Analytic energies and wave functions of the two-dimensional Schrödinger equation: ground state of two-dimensional quartic potential and classification of solutions

2012 ◽  
Vol 90 (6) ◽  
pp. 503-513 ◽  
Author(s):  
Vladimír Tichý ◽  
Aleš Antonín Kuběna ◽  
Lubomír Skála
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Analytic Solutions ◽  
Wave Functions ◽  
Two Dimensional ◽  
Quartic Potential ◽  
Ground State Energies ◽  
Fourth Order Polynomial

New analytic solutions of the two-dimensional Schrödinger equation with a two-dimensional fourth-order polynomial (i.e., quartic) potential are derived and discussed. The solutions represent the ground state energies and the corresponding wave functions. In general, the obtained results cannot be reduced to two one-dimensional cases.

scholarly journals Algebraic approach to non-separable two-dimensional Schrödinger equation: Ground states for polynomial and Morse-like potentials

Open Physics ◽  
2014 ◽  
Vol 12 (10) ◽  
Author(s):  
Vladimír Tichý ◽  
Lubomír Skála ◽  
René Hudec
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Ground States ◽  
Algebraic Approach ◽  
Algebraic Method ◽  
Analytic Solutions ◽  
Wave Functions ◽  
Two Dimensional ◽  
One Dimensional

AbstractThis paper presents a direct algebraic method of searching for analytic solutions of the two-dimensional time-independent Schrödinger equation that is impossible to separate into two one-dimensional ones. As examples, two-dimensional polynomial and Morse-like potentials are discussed. Analytic formulas for the ground state wave functions and the corresponding energies are presented. These results cannot be obtained by another known method.

Analytic Energies and Wave Functions of Two-Dimensional Schrödinger Equation: Two-Dimensional Fourth-Order Polynomial Potential

2008 ◽  
Vol 73 (10) ◽  
pp. 1327-1339 ◽  
Author(s):  
Vladimír Tichý ◽  
Lubomír Skála
Keyword(s):  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Direct Method ◽  
Analytic Solutions ◽  
Wave Functions ◽  
Two Dimensional ◽  
Polynomial Potential ◽  
Order Polynomial ◽  
Fourth Order Polynomial

Direct method for searching analytic solutions of the two-dimensional Schrödinger equation with a two-dimensional fourth-order polynomial potential is presented. Analytic formulas for the energies and wave functions of the ground state and excited state are found. Obtained results can not be in general reduced to two one-dimensional cases.

Analytical Solutions of the Schrödinger Equation. Ground State Energies and Wave Functions

1998 ◽  
Vol 63 (8) ◽  
pp. 1161-1176 ◽  
Author(s):  
Jan Dvořák ◽  
Lubomír Skála
Keyword(s):  
Schrödinger Equation ◽  
Excited States ◽  
Analytical Solutions ◽  
Ground State ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Exact Analytical Solutions ◽  
Ground State Energies ◽  
Solvable Problems

It is shown that there are two generalizations of some well-known analytically solvable problems leading to exact analytical solutions of the Schrödinger equation for the ground state and a few low lying excited states. In this paper, the ground state energies and wave functions are discussed.

scholarly journals Two-dimensional boson oscillator under a magnetic field in the presence of a minimal length in the non-commutative space

2021 ◽  
Vol 67 (2 Mar-Apr) ◽  
pp. 226
Author(s):  
Z. Selema ◽  
A. Boumal
Keyword(s):  
Magnetic Field ◽  
Schrödinger Equation ◽  
Momentum Space ◽  
Schrodinger Equation ◽  
Minimal Length ◽  
Wave Functions ◽  
Two Dimensional ◽  
Commutative Space ◽  
Klein Gordon

Minimal length in non-commutative space of a two-dimensional Klein-Gordon oscillator isinvestigated and illustrates the wave functions in the momentum space. The eigensolutionsare found and the system is mapping to the well-known Schrodinger equation in a Pöschl-Teller potential.

scholarly journals Q-Deformed Morse and Oscillator Potential

2017 ◽  
Vol 2017 ◽  
pp. 1-4 ◽  
Author(s):  
H. Hassanabadi ◽  
W. S. Chung ◽  
S. Zare ◽  
S. B. Bhardwaj
Keyword(s):  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Analytic Solutions ◽  
Wave Functions ◽  
Oscillator Potential ◽  
Energy Eigenvalues

We studied the q-deformed Morse and harmonic oscillator systems with appropriate canonical commutation algebra. The analytic solutions for eigenfunctions and energy eigenvalues are worked out using time-independent Schrödinger equation and it is also noted that these wave functions are sensitive to variation in the parameters involved.

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.

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