Related papers: Wallis formula from the harmonic oscillator
The Dunkl Laplacian is used to define the Hamiltonian of a modified quantum harmonic oscillator, associated with any finite reflection group. The potential is a sum of the inverse squares of the linear functions whose zero sets are the…
The damped harmonic oscillator is a workhorse for the study of dissipation in quantum mechanics. However, despite its simplicity, this system has given rise to some approximations whose validity and relation to more refined descriptions…
For systems whose classical dynamics is chaotic, it is generally believed that the local statistical properties of the quantum energy levels are well described by Random Matrix Theory. We present here two counterexamples - the hydrogen atom…
An asymptotic formula for the number of states of Boson gas whose Hamiltonian is given by a positive elliptic pseudo-differential operator of order one on a compact manifold is given under a integrality assumption on the spectrum of the…
It is shown that general solutions of the free-particle Schroedinger equation can be mapped onto solutions of the Schroedinger equation for the harmonic oscillator. This is done in such a way that the time evolution of a free particle…
We study the Wigner Function in non-commutative quantum mechanics. By solving the time independent Schr\"{o}dinger equation both on a non-commutative (NC) space and a non-commutative phase space, we obtain the Wigner Function for the…
We study the known coherent states of a quantum harmonic oscillator from the standpoint of the original developed noncommutative integration method for linear partial differential equations. The application of the method is based on the…
The ongoing discussion whether thermodynamic properties can be extracted from a (possibly approximate) quantum mechanical time evolution using time averages is fed with an instructive example. It is shown for the harmonic oscillator how the…
The numerical version of the Hamilton-Jacobi quantization method, recently proposed, is applied to the one dimensional quartic oscillator. A suitable quantization condition is formulated and various energy levels and wave functions are…
This work is mainly based on some theoretical surveys on two dimensional quantum gravitational well, considering harmonic oscillator potential causes an effective plank constant. We find that there is a similarity between two different…
This paper presents a detailed analysis of the radial uncertainty product for quantum systems with spherically symmetric potentials. Using the principles of quantum mechanics, the study derives the radial uncertainty relation analogous to…
Today it still remains a challenge whether quantum mechanics has an underlying statistical explanation or not. While there are and were a lot of models trying to explain quantum phenomena with statistical methods these all failed on certain…
We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials. This problem is of fundamental interest in electronic…
We present the Hamiltonian formulation of a relativistic point-particle coupled to Einstein gravity and its canonical quantization \`a la Wheeler-DeWitt. In the resulting quantum theory, the wave functional is a function of the particle…
This paper presents a comprehensive investigation of the problem of a harmonic oscillator with time-depending frequencies in the framework of the Vlasov theory and the Wigner function apparatus for quantum systems in the phase space. A new…
We continue our work on the study of spherically symmetric loop quantum gravity coupled to two spherically symmetric scalar fields, one that acts as a clock. As a consequence of the presence of the latter, we can define a true Hamiltonian…
A harmonic oscillator model in four dimensions is presented for the helium atom to estimate the distance to the inner and outer electron from the nucleus, the angle between electrons and the energy levels. The method is algebraic and is not…
We show that this problem gives rise to the same differential equation of a well known potential of ordinary quantum mechanics. However there is a subtle difference in the choice of the parameters of the hypergeometric function solving the…
We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics…
We show that the Schr\"{o}dinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals $O( m / \hbar)$,…