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Formulation of the q-Ruark-Devol Differential Equation

Shawn Eastmond

Published Apr 14, 2025• 0 citations
Publisher

Abstract

In this paper, we introduce, for the first time ever, the formulation of the q-Ruark-Devol density function, a q-deformed generalization of the classical Ruark-Devol density function used to model radioactive decay processes. By employing quantum calculus (q-calculus), a branch of mathe- matics that generalizes classical calculus through the introduction of a deformation parameter q, we derive the q-Ruark-Devol density differential equation and demonstrate its reduction to the classical result in the limit q → 1. We provide a rigorous theoretical background on q -calculus, including the q - derivative, q -Leibniz product rule , and q -exponential, which are requirements for modeling deformed radioactive decay processes, in our framework. Furthermore, we explore the physical implications of the q-Ruark-Devol density function and its potential applications in quantum sensing, quantum op- tics, and neutrino detection. This work bridges the gap between classical and quantum-inspired mod- els of radioactive decay, offering a new perspective on the statistical behavior of decay processes in deformed or non-standard environments.

DOI
10.31219/osf.io/t8qrz_v1

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In this paper, we introduce, for the first time ever, the formulation of the q-Ruark-Devol density function, a q-deformed generalization of the classical Ruark-Devol density function used to model radioactive decay processes. By employing quantum calculus (q-calculus), a branch of mathe- matics that generalizes classical calculus through the introduction of a deformation parameter q, we derive the q-Ruark-Devol density differential equation and demonstrate its reduction to the classical result in the limit q → 1. We provide a rigorous theoretical background on q -calculus, including the q - derivative, q -Leibniz product rule , and q -exponential, which are requirements for modeling deformed radioactive decay processes, in our framework. Furthermore, we explore the physical implications of the q-Ruark-Devol density function and its potential applications in quantum sensing, quantum op- tics, and neutrino detection. This work bridges the gap between classical and quantum-inspired mod- els of radioactive decay, offering a new perspective on the statistical behavior of decay processes in deformed or non-standard environments.
In this paper, we use the q-derivative emerging in the non-extensive statistical physics to formulate the q-deformed quantum mechanics. We find the algebraic structure related to this deformed theory and investigate some properties of the q-deformed elementary functions. Using this mathematical background, we formulate the q-deformed Heisenberg algebra and q-deformed time-dependent Schrödinger equation. As an example, we deal with the infinite potential well and compute the Fermi energy in the q-deformed theory.
Abstract Starting on the basis of Fibonacci calculus and Fibonacci oscillator algebra, we introduce the main properties to develop a new formalism for the two-parameter (q1,q2)({q}_{1},{q}_{2})(q1​,q2​) ( q 1 , q 2 ) -deformed quantum mechanics, where q1{q}_{1}q1​ q 1 and q2{q}_{2}q2​ q 2 are real positive independent deformation parameters. As applications of such a two-parameter deformed formalism, we investigate the behavior of a quantum particle in some different physical phenomena covering the free particle and the inverse-harmonic potential case. The effect of two deformation parameters on the wave functions for these applications is studied. Another application is carried out onto the quantum statistics of lattice oscillations through a model of the (q1,q2)({q}_{1},{q}_{2})(q1​,q2​) ( q 1 , q 2 ) -deformed phonon gas, and it is shown that the high- and low-temperature behavior of the model specific heat differs notably from the classical theories for the interval 0<(q1,q2)<∞0<({q}_{1},{q}_{2})<\infty 0<(q1​,q2​)<∞ 0 < ( q 1 , q 2 ) < ∞ . We also construct a two-parameter deformed non-extensive entropy based on some elements of the Fibonacci calculus and discuss its possible connection with the Tsallis entropy in non-extensive statistical mechanics. Finally, other possible application areas of the present two-parameter (q1,q2)({q}_{1},{q}_{2})(q1​,q2​) ( q 1 , q 2 ) -deformed construction on quantum mechanics are discussed.
Various phenomenological generalizations of the foundational equation in quantum physics have been proposed in prior studies. This paper presents a rigorous analytical derivation, grounded in first principles, that elucidates the impact of quantum fluctuations on the evolution of quantum systems. Furthermore, it demonstrates how this essential generalization can be achieved through statistical methods. The paper reveals that standard linear equations of quantum mechanics are recovered under specific limits of the parameter that governs nonlinear behavior. It establishes a direct correlation between the decay of quantum waves and the magnitude of these fluctuations. This connection provides critical insights into the dynamic properties of quantum systems and their susceptibility to underlying stochastic influences. Moreover, this work successfully formulates a comprehensive approach to a complete family of nonlinear quantum evolution equations. This framework expands our theoretical arsenal and enhances our ability to model and predict the behavior of complex quantum systems under various conditions. This research represents a significant advancement in our understanding of quantum mechanics, offering a more nuanced view of how quantum systems evolve under the influence of intrinsic fluctuations. It paves the way for future explorations into the stability, coherence, and dynamical evolution of quantum states, potentially impacting quantum computing, information processing, and other applications in quantum technology.
