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Problems concerning with application of quantum rules on classical phenomena have been widely studied, for which lifted up the idea about quantization and uncertainty principle. Energy quantization on classical example of simple harmonic…
By factorization of the Hamiltonian describing the quantum mechanics of the continuous q-Hermite polynomial, the creation and annihilation operators of the q-oscillator are obtained. They satisfy a q-oscillator algebra as a consequence of…
In this work we address the problem of the quantization of a simple harmonic oscillator that is perturbed by a time dependent force. The approach consists of removing the perturbation by a canonical change of coordinates. Since the…
In this article we obtained the harmonic oscillator solution for quaternionic quantum mechanics ($\mathbbm{H}$QM) in the real Hilbert space, both in the analytic method and in the algebraic method. The quaternionic solutions have many…
We consider the relativistic generalization of the harmonic oscillator problem by addressing different questions regarding its classical aspects. We treat the problem using the formalism of Hamiltonian mechanics. A Lie algebraic technique…
The problem of the quantum harmonic oscillator is investigated in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero divisors. Starting with the commutator of the bicomplex position…
We present a full algebraic derivation of the wavefunctions of the simple harmonic oscillator in coordinate and momentum space. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation…
We have studied the underlying algebraic structure of the anharmonic oscillator by using the variational perturbation theory. To the first order of the variational perturbation, the Hamiltonian is found to be factorized into a…
We perform a perturbative calculation of the physical observables, in particular pseudo-Hermitian position and momentum operators, the equivalent Hermitian Hamiltonian operator, and the classical Hamiltonian for the PT-symmetric cubic…
In this work we show how to complete some Hamilton-Jacobi solutions of linear, nonconservative classical oscillatory systems which appeared in the literature and we extend these complete solutions to the quantum mechanical case. In…
The response of a test particle, both for the free case and under the harmonic oscillator potential, to circularly polarized gravitational waves is investigated in a noncommutative quantum mechanical setting. The system is quantized…
A linear quantum harmonic oscillator factors into one dimensional oscillators and can be solved using creation and annihilation operators. We consider a spherical analogue. This analogue does not factor. The two dimensional case is…
Classical oscillators of sextic and octic anharmonicities are solved analytically up to the linear power of \lambda (Anharmonic Constant) by using Taylor series method. These solutions exhibit the presence of secular terms which are summed…
We discuss the classical and quantum mechanical evolution of systems described by a Hamiltonian that is a function of a solvable one, both classically and quantum mechanically. The case in which the solvable Hamiltonian corresponds to the…
We show that the algebraic method solving the ordinary harmonic oscillator can be adapted to the non-commutative case.
We show how several important classical problems, with positive definite potential energy, can be solved by starting from the factorization of the total mechanical energy using complex numbers. In particular, we derive in a new way exact…
Keeping in view the ordering ambiguity that arises due to the presence of position-dependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with…
We consider a set of N linearly coupled harmonic oscillators and show that the diagonalization of this problem can be put in geometrical terms. The matrix techniques developed here allowed for solutions in both the classical and quantum…
Courses on undergraduate quantum mechanics usually focus on solutions of the Schr\"odinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced only academic examples are used, such as the…
We construct a generalised expression for the normal ordering of (a+a^{\dagger})^{m} for integral values of m and use the result to study the quantum anharmonic oscillator problem in the Heisenberg approach. In particular, we derive…