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Open AccessArticle

A New Approach to Understanding Quantum Mechanics: Illustrated Using a Pedagogical Model over ℤ₂

by David Ellerman

AppliedMath 2024, 4(2), 468-494; https://doi.org/10.3390/appliedmath4020025 - 9 Apr 2024

Viewed by 2728

The new approach to quantum mechanics (QM) is that the mathematics of QM is the linearization of the mathematics of partitions (or equivalence relations) on a set. This paper develops those ideas using vector spaces over the field [...] Read more.

Z_{2} = {0.1}

as a pedagogical or toy model of (finite-dimensional, non-relativistic) QM. The

0, 1

-vectors are interpreted as sets, so the model is “quantum mechanics over sets” or QM/Sets. The key notions of partitions on a set are the logical-level notions to model distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. Those pairs of concepts are the key to understanding the non-classical ‘weirdness’ of QM. The key non-classical notion in QM is the notion of superposition, i.e., the notion of a state that is indefinite between two or more definite- or eigen-states. As Richard Feynman emphasized, all the weirdness of QM is illustrated in the double-slit experiment, so the QM/Sets version of that experiment is used to make the key points. Full article

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15 pages, 312 KB

Open AccessArticle

A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes

by Valeriy Naumov

Mathematics 2024, 12(3), 444; https://doi.org/10.3390/math12030444 - 30 Jan 2024

Cited by 1 | Viewed by 1836

Abstract

We consider an irreducible positive-recurrent discrete-time Markov process on the state space

X = ℤ_{+}^{M} \times J

, where

ℤ_{+}

is the set of non-negative integers and

J = {1, 2, \dots, n}

. The [...] Read more.

We consider an irreducible positive-recurrent discrete-time Markov process on the state space

X = ℤ_{+}^{M} \times J

, where

ℤ_{+}

is the set of non-negative integers and

J = {1, 2, \dots, n}

. The number of states in

J

may be either finite or infinite. We assume that the process is a homogeneous quasi-birth-and-death process (QBD). It means that the one-step transition probability between non-boundary states

(k, i)

and

(n, j)

may depend on

i, j

, and

n - k

but not on the specific values of

k

and

n

. It is shown that the stationary probability vector of the process is expressed through square matrices of order

n

, which are the minimal non-negative solutions to nonlinear matrix equations. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

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