On fractional fourier transform and the schrodinger equation of harmonic oscillation

G. S. Agarwal, R. Simon

Joachim Toft, Divyang G. Bhimani, Ramesh Manna

We show that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties for such operators on modulation spaces. Especially we extend some known results in the literature. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable Pilipovic spaces.

Nga T. T. Nguyen, Rolando D. Somma

Ari K. Ahmed, Hemn G. Hussein

Laplace transformation has been used for solving Schrödinger equation in momentum space without using integral equation (Fourier transformation) because it is much harder and more complicated. Furthermore, energy spectra and particle distributions have been demonstrated by using Schrödinger differential equation without ignoring dimensions or units. In addition, a maple program has also been considered to solve the method since it can help to pave a way for other models to work in quantum mechanics.

Muhammad I. Bhatti

Natig M. Atakishiyev, Anatoliy U. Klimyk, Kurt Bernardo Wolf

The finite q -oscillator is a model that obeys the dynamics of the harmonic oscillator, with the operators of position, momentum and Hamiltonian being functions of elements of the q -algebra su q (2). The spectrum of position in this discrete system, in a fixed representation j , consists of 2 j + 1 'sensor'-points , and similarly for the momentum observable. The spectrum of energies is finite and equally spaced, so the system supports coherent states. The wavefunctions involve dual q -Kravchuk polynomials, which are solutions to a finite-difference Schrödinger equation. Time evolution (times a phase) defines the fractional Fourier– q -Kravchuk transform. In the classical limit as q → 1 we recover the finite oscillator Lie algebra, the N = 2 j → ∞ limit returns the Macfarlane–Biedenharn q -oscillator and both limits contract the generators to the standard quantum-mechanical harmonic oscillator.