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Quantum Optics for Engineers by F.J. Duarte

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7

Dirac Notation Identities

7.1    Useful Identities

As described by Dirac (1978) and Feynman et al. (1965), the Dirac notation includes various mathematical properties and allows for various abstractions and permutations. Here, a few useful set of identities and properties of the notation are described.

First, the complex conjugate of 〈ϕ|ψ〉 is defined as

ϕ|ψ=ψ|ϕ*

(7.1)

Also, the probability amplitude

ϕ|ψ=ϕ|jj|ψ

(7.2)

can be expressed in abstract form as

|ψ=|jj|ψ

(7.3)

An additional form of abstract notation is

χ|A|ϕ=χ|ii|A|jj|ϕ

(7.4)

where A is

A=|ii|A|jj|

(7.5)

Another abstraction is illustrated by

A|ϕ=|ii|A|jj|ϕ

(7.6)

Further, A can be multiplied by B so that

χ|BA|ϕ=χ|ii

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