delta and observable algebras Dirac notations,DiracNotation2006-05-30 11:18:212009-01-31 11:39:43TopicDirac notationDirac relativistic field equationsnotationand non-commutative observable algebra% this is the default PlanetPhysics preamble. as your knowledge
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% define commands here[{\bf This is a contributed topic entry in progress on Dirac notations and quantum observable algebras.}]
\subsection{Introduction}
(In progress.)
\subsection{Non-Abelian (or non-commutative) Observable (Clifford) Algebra}
(In progress.)
\subsection{Dirac notations:$<{bra}|$ c $|{ket}>$ and delta}
The {\em Dirac notation} (or ``bra-ket'' notation as commonly known in physics) is used to represent quantum states in quantum mechanics. It was invented by Physics Nobel Laureate Paul A. M. Dirac, and since then has been established as one of the preferred notations in quantum mechanics.
The Dirac notation denotes both the ``ket'' vector-- defined as $|{\psi}>$-- and its {\em transpose vector}-- defined as $<{\phi}|$ (or {\em ``bra'' vector}). Thus, a ``{\em bra-ket}'' is defined as the inner product of the two vectors defined above, which is denoted as $<{\phi}|{\psi}>$.
Then, the Dirac notation also satisifies the following identities:
$$<{\phi}|{\omega}|{\psi}>{\equiv}<{\phi}|{\omega}{\psi}>$$
$$<{\phi}|{\psi}>{\equiv}\int_{-{\infty}}^{\infty} \widetilde{\phi} {\psi} dx \quad,$$
where $\widetilde{\phi}$ is the ``complex conjugate'' of ${\phi}$.
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\bibitem{PAMD68}
Paul A.M. Dirac. 1968. {\em Principles of Quantum Mechanics},
Cambridge University Press, Cambridge, UK
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