Difference between revisions of "Template:Braket/doc"

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MT
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{{Documentation subpage}}
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<!-- Categories and interwikis go at the bottom of this page. -->
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:''"Template:Dirac notation" redirects here.''
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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>.
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==Application==
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There are three parameters, use as many as needed in this order:
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<!---THESE ARE NOT SPELLING MISTAKES - THEY ARE ACTUALLY USED IN THE TEMPLATE PARAMETERS - PLEASE DON'T "CORRECT" THE SPELLING (SAY) "BRAKET" TO "BRACKET" - THANKS!!!--->
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<ol>
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<li>''Brackets:'' choose one of:
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*''bra'' (for a bra vector),
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*''ket'' (for a ket vector),
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*''bra-ket'' (for the [[Bra–ket notation#Inner products and bras|inner product]]), or
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</li>
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<li>''Symbol 1:''
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*if 1 is set to ''bra'' or ''ket'': enter the first symbol for the bra or ket,
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*if 1 is set to ''bra-ket'': enter the symbol for the ''bra'' part of the inner product
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</li>
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<li>''Symbol 2:''
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*if 1 is set to ''bra'' or ''ket'': this parameter is not needed.
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*if 1 is set to ''bra-ket'': enter the symbol for the ''ket'' part of the inner product
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</li>
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</ol>
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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'', <u>underlined<u>, any [[unicode]] symbol, etc.
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==Examples==
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;Ket
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A ket can be written: {{braket|ket|ψ}}, that is <code><nowiki>{{braket|ket|ψ}}</nowiki></code>.
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Using {{tl|math}}, a ket can be written: {{math|{{braket|ket|ψ}}}}, that is <code><nowiki>{{math|{{braket|ket|ψ}}}}</nowiki></code>.
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;Bra
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A bra can be written: {{braket|bra|ψ}} = {{braket|ket|ψ}}<sup>†</sup>, that is <code><nowiki>{{braket|bra|ψ}} = {{braket|ket|ψ}}<sup>†</sup></nowiki></code>.
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Using {{tl|math}}, a bra can be written: {{math|{{braket|bra|ψ}} {{=}} {{braket|ket|ψ}}<sup>†</sup>}}, that is <code><nowiki>{{math|{{braket|bra|ψ}} {{=}} {{braket|ket|ψ}}<sup>†</sup>}}</nowiki></code>.
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;Bra-ket
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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}}<sup>†</sup>, that is <code><nowiki>{{braket|bra-ket|ψ|ξ}} = {{braket|bra-ket|ξ|ψ}}<sup>†</sup></nowiki></code>.
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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}}<sup>†</sup>}}, that is <code><nowiki>{{math|{{braket|bra-ket|ψ|ξ}} {{=}} {{braket|bra-ket|ξ|ψ}}<sup>†</sup>}}</nowiki></code>.
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;Outer products
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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}}]<sup>†</sup>, that is <code><nowiki>{{braket|ket|ψ}}{{braket|bra|ξ}} = [{{braket|ket|ξ}}{{braket|bra|ψ}}]<sup>†</sup></nowiki></code>.
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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}}]<sup>†</sup>}}, that is <code><nowiki>{{braket|ket|ψ}}{{braket|bra|ξ}} {{=}} [{{braket|ket|ξ}}{{braket|bra|ψ}}]<sup>†</sup></nowiki></code>.
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;Inner products including [[operator (physics)|operator]]s
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The inner product of the kets {{braket|ket|ξ}} and ''Ĥ''{{braket|ket|ψ}} is written using a bra and ket separately between the operator (there is no third parameter for the operator symbol):
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:{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} = {{braket|bra|ξ}}''Ĥ''<sup>†</sup>{{braket|ket|ψ}},
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that is
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:<code><nowiki>{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} = {{braket|bra|ξ}}''Ĥ''<sup>†</sup>{{braket|ket|ψ}}</nowiki></code>.
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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 separately between the operator:
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:{{math|{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} {{=}} {{braket|bra|ξ}}''Ĥ''<sup>†</sup>{{braket|ket|ψ}}}},
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that is
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:<code><nowiki>{{math|{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} {{=}} {{braket|bra|ξ}}''Ĥ''<sup>†</sup>{{braket|ket|ψ}}}}</nowiki></code>.
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;[[Schrödinger equation]]:
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In wiki-markup rather than LaTeX:
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:{{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>}}
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that is,
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:<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''<sup>†</sup>}}</nowiki></code>
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;[[Bra–ket notation#Composite bras and kets|Tensor products]]:
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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):
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:{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}}&otimes;{{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}},
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that is
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:<code><nowiki>{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}}&amp;otimes;{{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}}</nowiki></code>.
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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:
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:{{math|{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}}&otimes;{{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}}}},
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that is
 +
:<code><nowiki>{{math|{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}}&amp;otimes;{{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}}}}</nowiki></code>.
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==See also==
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* {{tl|Angle bracket}}: surrounding angle brackets
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* {{tl|Langle}}, {{tl|Rangle}}: single angle brackets
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{{Math templates|notation}}
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<includeonly>
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{{#ifeq:{{PAGENAME}}|Braket|
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<!-- Categories go here -->
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[[Category:Mathematical formatting templates]]
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[[Category:Physics templates]]
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}}</includeonly>

