Template:Braket

"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:

Brackets: choose one of:
- bra (for a bra vector),
- ket (for a ket vector),
- bra-ket (for the inner product), or
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
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}†.

Using {math}, a bra can be written: $⟨ ψ| = |ψ ⟩ †$ , that is {math|{braket|bra|ψ} {=} {braket|ket|ψ}†}.

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}†.

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}†}.

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}]†.

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}]†.

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''†{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''†{braket|ket|ψ}.

Schrödinger equation

In wiki-markup rather than LaTeX:

iħ d / dt |Ψ(t) ⟩ = Ĥ |Ψ(t) ⟩ \leftrightarrow - 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}&otimes;{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 ⟩ \equiv |ξ ⟩ \otimes |ψ ⟩ \equiv |ξ, ψ ⟩

,

that is

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

参见

上述文档嵌入自Template:Braket/doc。
编者可以在本模板的沙盒和测试样例页面进行实验。
请在/doc子页面中添加分类。本模板的子页面。