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Journal of Quantum Information Science

Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential

Saeed Naif Turki Al-Rashid, Mohammed A. Z. Habeeb, Khalid A. Ahmed

Published Jan 1, 2017• 1 citation
Publisher

Abstract

In the present work, Scale Relativity (SR) is applied to a particle in a simple harmonic oscillator (SHO) potential. This is done by utilizing a novel mathematical connection between SR approach to quantum mechanics and the well-known Riccati equation. Then, computer programs were written using the standard MATLAB 7 code to numerically simulate the behavior of the quantum particle utilizing the solutions of the fractal equations of motion obtained from SR method. Comparison of the results with the conventional quantum mechanics probability density is shown to be in very precise agreement. This agreement was improved further for some cases by utilizing the idea of thermalization of the initial particle state and by optimizing the parameters used in the numerical simulations such as the time step and number of coordinate divisions. It is concluded from the present work that SR method can be used as a basis for description the quantum behavior without reference to conventional formulation of quantum mechanics. Hence, it can also be concluded that the fractal nature of space-time implied by SR, is at the origin of the quantum behavior observed in these problems. The novel mathematical connection between SR and the Riccati equation, which was previously used in quantum mechanics without reference to SR, needs further investigation in future work.

DOI
10.4236/jqis.2017.73008

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In the present work, Scale Relativity (SR) is applied to a particle in a simple harmonic oscillator (SHO) potential. This is done by utilizing a novel mathematical connection between SR approach to quantum mechanics and the well-known Riccati equation. Then, computer programs were written using the standard MATLAB 7 code to numerically simulate the behavior of the quantum particle utilizing the solutions of the fractal equations of motion obtained from SR method. Comparison of the results with the conventional quantum mechanics probability density is shown to be in very precise agreement. This agreement was improved further for some cases by utilizing the idea of thermalization of the initial particle state and by optimizing the parameters used in the numerical simulations such as the time step and number of coordinate divisions. It is concluded from the present work that SR method can be used as a basis for description the quantum behavior without reference to conventional formulation of quantum mechanics. Hence, it can also be concluded that the fractal nature of space-time implied by SR, is at the origin of the quantum behavior observed in these problems. The novel mathematical connection between SR and the Riccati equation, which was previously used in quantum mechanics without reference to SR, needs further investigation in future work.
In the present work, and along the lines of Hermann, ScR theory is applied to a finite one-dimensional square well potential problem. The aim is to show that scale relativity theory can reproduce quantum mechanical results without employing the Schrödinger equation. Some mathematical difficulties that arise when obtaining the solution to this problem were overcome by utilizing a novel mathematical connection between ScR theory and the well-known Riccati equation. Computer programs were written using the standard MATLAB 7 code to numerically simulate the behavior of the quantum particle in the above potential utilizing the solutions of the fractal equations of motion obtained from ScR theory. Several attempts were made to fix some of the parameters in the numerical simulations to obtain the best possible results in a practical computer CPU time within limited local computer facilities [1,2]. Comparison of the present results with the corresponding results obtained from conventional quantum mechanics by solving the Schrödinger equation, shows very good agreement. This agreement was improved further by optimizing the parameters used in the numerical simulations [1,3]. This represents a new example where scale relativity theory, based on a fractal space-time concept, can accurately reproduce quantum mechanical results without invoking the Schrödinger equation.
Another "simple" potential is one experienced by a spring, the simple harmonic oscillator potential (SHO). To understand how a quantum particle behaves in such a potential, first we study how a classical particle, such as a mass on a spring, behaves. Then we change to using the TISE with the above potential. Solving this TISE is not a simple task. One way of solving a differential equation is to see if someone else has already solved it. Fortunately, a French mathematician named Hermite solved a related equation in the 19 th century. His solutions can be multiplied by a simple exponential function, and these modified Hermite polynomials can be shown to solve the TISE for the SHO. Interestingly, the resulting trapped particles have quantized energies that are evenly spaced. It is for this reason that bond vibrations, which can be modeled as springs on a mass, are found to be quantized in nearly evenly spaced energy levels, at least for relatively small vibrations. The SHO TISE also provides an opportunity to introduce two new operators: the creation and annihilation operators.
Abstract The special and recurring role that the simple harmonic oscillator (SHO) plays in quantum mechanics can be attributed both to its physical relevance and its simple solutions. The fact that the ground state solution is the minimum uncertainty wave packet (Section 12.4) and the highly constrained connection between the position-space and momentum-space wavefunctions (P4.23 and Section 9.2.2) also indicates that this problem occupies a special niche. The symmetry between x and p present in the solutions is obviously a reflection of the fact that only for the SHO is the potential energy function quadratic in x. It is often the case in physics where systems with a high degree of symmetry are amenable to solution in a variety of ways, sometimes quite unexpected. In this chapter, we discuss a powerful method of solving the harmonic oscillator problem involving the factorization of the differential equation using differential operators; we then (briefly) discuss extensions and applications of operator methodsÉ to other physical problems.
Complete description of the classical and quantum dynamics of a particle in an anisotropic, rotating, harmonic trap is given. The problem is studied in three dimensions and no restrictions on the geometry are imposed. In the generic case, for an arbitrary orientation of the rotation axis, there are two regions of instability with different characteristics. The analysis of the quantum-mechanical problem is made simple due to a direct connection between the classical mode vectors and the quantum-mechanical wave functions. This connection is obtained via the matrix Riccati equation that governs the time evolution of squeezed states of the harmonic oscillator. It is also shown that the inclusion of gravity leads to a resonant behavior -- the particles are expelled from the trap.
In the present work, we extend the Hermann and Al-Rashid works to the problem of particle in a double oscillators potential. In this problem, one can take a special case when oscillation quantum number ( ) is none negative integer. Computer programming is built to make numerical simulations to this problem. The probability density of finding particle in a double oscillators potential is calculated without using Schrِdinger equation or any conventional quantum mechanics. This probability is compared with probability of conventional quantum mechanics.
We theoretically investigate the stability and linear oscillatory behavior of a naturally unstable particle whose potential energy is harmonically modulated. We find this fundamental dynamical system is analogous in time to a quantum harmonic oscillator. In a certain modulation limit, a.k.a. the Kapitza regime, the modulated oscillator can behave like an effective classic harmonic oscillator. But in the overlooked opposite limit, the stable modes of vibrations are quantized in the modulation parameter space. By analogy with the statistical interpretation of quantum physics, those modes can be characterized by the time-energy uncertainty relation of a quantum harmonic oscillator. Reducing the almost-periodic vibrational modes of the particle to their periodic eigenfunctions, one can transform the original equation of motion to a dimensionless Schrodinger stationary wave equation with a harmonic potential. This reduction process introduces two features reminiscent of the quantum realm: a wave-particle duality and a loss of causality that could legitimate a statistical interpretation of the computed eigenfunctions. These results shed new light on periodically time-varying linear dynamical systems and open an original path in the recently revived field of quantum mechanical analogs.