scholarly journals A class of exactly solvable models for the Schrödinger equation

Open Physics ◽  
2013 ◽  
Vol 11 (8) ◽  
Author(s):  
Charles Downing
Keyword(s):  
Differential Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Mathematical Physics ◽  
Transcendental Equation ◽  
Exactly Solvable Models ◽  
Solvable Models ◽  
One Dimensional ◽  
Exactly Solvable ◽  
The One

AbstractWe present a class of confining potentials which allow one to reduce the one-dimensional Schrödinger equation to a named equation of mathematical physics, namely either Bessel’s or Whittaker’s differential equation. In all cases, we provide closed form expressions for both the symmetric and antisymmetric wavefunction solutions, each along with an associated transcendental equation for allowed eigenvalues. The class of potentials considered contains an example of both cusp-like single wells and a double-well.

scholarly journals Quantization of the one-dimensional free harmonic oscillator as an example of the application of the node theorem and the Macdonald-Hylleraas-Undheim theorem

2020 ◽  
Vol 22 (1) ◽  
pp. 87-90
Author(s):  
Kunle Adegoke ◽  
A. Olatinwo
Keyword(s):  
Differential Equation ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Hermite Polynomials ◽  
Traditional Methods ◽  
One Dimensional ◽  
Energy Eigenvalues ◽  
The One

Using heuristic arguments alone, based on the properties of the  wavefunctions, the energy eigenvalues and the corresponding eigenfunctions of the one-dimensional harmonic oscillator are obtained. This approach is considerably simpler and is perhaps more intuitive than the traditional methods of solving a differential equation and manipulating operators. Keywords: Time-independent Schrödinger equation, MacDonald-Hylleraas-Undheim theorem, Node theorem, Hermite polynomials,  energy eigenvalues

scholarly journals SUPERSYMMETRIC APPROACH TO EXACTLY SOLVABLE SYSTEMS WITH POSITION-DEPENDENT EFFECTIVE MASSES

2002 ◽  
Vol 17 (31) ◽  
pp. 2057-2066 ◽  
Author(s):  
BEŞİ GÖNÜL ◽  
BÜLENT GÖNÜL ◽  
DİLEK TUTCU ◽  
OKAN ÖZER
Keyword(s):  
Schrödinger Equation ◽  
Effective Mass ◽  
Schrodinger Equation ◽  
Hamiltonian Operator ◽  
One Dimensional ◽  
Exact Solvability ◽  
Exactly Solvable ◽  
The One ◽  
Dependent Mass ◽  
The Relationship

We discuss the relationship between exact solvability of the Schrödinger equation with a position-dependent mass and the ordering ambiguity in the Hamiltonian operator within the framework of supersymmetric quantum mechanics. The one-dimensional Schrödinger equation, derived from the general form of the effective mass Hamiltonian, is solved exactly for a system with exponentially changing mass in the presence of a potential with similar behaviour, and the corresponding supersymmetric partner Hamiltonians are related to the effective-mass Hamiltonians proposed in the literature.

A TWELFTH-ORDER FOUR-STEP FORMULA FOR THE NUMERICAL INTEGRATION OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1087-1105 ◽  
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Numerical Test ◽  
New Method ◽  
Step Method ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Order Four

A new twelfth-order four-step formula containing fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation has been developed. It was found that by adding multi-derivative terms, the stability of a linear multi-step method can be improved and the interval of periodicity of this new method is larger than that of the Numerov's method. The numerical test shows that the new method is superior to the previous lower orders in both accuracy and efficiency and it is specially applied to the problem when an increasing accuracy is requested.

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