scholarly journals COMPLEXIFIER VERSUS FACTORIZATION AND DEFORMATION METHODS FOR GENERATION OF COHERENT STATES OF A 1D NLHO I: MATHEMATICAL CONSTRUCTION

2013 ◽  
Vol 10 (10) ◽  
pp. 1350056 ◽  
Author(s):  
R. ROKNIZADEH ◽  
H. HEYDARI
Keyword(s):  
Exact Solution ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Jacobi Polynomials ◽  
One Dimensional ◽  
The Difference ◽  
Mathematical Construction

Three methods: complexifier, factorization and deformation, for construction of coherent states are presented for one-dimensional nonlinear harmonic oscillator (1D NLHO). Since by exploring the Jacobi polynomials [Formula: see text], bridging the difference between them is possible, we give here also the exact solution of Schrödinger equation of 1D NLHO in terms of Jacobi polynomials.

scholarly journals Schrödinger equation of a particle in a uniformly accelerated frame and the possibility of a new kind of quanta

2015 ◽  
Vol 30 (38) ◽  
pp. 1550182 ◽  
Author(s):  
Sanchari De ◽  
Sutapa Ghosh ◽  
Somenath Chakrabarty
Keyword(s):  
Exact Solution ◽  
Hydrogen Atom ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Minkowski Spacetime ◽  
Quantum Harmonic Oscillator ◽  
One Dimensional ◽  
Spacetime Geometry

In this paper, we have developed a formalism to obtain the Schrödinger equation for a particle in a frame undergoing a uniform acceleration in an otherwise flat Minkowski spacetime geometry. We have presented an exact solution of the equation and obtained the eigenfunctions and the corresponding eigenvalues. It has been observed that the Schrödinger equation can be reduced to a one-dimensional hydrogen atom problem. Whereas, the quantized energy levels are exactly identical with that of a one-dimensional quantum harmonic oscillator. Hence, considering transitions, we have predicted the existence of a new kind of quanta, which will either be emitted or absorbed if the particles get excited or de-excited, respectively.

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

Eigenvalues of the Potential Function V=z4±Bz2 and the Effect of Sixth Power Terms

1970 ◽  
Vol 24 (1) ◽  
pp. 73-80 ◽  
Author(s):  
Jaan Laane
Keyword(s):  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Potential Function ◽  
Schrodinger Equation ◽  
Harmonic Potential ◽  
Reduced Form ◽  
Approximation Formula ◽  
One Dimensional ◽  
Ring Puckering ◽  
The One

The one-dimensional Schrödinger equation in reduced form is solved for the potential function V = z4+ Bz2 where B may be positive or negative. The first 17 eigenvalues are reported for 58 values of B in the range −50⩽ B⩽100. The interval of B between the tabulated values is sufficiently small so that the eigenvalues for any B in this range can be found by interpolation. At the limits of the range of B the potential function approaches that of a harmonic oscillator with only small anharmonicity. The effect of a small Cz6 term on this potential is studied and it is concluded that a previously reported approximation formula is quite applicable but only for positive values of B. The success of the quartic—harmonic potential function for the analysis of the ring-puckering vibration is shown; it is also demonstrated that the same potential serves as a useful approximation for many other systems, especially those of the double minimum type.

THE q-SCHRÖDINGER EQUATION

1990 ◽  
Vol 05 (31) ◽  
pp. 2625-2632 ◽  
Author(s):  
JOSEPH A. MINAHAN
Keyword(s):  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Hermite Polynomials ◽  
Normal Derivative ◽  
One Dimensional ◽  
The One

We consider the Schrödinger equation for the one-dimensional harmonic oscillator, but with the normal derivative replaced by a q-derivative. The normalizable solutions are found and the q-generalization of the Hermite polynomials is given. The free equation is also considered, but no normalizable eigenstates exist even if the system is in a box.

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