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{{缺乏中文说明}}
This is the left-handed angular bracket used for writing [[average]]s or [[bra–ket notation]], with other applications primarily in [[mathematics]] and [[physics]], for use when inline html rendering is desired rather than {{TᴇX}} rendering.

This is used in the {{tl|braket}} template. When creating bra or ket vectors, or inner products, use {{tl|Braket}} to save the trouble of typing &#124; (for the pipe symbol), {{tl|langle}}, or {{tl|rangle}} every time.

==Examples==

;Bras

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.

<pre>
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.
</pre>

;Tables (also hidden boxes)

Due to the vertical bar | used in template coding, the html code '''&#124;''' 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:

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

and the wrong way:

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

<pre>
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|
|}
</pre>

;In conjunction with {{tl|rangle}}:

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.

<pre>
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.
</pre>

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.

<pre>
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.
</pre>

== 参见 ==
{{math templates|notation}}
<includeonly>
[[Category:数学格式模板]]</includeonly>

Template:Langle/doc - 版本历史

imported>SolidBlock 来自 2024年10月7日 (一) 12:25