Template:Braket/doc

"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: |ψ⟩, that is {{braket|ket|ψ}}.

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

Bra

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

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

Bra-ket

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

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

Outer products

The outer product of the kets |ξ⟩ and |ψ⟩ can be written: |ψ⟩⟨ξ| = [|ξ⟩⟨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 |ξ⟩ and |ψ⟩ can be written: |ψ⟩⟨ξ| = [|ξ⟩⟨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 |ξ⟩ and Ĥ|ψ⟩ is written using a bra and ket seperatley between the operator (there is no third parameter for the operator symbol):

⟨ψ|Ĥ|ξ⟩ = ⟨ξ|Ĥ^†|ψ⟩,

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 |ξ⟩ and Ĥ|ψ⟩ is written using a bra and ket seperatley between the operator:

⟨ψ|Ĥ|ξ⟩ = ⟨ξ|Ĥ^†|ψ⟩,

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:

iħd/dt|Ψ(t)⟩ = Ĥ|Ψ(t)⟩ ↔ −iħ⟨Ψ(t)|d/dt = ⟨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 |ξ⟩ and |ψ⟩ is written using the ket mode only (there is no paramter for tensor products):

|ξ⟩|ψ⟩ ≡ |ξ⟩⊗|ψ⟩ ≡ |ξ, ψ⟩,

that is

{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}}⊗{{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}}.

Using {{math}}, the tensor product of the kets |ξ⟩ and |ψ⟩ is written using the ket mode only:

|ξ⟩|ψ⟩ ≡ |ξ⟩⊗|ψ⟩ ≡ |ξ, ψ⟩,

that is

{{math|{{braket|ket|ξ}}{{braket|ket|ψ}} ≡ {{braket|ket|ξ}}⊗{{braket|ket|ψ}} ≡ {{braket|ket|ξ, ψ}}}}.

参见

• {{angle bracket}}
• {{rangle}}
• {{langle}}