EXACT WAVE FUNCTIONS AND ENERGIES OF A NON-RELATIVISTIC FREE QUANTUM PARTICLE ON THE SURFACE OF A DEGENERATE TORUS

2004 ◽  
Vol 19 (23) ◽  
pp. 1759-1766 ◽  
Author(s):  
AXEL SCHULZE-HALBERG
Keyword(s):  
Schrödinger Equation ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Azimuthal Angle ◽  
Polar Angle ◽  
Wave Functions ◽  
Exactly Solvable

We study the non-relativistic Schrödinger equation for a free quantum particle constrained to the surface of a degenerate torus, parametrized by its polar and azimuthal angle. On restricting to wave functions that depend on the polar angle only, the Schrödinger equation becomes exactly-solvable. We compute its physical solutions (continuous, normalizable and 2π-periodic) and the associated energies in closed form.

EXACT SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH SPHERICALLY SYMMETRIC OCTIC POTENTIAL

2012 ◽  
Vol 27 (20) ◽  
pp. 1250112 ◽  
Author(s):  
DAVIDS AGBOOLA ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Spherically Symmetric ◽  
Algebraic Equations ◽  
Potential Parameters

We present exact solutions of the Schrödinger equation with spherically symmetric octic potential. We give closed-form expressions for the energies and the wave functions as well as the allowed values of the potential parameters in terms of a set of algebraic equations.

Nonrelativistic particles in the presence of a Cariñena–Perelomov–Rañada–Santander oscillator and a disclination

2020 ◽  
Vol 35 (17) ◽  
pp. 2050071 ◽  
Author(s):  
Soroush Zare ◽  
Hassan Hassanabadi ◽  
Marc de Montigny
Keyword(s):  
Schrödinger Equation ◽  
Elastic Medium ◽  
Schrodinger Equation ◽  
Nonlinear Oscillator ◽  
Wave Functions ◽  
Oscillator Potential ◽  
Energy Eigenvalues ◽  
Radial Schrödinger Equation ◽  
Exactly Solvable ◽  
Heun Functions

We examine an elastic medium with a disclination and consider the topological effects in the presence of a nonpolynomial quantum exactly solvable nonlinear oscillator potential related to the isotonic oscillator, and to which we refer as the Cariñena–Perelomov–Rañada–Santander (CPRS) potential. We obtain the wave functions, which are related to the confluent Heun functions, as well as the energy eigenvalues by solving exactly the corresponding radial Schrödinger equation.

scholarly journals Hidden symmetry of the 16D oscillator and its 9D coulomb analogue

2020 ◽  
Vol 56 (2) ◽  
pp. 206-216
Author(s):  
А. N. Lavrenov ◽  
I. А. Lavrenov
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Separation Method ◽  
Cylindrical Coordinates ◽  
Hidden Symmetry ◽  
Solvable Potentials ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable

We present the quadratic Hahn algebra QH(3) as an algebra of the hidden symmetry for a certain class of exactly solvable potentials, generalizing the 16D oscillator and its 9D coulomb analogue to the generalized version of the Hurwitz transformation based on SU (1,1)⊕ SU (1,1)  . The solvability of the Schrodinger equation of these problems by the variables separation method are discussed in spherical and parabolic (cylindrical) coordinates. The overlap coefficients between wave functions in these coordinates are shown to coincide with the Clebsch – Gordan coefficients for the SU(1,1) algebra.

Quantum Boxes, Stringed Instruments

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Energy Level ◽  
Schrödinger Equation ◽  
Quantum System ◽  
Schrodinger Equation ◽  
Probability Distributions ◽  
Wave Functions ◽  
Energy Level Diagram ◽  
One Dimensional ◽  
Stringed Instruments ◽  
Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

A Student's Guide to the Schrödinger Equation

2020 ◽  
Author(s):  
Daniel A. Fleisch
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Plain Language ◽  
Science And Engineering ◽  
Popular Approach ◽  
Matlab Code ◽  
Mathematical Techniques

Quantum mechanics is a hugely important topic in science and engineering, but many students struggle to understand the abstract mathematical techniques used to solve the Schrödinger equation and to analyze the resulting wave functions. Retaining the popular approach used in Fleisch's other Student's Guides, this friendly resource uses plain language to provide detailed explanations of the fundamental concepts and mathematical techniques underlying the Schrödinger equation in quantum mechanics. It addresses in a clear and intuitive way the problems students find most troublesome. Each chapter includes several homework problems with fully worked solutions. A companion website hosts additional resources, including a helpful glossary, Matlab code for creating key simulations, revision quizzes and a series of videos in which the author explains the most important concepts from each section of the book.

scholarly journals On the position-dependent mass Schrödinger equation for Mie-type potentials

2021 ◽  
Vol 2090 (1) ◽  
pp. 012165
Author(s):  
G Ovando ◽  
J J Peña ◽  
J Morales ◽  
J López-Bonilla
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Operator ◽  
Constant Mass ◽  
Point Canonical Transformation ◽  
Mass Distributions ◽  
Exactly Solvable ◽  
Reference Problem ◽  
Dependent Mass ◽  
Position Dependent Mass

Abstract The exactly solvable Position Dependent Mass Schrödinger Equation (PDMSE) for Mie-type potentials is presented. To that, by means of a point canonical transformation the exactly solvable constant mass Schrödinger equation is transformed into a PDMSE. The mapping between both Schrödinger equations lets obtain the energy spectra and wave functions for the potential under study. This happens for any selection of the O von Roos ambiguity parameters involved in the kinetic energy operator. The exactly solvable multiparameter exponential-type potential for the constant mass Schrödinger equation constitutes the reference problem allowing to solve the PDMSE for Mie potentials and mass functions of the form given by m(x) = skx s-1/(xs + 1))2. Thereby, as a useful application of our proposal, the particular Lennard-Jones potential is presented as an example of Mie potential by considering the mass distribution m(x) = 6kx 5/(x 6 + 1))2. The proposed method is general and can be straightforwardly applied to the solution of the PDMSE for other potential models and/or with different position-dependent mass distributions.

Solution of the Schrödinger equation for bound states in closed form

1982 ◽  
Vol 26 (1) ◽  
pp. 662-664 ◽  
Author(s):  
Edgardo Gerck ◽  
Jason A. C. Gallas ◽  
Augusto B. d'Oliveira
Keyword(s):  
Schrödinger Equation ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Bound States
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