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

$\langle \text{\varphi}|\text{\psi}\rangle ={\langle \text{\psi}|\text{\varphi}\rangle}^{*}$ |
(7.1) |

Also, the probability amplitude

$\langle \text{\varphi}|\text{\psi}\rangle =\langle \text{\varphi}|j\rangle \langle j|\text{\psi}\rangle $ |
(7.2) |

can be expressed in abstract form as

$|\text{\psi}\rangle =|j\rangle \langle j|\text{\psi}\rangle $ |
(7.3) |

An additional form of abstract notation is

$\langle \text{\chi}\left|A\right|\text{\varphi}\rangle =\langle \text{\chi}|i\rangle \langle i|A|j\rangle \langle j|\text{\varphi}\rangle $ |
(7.4) |

where *A* is

$A=|i\rangle \langle i|A|j\rangle \langle j|$ |
(7.5) |

Another abstraction is illustrated by

$A|\text{\varphi}\rangle =|i\rangle \langle i\left|A\right|j\rangle \langle j|\text{\varphi}\rangle $ |
(7.6) |

Further, *A* can be multiplied by *B* so that

$\langle \text{\chi}\left|BA\right|\text{\varphi}\rangle =\langle \text{\chi}|i\rangle \langle i$ |

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