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Constantin Rasinariu

Publications and source records attributed to Constantin Rasinariu.

18 recordsLinked to original sources
arXiv2024

Recent Advances in Semiclassical Methods Inspired by Supersymmetric Quantum Mechanics

Semiclassical methods are essential in analyzing quantum mechanical systems. Although they generally produce approximate results, relatively rare potentials exist for which these methods are exact. Such intriguing potentials serve as crucial test cases for semiclassical approximations. Recent research has demonstrated a deep connection between supersymmetric quantum mechanics and the exactness of semiclassical methods. Specifically, the mathematical form of conventional shape-invariant potentials guarantees exactness in several related situations. In this manuscript, we review these recent results and discuss their significance.

quant-ph↗
arXiv2023

Generalized Langer Correction and the Exactness of WKB for all Conventional Potentials

In this paper we investigate the exactness of the WKB quantization condition for translationally shape invariant systems. In particular, using the formalism of supersymmetric quantum mechanics, we generalize the Langer correction and show that it generates the exact quantization condition for all conventional potentials. We also prove that this correction is related to the previously proven exactness of SWKB for these potentials.

quant-ph↗
arXiv2021

Looping Pendulum: Theory, Simulation, and Experiment

The looping pendulum is a simple physical system consisting of two masses connected by a string that passes over a rod. We derive equations of motion for the looping pendulum using Newtonian mechanics, and show that these equations can be solved numerically to give a good description of the system's dynamics. The numerical solution captures complex aspects of the looping pendulum's behavior, and is in good agreement with the experimental results.

physics.class-ph↗
arXiv2021

Exactness of Semiclassical Quantization Rule for Broken Supersymmetry

Semiclassical methods provide important tools for approximating solutions in quantum mechanics. In several cases these methods are intriguingly exact rather than approximate, as has been shown by direct calculations on particular systems. In this paper we prove that the long-conjectured exactness of the supersymmetry-based semiclassical quantization condition for broken supersymmetry is a consequence of the additive shape invariance for the corresponding potentials.

quant-ph↗
arXiv2020

Building Confidence in the Dirac $δ$-function

In this note we present an example from undergraduate quantum mechanics designed to highlight the versatility of the Dirac $δ$-function. Namely, we compute the expectation value of the Hamiltonian of a free-particle in a state described by a triangular wave function $ψ(x)$. Since the first derivative of $ψ(x)$ is piecewise constant, and because this Hamiltonian is proportional to the second order spatial derivative, students often end up finding the expectation value to be zero --an unphysical answer. This problem provides a pedagogical application of the Dirac $δ$-function. By arriving at the same result via alternate pathways, this exercise reinforces students' confidence in the Dirac $δ$-function and highlights its efficiency and elegance.

physics.gen-ph↗
arXiv2020

Exactness of SWKB for Shape Invariant Potentials

The supersymmetry based semiclassical method (SWKB) is known to produce exact spectra for conventional shape invariant potentials. In this paper we prove that this exactness follows from their additive shape invariance.

quant-ph↗
arXiv2019

Inter-relations between additive shape invariant superpotentials

All known additive shape invariant superpotentials in nonrelativistic quantum mechanics belong to one of two categories: superpotentials that do not explicitly depend on $\hbar$, and their $\hbar$-dependent extensions. The former group themselves into two disjoint classes, depending on whether the corresponding Schrödinger equation can be reduced to a hypergeometric equation (type-I) or a confluent hypergeometric equation (type-II). All the superpotentials within each class are connected via point canonical transformations. Previous work showed that type-I superpotentials produce type-II via limiting procedures. In this paper we develop a method to generate a type I superpotential from type II, thus providing a pathway to interconnect all known additive shape invariant superpotentials.

quant-ph↗
arXiv2019

The Supersymmetric WKB Formalism is Not Exact for All Additive Shape Invariant Potentials

Following the verification of the conjecture made by Comtet, Bandrauk and Campbell that the supersymmetry-inspired semiclassical method known as SWKB is exact for the conventional additive shape invariant potentials, it was widely believed that SWKB yields exact results for all additive shape invariant potentials. In this paper we present a concrete example of an additive shape invariant potential for which the SWKB method fails to produce exact results.