Revision as of 22:05, 13 August 2016

"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}}.

Application

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

  1. Brackets: choose one of:
    • bra (for a bra vector),
    • ket (for a ket vector),
    • bra-ket (for the inner product), or
  2. 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
  3. 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.

Examples

Ket

A ket can be written: Template:Rangle, that is {{braket|ket|ψ}}.

Using {{math}}, a ket can be written: Template:Rangle, that is {{math|{{braket|ket|ψ}}}}.

Bra

A bra can be written: ⟨ψ| = Template:Rangle, that is {{braket|bra|ψ}} = {{braket|ket|ψ}}<sup>†</sup>.

Using {{math}}, a bra can be written: ⟨ψ| = Template:Rangle, that is {{math|{{braket|bra|ψ}} {{=}} {{braket|ket|ψ}}<sup>†</sup>}}.

Bra-ket

The inner product of the kets Template:Rangle and Template:Rangle can be written: ⟨ψ|ξTemplate:Rangle = ⟨ξ|ψTemplate:Rangle, that is {{braket|bra-ket|ψ|ξ}} = {{braket|bra-ket|ξ|ψ}}<sup>†</sup>.

Using {{math}}, the inner product of the kets Template:Rangle and Template:Rangle can be written: ⟨ψ|ξTemplate:Rangle = ⟨ξ|ψTemplate:Rangle, that is {{math|{{braket|bra-ket|ψ|ξ}} {{=}} {{braket|bra-ket|ξ|ψ}}<sup>†</sup>}}.

Outer products

The outer product of the kets Template:Rangle and Template:Rangle can be written: Template:Rangle⟨ξ| = [Template:Rangle⟨ψ|], that is {{braket|ket|ψ}}{{braket|bra|ξ}} = [{{braket|ket|ξ}}{{braket|bra|ψ}}]<sup>†</sup>.

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

Inner products including operators

The inner product of the kets Template:Rangle and ĤTemplate:Rangle is written using a bra and ket separately between the operator (there is no third parameter for the operator symbol):

⟨ψ|ĤTemplate:Rangle = ⟨ξ|ĤTemplate:Rangle,

that is

{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} = {{braket|bra|ξ}}''Ĥ''<sup>†</sup>{{braket|ket|ψ}}.

Using {{math}}, the inner product of the kets Template:Rangle and ĤTemplate:Rangle is written using a bra and ket separately between the operator:

⟨ψ|ĤTemplate:Rangle = ⟨ξ|ĤTemplate:Rangle,

that is

{{math|{{braket|bra|ψ}}''Ĥ''{{braket|ket|ξ}} {{=}} {{braket|bra|ξ}}''Ĥ''<sup>†</sup>{{braket|ket|ψ}}}}.
Schrödinger equation

In wiki-markup rather than LaTeX:

Template:Sfrac|Ψ(t)Template:Rangle = Ĥ|Ψ(t)Template:Rangle ↔ −⟨Ψ(t)|Template:Sfrac = ⟨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 Template:Rangle and Template:Rangle is written using the ket mode only (there is no paramter for tensor products):

Template:RangleTemplate:RangleTemplate:RangleTemplate:Rangle|ξ, ψTemplate:Rangle,

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 Template:Rangle and Template:Rangle is written using the ket mode only:

Template:RangleTemplate:RangleTemplate:RangleTemplate:Rangle|ξ, ψTemplate:Rangle,

that is

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

See also