en>ChrisGualtieri: General Fixes using AWB

2013-10-23T00:42:48Z

General Fixes using AWB

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:''"Template:Dirac notation" redirects here.''

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

==Application==

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

<ol>
<li>'''''Brackets:''''' choose one of:
*'''bra''' (for a bra vector),
*'''ket''' (for a ket vector),
*'''bra-ket''' (for the [[Bra–ket notation#Inner products and bras|inner product]]), or
<li>'''''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</li>
<li>'''''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</li>
</ol>

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: {{braket|ket|ψ}}, that is <code><nowiki>{{braket|ket|ψ}}</nowiki></code>.

Using {{tl|math}}, a ket can be written: {{math|{{braket|ket|ψ}}}}, that is <code><nowiki>{{math|{{braket|ket|ψ}}}}</nowiki></code>.

;Bra

A bra can be written: {{braket|bra|ψ}} = {{braket|ket|ψ}}†, that is <code><nowiki>{{braket|bra|ψ}} = {{braket|ket|ψ}}†</nowiki></code>.

Using {{tl|math}}, a bra can be written: {{math|{{braket|bra|ψ}} {{=}} {{braket|ket|ψ}}†}}, that is <code><nowiki>{{math|{{braket|bra|ψ}} {{=}} {{braket|ket|ψ}}†}}</nowiki></code>.

;Bra-ket

The [[Bra–ket notation#Inner products and bras|inner product]] of the kets {{braket|ket|ξ}} and {{braket|ket|ψ}} can be written: {{braket|bra-ket|ψ|ξ}} = {{braket|bra-ket|ξ|ψ}}†, that is <code><nowiki>{{braket|bra-ket|ψ|ξ}} = {{braket|bra-ket|ξ|ψ}}†</nowiki></code>.

Using {{tl|math}}, the inner product of the kets {{math|{{braket|ket|ξ}}}} and {{math|{{braket|ket|ψ}}}} can be written: {{math|{{braket|bra-ket|ψ|ξ}} {{=}} {{braket|bra-ket|ξ|ψ}}†}}, that is <code><nowiki>{{math|{{braket|bra-ket|ψ|ξ}} {{=}} {{braket|bra-ket|ξ|ψ}}†}}</nowiki></code>.

;Outer products

The [[Bra–ket notation#Outer products|outer product]] of the kets {{braket|ket|ξ}} and {{braket|ket|ψ}} can be written: {{braket|ket|ψ}}{{braket|bra|ξ}} = [{{braket|ket|ξ}}{{braket|bra|ψ}}]†, that is <code><nowiki>{{braket|ket|ψ}}{{braket|bra|ξ}} = [{{braket|ket|ξ}}{{braket|bra|ψ}}]†</nowiki></code>.

Using {{tl|math}}, the outer product of the kets {{math|{{braket|ket|ξ}}}} and {{math|{{braket|ket|ψ}}}} can be written: {{math|{{braket|ket|ψ}}{{braket|bra|ξ}} {{=}} [{{braket|ket|ξ}}{{braket|bra|ψ}}]†}}, that is <code><nowiki>{{braket|ket|ψ}}{{braket|bra|ξ}} {{=}} [{{braket|ket|ξ}}{{braket|bra|ψ}}]†</nowiki></code>.

;Inner products including [[operator (physics)|operator]]s

The inner product of the kets {{braket|ket|ξ}} and ''Ĥ''{{braket|ket|ψ}} is written using a bra and ket seperatley between the operator (there is no third parameter for the operator symbol):
:{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} = {{braket|bra|ξ}}''Ĥ''†{{braket|ket|ψ}},
that is
:<code><nowiki>{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} = {{braket|bra|ξ}}''Ĥ''†{{braket|ket|ψ}}</nowiki></code>.

Using {{tl|math}}, the inner product of the kets {{math|{{braket|ket|ξ}}}} and {{math|''Ĥ''{{braket|ket|ψ}}}} is written using a bra and ket seperatley between the operator:
:{{math|{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} {{=}} {{braket|bra|ξ}}''Ĥ''†{{braket|ket|ψ}}}},
that is
:<code><nowiki>{{math|{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} {{=}} {{braket|bra|ξ}}''Ĥ''†{{braket|ket|ψ}}}}</nowiki></code>.

;[[Schrödinger equation]]:
In wiki-markup rather than LaTeX:
:{{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''†}}
that is,
:<code><nowiki>{{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''†}}</nowiki></code>

;[[Bra–ket notation#Composite bras and kets|Tensor products]]:

The [[tensor product]] of the kets {{braket|ket|ξ}} and {{braket|ket|ψ}} is written using the ket mode only (there is no paramter for tensor products):
:{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}}&otimes;{{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}},
that is
:<code><nowiki>{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}}&otimes;{{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}}</nowiki></code>.

Using {{tl|math}}, the tensor product of the kets {{math|{{braket|ket|ξ}}}} and {{math|{{braket|ket|ψ}}}} is written using the ket mode only:
:{{math|{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}}&otimes;{{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}}}},
that is
:<code><nowiki>{{math|{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}}&otimes;{{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}}}}</nowiki></code>.

==See also==
*{{tl|rangle}}
*{{tl|langle}}
*{{tl|bra-ket}}

<includeonly>
{{#ifeq:{{PAGENAME}}|Braket|

[[Category:Mathematical formatting templates]]
[[Category:Physics templates]]
}}</includeonly>

Lyapunov optimization - Revision history

en>ChrisGualtieri: General Fixes using AWB