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Born’s rule and Planck’s energy from quantum interferometry

Francisco Duarte

Published Nov 25, 2024• 0 citations
Publisher

Abstract

Born’s rule and Planck’s quantum energy equation are derived from quantum interferometry principles. The derivation of Born’s rule is transparent, brief, and is based on the Dirac-Feynman interferometric principle. The derivation of Planck’s equation relies on N-slit interferometric experimental observations and a heuristic approach.

DOI
10.1364/opticaopen.27896898

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Born’s rule and Planck’s quantum energy equation are derived from quantum interferometry principles. The derivation of Born’s rule is transparent, brief, and is based on the Dirac-Feynman interferometric principle. The derivation of Planck’s equation relies on N-slit interferometric experimental observations and a heuristic approach.
The Dirac–Feynman probability amplitude is introduced from first principles. This introduction starts from the basic two intra-interferometric spaces and continues to multiple intra-interferometric spaces. The probability amplitudes are then multiplied by their complex conjugate, via Born’s rule, to yield practical usable generalized interferometric quantum probabilities. A series of mathematically explicit quantum probabilities for N = 2, 3, 4, 5 are given. Various measurements and practical examples for N = 2 up to N = 100 are discussed in detail. Generalized N -slit quantum probability equations for two dimensions and three dimensions are also given.
In quantum mechanics, the physics of probability amplitudes, that exists in the imaginary and complex mathematical realm, is brought to the measurable realm by the iconic Born rule https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 that convert the superposition probability amplitude https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 into a measurable interferometric probability distribution. When Born first introduced his rule, in 1926, he did so in the absence of a physical derivation. In fact, Born never offered a derivation for his most crucial rule. In this chapter a transparent interferometric derivation of Born's rule is given. Furthermore, it is indicated that far from a ‘collapse’ the https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 → | ψ 〉 | ψ 〉 ∗ transition, for interferometric configurations, clearly involves a mathematical expansion . The https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 → | ψ 〉 | ψ 〉 ∗ transition for quantum entanglement is also discussed.
The quantum mechanics postulate called the Born Rule attributes a probabilistic meaning to a wave function. This paper derives the Born Rule from other quantum principles along with a model of the measurement process. The nondeterministic nature of quantum measurements is hypothesized to arise from an ignorance of the quantum states of a measuring device's microscopic components. Their interactions with a system to be measured are modeled heuristically with any member of a particular class of stochastic processes, each of which generate the Born Rule. One member of the class appears particularly compatible with properties expected of quantum interactions.
The Born rule stands as the fundamental probability postulate of quantum mechanics, yet its first-principle derivation remains a longstanding open problem. This paper presents a rigorous, collapse-free, and observer-independent derivation of the Born rule. Starting from four physically natural and minimal axioms—unitary dynamics, uniqueness of classical records, unitary invariance, and consistency with the classical limit and ergodicity—we uniquely deduce the Born rule from first principles. We prove that the Born rule is not an independent postulate, but a necessary mathematical consequence of the core axiomatic structure. Our approach is self-contained, assumption-free, and purely unitary, establishing that the Born rule is canonically determined by the fundamental mathematical and physical principles of quantum mechanics.