PlanetPhysics/Dirac Notation

[{\mathbf This is a contributed topic entry in progress on Dirac notations and quantum observable algebras.}]


 * 1) {\mathbf Introduction}

(In progress.)


 * 1) {\mathbf non-Abelian (or non-commutative) observable (Clifford) Algebra}

(In progress.)


 * 1) {\mathbf Dirac notations:$$<{bra}|$$ c $$|{ket}>$$ and delta}

The 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 transpose vector-- defined as $$<{\phi}|$$ (or "bra" vector ). Thus, a ""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 $${\phi}$$.