Probability Bracket Notation: the Unified Expressions of Conditional Expectation and Conditional Probability in Quantum Modeling

TL;DR

This paper introduces Probability Bracket Notation (PBN) as a unified framework to express conditional expectation and probability in quantum systems, covering various scenarios including multiple observables and many-particle systems.

Contribution

It develops a unified expression framework for conditional expectation and probability in quantum models using PBN, applicable to diverse quantum systems and observables.

Findings

Unified expressions for CE, CP, and AP in quantum systems.

Applicable to discrete and continuous spectra.

Useful for non-commutative observable analysis.

Abstract

After a brief introduction to Probability Bracket Notation (PBN), indicator operator and conditional density operator (CDO), we investigate probability spaces associated with various quantum systems: system with one observable (discrete or continuous), system with two commutative observables (independent or dependent) and a system of indistinguishable non-interacting many-particles. In each case, we derive unified expressions of conditional expectation (CE), conditional probability (CP), and absolute probability (AP): they have the same format for discrete or continuous spectrum; they are defined in both Hilbert space (using Dirac notation) and probability space (using PBN); and they may be useful to deal with CE of non-commutative observables.