {{langle}}Ψ&#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.

参见