The description of a particle in the quantum system is probabilistic. In the ordinary quantum mechanics the total probability of finding the particle is conserved, i.e. the probability is normalized for all the times. To find a non-constant total probability an imaginary term should be added to the potential energy which is not physical. Recently, generalizations of the ordinary Schrödinger equation have been proposed by using the Feynman path integral and analogy between the Schrödinger equation and diffusion equation. In this work, an integro-differential Schrödinger equation is proposed by using analogy between the Schrödinger equation and diffusion equation. The equation is obtained from the continuous time random walk model with diverging jump length variance and generic waiting time probability density. The equation generalizes the ordinary and fractional Schrödinger equations. One can show that the integro-differential Schrödinger equation can describe a non-constant total probability for a free particle, and it includes the exponential decay which is fundamental for the description of radioactive decay.
Elementary parameter-differentiation techniques are developed to systematically derive a wide variety of operator identities, expansions, and solutions to differential equations of interest to quantum physics. The treatment is largely centered around a general closed formula for the derivative of an exponential operator with respect to a parameter. Derivations are given of the Baker-Campbell-Hausdorff formula and its dual, the Zassenhaus formula. The continuous analogs of these formulas which solve the differential equation dY(t)/dt = A(t) Y(t), the solutions of Magnus and Fer, respectively, are similarly derived in a recursive manner which manifestly displays the general repeated-commutator nature of these expansions and which is quite suitable for computer programming. An expansion recently obtained by Kumar and another new expansion are shown to be derivable from the Fer and Magnus solutions, respectively, in the same way. Useful similarity transformations involving linear combinations of elements of a Lie algebra are obtained. Some cases where the product eAeB can be written as a closed-form single exponential are considered which generalize results of Sack and of Weiss and Maradudin. Closed-form single-exponential solutions to the differential equation dY(t)/dt = A(t) Y(t) are obtained for two cases and compared with the corresponding multiple-exponential solutions of Wei and Norman. Normal ordering of operators is also treated and derivations, corollaries, or generalization of a number of known results are efficiently obtained. Higher derivatives of exponential and general operators are discussed by means of a formula due to Poincaré which is the operator analog of the Cauchy integral formula of complex variable theory. It is shown how results obtained by Aizu for matrix elements and traces of derivatives may be readily derived from the Poincaré formula. Some applications of the results of this paper to quantum statistics and to the Weyl prescription for converting a classical function to a quantum operator are given. A corollary to a theorem of Bloch is obtained which permits one to obtain harmonic-oscillator canonical-ensemble averages of general operators defined by the Weyl prescription. Solutions of the density-matrix equation are also discussed. It is shown that an initially canonical ensemble behaves as though its temperature remains constant with a ``canonical distribution'' determined by a certain fictitious Hamiltonian.
Abstract In this paper, we consider a system of the q-deformed bosonic Tamm-Dancoff oscillators, whose spectrum has some exponential cutoff factors at high energies. We first investigate the q-calculus in the Tamm-Dancoff (TD) boson algebra, and within this framework, the q-derivative, q-integral and q-exponential function are introduced. Using these properties, we construct a new formalism for the q-deformed quantum mechanics, which accordingly involve the q-adjoint operator and the q-Hermitian operator properties. We then derive the q-deformed Heisenberg relation, and develop the q-Hermitian momentum operator. The q-deformed Schrödinger equation is introduced, and as applications, we study the momentum eigenfunction and one-dimensional box problem. Another application of the TD type deformation onto lattice oscillations is also discussed through a model of the q-deformed Debye solid. Finally, other potential applications of the TD-oscillators gas model are concisely pointed out.