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- W1898585595 abstract "page1 Abstract Dirac notation has been widely used for vectors in Hilbert spaces of Quantum Theories. In this paper, we propose to use the Probability Bracket Notation (PBN), a new set of symbols defined similarly (but not identically) as in Dirac notation. By applying PBN to fundamental definitions and theorems for discrete and continuous random variables, we show that PBN could play a similar role in probability sample space as Dirac notation in Hilbert vector space. Applying PBN to homogeneous Markov chains (MC) with discrete time, we show that our system state P-kets are identified with the probability vectors in Markov chains (MC). Then we apply PBN to general stochastic processes (SP). The master equation of time-continuous homogeneous MC in the Schrodinger pictures is discussed. Our system state P-bra is identified with Doi's state function and Peliti's standard bra. In the end, we investigated the transition of probability density from the Schrodinger picture to the Heisenberg picture for time-continuous homogeneous MC. We" @default.
- W1898585595 created "2016-06-24" @default.
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- W1898585595 date "2007-02-04" @default.
- W1898585595 modified "2023-09-27" @default.
- W1898585595 title "Probability Bracket Notation, Probability Vectors, Markov Chains and Stochastic Processes" @default.
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