Revisiting the quantum harmonic oscillator via unilateral Fourier transforms (original) (raw)
Quantum states of a particle in a box via unilateral Fourier transform
Revista Brasileira de Ensino de Física, 2020
The quantum problem of stationary states of a particle in a box is revisited by means of the unilateral Fourier transform. Homogeneous Dirichlet boundary conditions demand a finite Fourier sine transform which is actually the Fourier sine series.
One dimensional (1-D) harmonic oscillators revisited
Leonardo Journal of Sciences, 2018
The concept of harmonic oscillator particularly one dimensional (1-D) is mentioned in literature repeatedly and is explained in more complex manner by using various methods. This creates difficulties in understanding the description of the concept for new learners. The purpose of this article was to enlighten different methods to formulate harmonic oscillator in improving knowledge about detailed steps to derive eigen energy values in more comprehensible manner for the beginners. The energy values are derived by using classical method, quantum mechanically, Schrodinger time independent equation, perturbation technique, variation method, WKB approximation etc. A coherent way of derivation of eigen values using various approaches makes this article as unique.
One-dimensional model of a quantum nonlinear harmonic oscillator
Reports on Mathematical Physics, 2004
In this paper we study the quantization of the nonlinear oscillator introduced by Mathews and Lakshmanan. This system with positiondependent mass allows a natural quantization procedure and is shown to display shape invariance. Its energy spectrum is found by factorization. The linear harmonic oscillator appears as the λ → 0 limit of this nonlinear oscillator, whose energy spectrum and eigenfunctions are compared to the linear ones.
On The Exact and JWKB Solution of 1D Quantum Harmonic Oscillator by Mathematica
Although being the fundamental semiclassical approximation method mainly used in quantum mechanics and optical waveguides, JWKB method along with the application of the associated JWKB asymptotic matching rules is known to give exact solutions for the Quantum Harmonic Oscillator (QHO). Asymptotically matched JWKB solutions are typically accurate or exact in the entire domain except for a narrow domain around the classical turning points where potential energy equals the total energy of the related quantum mechanical system. So, one has to cope with this diverging behavior at the classical turning points since it prohibits us from using continuity relations at the related boundaries to determine the required JWKB coefficients. Here, a computational diagram and related mathematica codes to surmount the problem by applying parity matching for even and odd JWKB solutions rather than boundary continuities are being presented. In effect, JWKB coefficients as well as the conversion factor for the dimensionless form of the Schrodingers equation, which is common to both exact and JWKB solutions, is being successfully obtained. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
An exact solution of the energy shift in each quantum mechanical energy levels in a one dimensional symmetrical linear harmonic oscillator has been investigated. The solution we have used here is firstly derived by manipulating Schrödinger differential equation to be confluent hypergeometric differential equation. The final exact numerical results of the energy shifts are then found by calculating the final analytical solution of the confluent hypergeometric equation with the use of a software (Mathcad Plus 6.0) or a program programmed by using Turbo Pascal 7.0. We find that the results of the energy shift in our exact solution method are almost the same as that in Barton et. al. approximation method. Thus, the approximation constants appeared in Barton et. al. method can also be calculated by using the results of the exact method.
Fourier transform and quantum mechanics
Buenos Aires Herald, 2024
The central objective of this article is to introduce a series of mathematical problems of the domain of quantum mechanics in order to be developed with Fourier Analysis.
Journal of Physics A: Mathematical and General, 2003
An extension of the classical orthogonal functions invariant to the quantum domain is presented. This invariant is expressed in terms of the Hamiltonian. Unitary transformations which involve the auxiliary function of this quantum invariant are used to solve the time-dependent Schrödinger equation for a harmonic oscillator with time-dependent parameter. The solution thus obtained is in agreement with the results derived using other methods which invoke the Lewis invariant in their procedures.
Cornell University - arXiv, 2016
In this article, we try to test the influence of the modification of the scalar product, found in the problems of the energy-dependent potential, on the physical properties of the harmonic oscillator in one dimension. For this, we at first discuss the effect of this change on the thermodynamic properties of this oscillator, and then on the parameters of Fisher and Shannon of quantum information. For the second problem, we are an obligation to redefine this parameters. Finally, the uncertainly relation of Cramer-Rao is well recovered in our problem in question.
The Quantum Harmonic Oscillator
UNITEXT for Physics, 2018
Think of a sliding block, constrained to move along one direction on an idealized frictionless surface, attached to an idealized spring. The block has mass and the spring has spring constant. This is an example of a classical one-dimensional harmonic oscillator.
Quantum anharmonic oscillators: a new approach
Journal of Physics A: Mathematical and General, 2005
The determination of the eigenenergies of a quantum anharmonic oscillator consists merely in finding the zeros of a function of the energy, namely the Wronskian of two solutions of the Schrödinger equation which are regular respectively at the origin and at infinity. We show in this paper how to evaluate that Wronskian exactly, except for numerical rounding errors. The procedure is illustrated by application to the gx 2 + x 2N (N a positive integer) oscillator.