quant-ph↗
arXiv2018

Emergence of Benford's Law in Music

We analyzed a large selection of classical musical pieces composed by Bach, Beethoven, Mozart, Schubert and Tchaikovsky, and found a surprising connection with mathematics. For each composer, we extracted the time intervals each note was played in each piece and found that the corresponding data sets are Benford distributed. Remarkably, the logarithmic distribution is present not only for the leading digits, but also for all digits.

physics.soc-ph↗
arXiv2016

The motion of two identical masses connected by an ideal string symmetrically placed over a corner

We introduce a novel, two-mass system that slides up an inclined plane while its center of mass moves down. The system consists of two identical masses connected by an ideal string symmetrically placed over a corner-shaped support. This system is similar to a double-cone that rolls up an inclined set of V-shaped rails. We find the double-cone's motion easy to demonstrate but difficult to analyze. Our example here is more straightforward to follow, and the experimental observations are in good agreement with the theoretical predictions.

physics.pop-ph↗
arXiv2016

$n$-digit Benford converges to Benford

Using the sum invariance property of Benford random variables, we prove that an $n$-digit Benford variable converges to a Benford variable as $n$ approaches infinity.

math.PR↗
arXiv2015

Generation of a Novel Exactly Solvable Potential

We report a new shape invariant (SI) isospectral extension of the Morse potential. Previous investigations have shown that the list of "conventional" SI superpotentials that do not depend explicitly on Planck's constant $\hbar$ is complete. Additionally, a set of "extended" superpotentials has been identified, each containing a conventional superpotential as a kernel and additional $\hbar$-dependent terms. We use the partial differential equations satisfied by all SI superpotentials to find a SI extension of Morse with novel properties. It has the same eigenenergies as Morse but different asymptotic limits, and does not conform to the standard generating structure for isospectral deformations.

quant-ph↗
arXiv2013

n-digit Benford distributed random variables

The scope of this paper is twofold. First, to emphasize the use of the mod 1 map in exploring the digit distribution of random variables. We show that the well-known base- and scale-invariance of Benford variables are consequences of their associated mod 1 density functions being uniformly distributed. Second, to introduce a new concept of the $n$-digit Benford variable. Such a variable is Benford in the first $n$ digits, but it is not guaranteed to have a logarithmic distribution beyond the $n$-th digit. We conclude the paper by giving a general construction method for $n$-digit Benford variables, and provide a concrete example.

math.PR↗
arXiv2012

Shape invariance in phase space

Shape invariance is a powerful solvability condition, that allows for complete knowledge of the energy spectrum, and eigenfunctions of a system. After a short introduction into the deformation quantization formalism, this paper explores the implications of the supersymmetric quantum mechanics and shape invariance techniques to the phase space formalism. We show that shape invariance induces a new set of relations between the Wigner functions of the system, that allows for their direct calculation, once we know one of them. The simple harmonic oscillator and the Morse potential are solved as examples.

quant-ph↗
arXiv2008

Numeric Experiments in Relativistic Thermodynamics: A Moving System Appears Cooler

In this paper we simulate a two dimensional relativistic ideal gas by implementing a relativistic elastic binary collision algorithm. We show that the relativistic gas faithfully obeys Jüttner's speed distribution function. Furthermore, using this numeric simulation in conjunction with the energy equipartition theorem for a relativistic gas, we conclude that a moving system appears cooler.

gr-qc↗
arXiv2001

Baxter T-Q Equation for Shape Invariant Potentials. The Finite-Gap Potentials Case

The Darboux transformation applied recurrently on a Schroedinger operator generates what is called a {\em dressing chain}, or from a different point of view, a set of supersymmetric shape invariant potentials. The finite-gap potential theory is a special case of the chain. For the finite-gap case, the equations of the chain can be expressed as a time evolution of a Hamiltonian system. We apply Sklyanin's method of separation of variables to the chain. We show that the classical equation of the separation of variables is the Baxter T-Q relation after quantization.

hep-th↗