Quantum mechanical derivation of the Wallis formula for <i>π</i>

Friedmann, Tamar; Hägen, C. R.

doi:10.1063/1.4930800

Cited by 27 publications

(32 citation statements)

References 10 publications

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“…The HNH is based on pure integer mathematical properties. It has recently been shown that π/2 can be derived from the orbital energy levels seen in the hydrogen atom [34]. We have shown that it is possible to the first approximation derive the properties of hydrogen from the frequency of the neutron only in conjunction with the prime numbers 2, 3, 5, 7, 11 [30].…”

Section: Assignment Of a Partial Harmonic Fraction 1/2 To ν Cmbmentioning

confidence: 91%

The Derivation of the Cosmic Microwave Background Radiation Peak Spectral Radiance, Planck Time, and the Hubble Constant from the Neutron and Hydrogen

Chakeres¹,

Andrianarijaona²

2016

JMP

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Purpose: The cosmic microwave background radiation, CMB, is fundamental to observational cosmology, and is believed to be a remnant from the Big Bang. The CMB, Planck time, t P , and the Hubble constant, H 0 , are important cosmologic constants. The goal is to accurately derive and demonstrate the inter-relationships of the CMB peak spectral radiance frequency, t P , and H 0 from neutron and hydrogen quantum data only. Methods: The harmonic neutron hypothesis, HNH, evaluates physical phenomena within a finite consecutive integer and exponential power law harmonic fraction series that are scaled by a fundamental frequency of the neutron as the exponent base. The CMB and the H 0 are derived from a previously published method used to derive The cosmic fundamental constants t P , H 0 , and CMB are mathematically inter-related constants all defined by gravity. They are also directly derivable from the quantum properties of the neutron and hydrogen within a harmonic power law.

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Section: Assignment Of a Partial Harmonic Fraction 1/2 To ν Cmbmentioning

confidence: 91%

The Derivation of the Cosmic Microwave Background Radiation Peak Spectral Radiance, Planck Time, and the Hubble Constant from the Neutron and Hydrogen

Chakeres¹,

Andrianarijaona²

2016

JMP

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“…π has an infinite number of digits in its decimal representation and no repeating pattern ever occurs. Magically, the formula for π appears in a basic calculation in the physics of the hydrogen atom (Friedmann and Hagen 2015). Then there is the claim that the distribution of the prime numbers follow the energy levels of a quantum system (Bender et al 2017).…”

Section: )mentioning

confidence: 99%

Philosophy and Science: What Can I Know?

Glattfelder¹

2019

The Frontiers Collection

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Philosophy is a thorny subject. Many philosophical statements cannot be formally proven, resulting in clever but endless debates. Scientists usually shy away from such ambiguity and retreat into their safe world of perceived clarity. Nevertheless, the philosophical study of nature is the wellspring of science. Simply asking "What is a law of nature?" poses a philosophical challenge. The philosophy of science is concerned with the foundations, methods, and implications of science and how scientists conduct their research. The first attempts to systematize the scientific method was based on common sense: from observations abstract laws are found. This view turned out to be untenable and both inductive and deductive reasoning suffer from conceptual problems. Indeed, science is not an incremental process accumulating knowledge but is greatly influenced by social and cultural conditions. It comes perhaps as no surprise that an animosity exists between science and philosophy. For instance, the controversial philosopher of science, Paul Feyerabend, continually challenged the scientific establishment. Historically, the emergence of modern physics overthrew nearly every postulate of classical science and replaced them with bizarre new concepts, from elusive quantum fluctuations to the fabric of space-time. The aftershocks of this fundamental transformation still echo to this day. On the one hand the universe is, miraculously, comprehensible to the human mind, but on the other hand scientific progress appears to be slowing down. Paradoxically, every question answered raises more and harder questions and theories appear to be losing meaning. If asked, some scientists will admit to these shortcomings: uncertainty and ignorance are inherent and ubiquitous in science. The final blow to a clear foundation of knowledge comes from the discoveries that incompleteness and randomness lurk at the heart of mathematics. Level of mathematical formality: low. Science works! This is attested by the spectacular display of human technological prowess. Indeed, technological advancements, made possible by the scientific understanding of the universe, are becoming ever more disruptive and frequent. How is it then justifiable to speak of the crisis of science and even allude to the end of science?

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“…In the recent article [1] the authors claim that a venerable asymptotic formula for π, the Wallis (1655) product, is related to quantum mechanics. More precisely to the variational method applied to the ground state of the hydrogen atom in the limit of large angular momentum (the classical limit ℓ → ∞).…”

Section: Introductionmentioning

confidence: 99%

“…So perhaps there are other potentials and trial functions that can give rise to the Wallis asymptotic formula for π or variants of it. The classical limit that we will take is just the classical limit used in [1].…”

Section: Introductionmentioning

confidence: 99%

Wallis formula from the harmonic oscillator

Cortese

Garcı́a

2018

Journal of Geometry and Physics

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We show that the asymptotic formula for π, the Wallis formula, that was related with quantum mechanics and the hydrogen atom in [1], can also be related to the harmonic oscillator using a quantum duality between these two systems. As a corollary we show that this very interesting asymptotic formula is not related with the hydrogen atom or quantum mechanics itself but with a clever choice of a trial function and a potential in the Schroedinger equation when we use the variational approach to calculate the ground state energy associated with the given potential function.

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Quantum mechanical derivation of the Wallis formula for π

Cited by 27 publications

References 10 publications

The Derivation of the Cosmic Microwave Background Radiation Peak Spectral Radiance, Planck Time, and the Hubble Constant from the Neutron and Hydrogen

The Derivation of the Cosmic Microwave Background Radiation Peak Spectral Radiance, Planck Time, and the Hubble Constant from the Neutron and Hydrogen

Philosophy and Science: What Can I Know?

Wallis formula from the harmonic oscillator

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