Starting from a discrete Heisenberg algebra we solve several representation problems for a discretized quantum oscillator in a weighted sequence space. The Schrödinger operator for a discrete harmonic oscillator is derived. The representation problem for a q‐oscillator algebra is studied in detail. The main result of the article is the fact that the energy representation for the discretized momentum operator can be interpreted as follows: It allows to calculate quantum properties of a large number of non‐interacting harmonic oscillators at the same time. The results can be directly related to current research on squeezed laser states in quantum optics. They reveal and confirm the observation that discrete versions of continuum Schrodinger operators allow more structural freedom than their continuum analogs do.Starting from a discrete Heisenberg algebra we solve several representation problems for a discretized quantum oscillator in a weighted sequence space. The Schrödinger operator for a discrete harmonic oscillator is derived. The representation problem for a q‐oscillator algebra is studied in detail. The main result of the article is the fact that the energy representation for the discretized momentum operator can be interpreted as follows: It allows to calculate quantum properties of a large number of non‐interacting harmonic oscillators at the same time. The results can be directly related to current research on squeezed laser states in quantum optics. They reveal and confirm the observation that discrete versions of continuum Schrodinger operators allow more structural freedom than their continuum analogs do.

Discrete Dynamics in Nature and SocietyJan 1, 20000 citations

Application study of virtual time evolution-split operator method to harmonic oscillator

CHEN Yan, ZHANG Ke-Zhi, GE Su-hong +3 more

We present virtual time evolution-split operator method for solving the two-dimensional time- dependent Schrodinger equation. In this method, the Hamiltonian is accessed by employing the two representations of the wave function. One is a coordinate representations, in which the coordinate dependence of the wave function is discretized using a discrete variable constructed from the Coulomb wave function. Another is the momentum representation, the time function is expanded in the corresponding. As an example, the present method is applied to the harmonic oscillator. It is found that there exists coulomb interaction when two electrons are stored in the harmonic oscillator potential well. We explore the qualitative change of two electrons ground state wave function by altering the strength of the harmonic oscillator potential. When harmonic oscillator potential is weak, the wave functions are not overlapped, which can be used an evidence for presentence of Wingner lattice. With the increase of harmonic oscillator potential, the wave functions begin to overlap, which is similar to the mokecules formation.We present virtual time evolution-split operator method for solving the two-dimensional time- dependent Schrodinger equation. In this method, the Hamiltonian is accessed by employing the two representations of the wave function. One is a coordinate representations, in which the coordinate dependence of the wave function is discretized using a discrete variable constructed from the Coulomb wave function. Another is the momentum representation, the time function is expanded in the corresponding. As an example, the present method is applied to the harmonic oscillator. It is found that there exists coulomb interaction when two electrons are stored in the harmonic oscillator potential well. We explore the qualitative change of two electrons ground state wave function by altering the strength of the harmonic oscillator potential. When harmonic oscillator potential is weak, the wave functions are not overlapped, which can be used an evidence for presentence of Wingner lattice. With the increase of harmonic oscillator potential, the wave functions begin to overlap, which is similar to the mokecules formation.

Jan 1, 20180 citations

HARMONIC OSCILLATOR

Mitchel Weissbluth

Elsevier eBooksJan 1, 19780 citations

A New Description of Quantum Behaviors for a Simple Harmonic Oscillator

Sangrak Kim

We describe quantum behaviors of a simple harmonic oscillator, starting from the classical mechanics. By imposing two conditions on the phase points generated from a symplectic algorithm, we obtain discrete energy levels, satisfying $E_n τ_n = h$ where $E_n$ is the total energy of the oscillator and $τ_n$ is the time step for the closed orbit of $n$ -polygon in phase space. We can thus successfully integrate classical and quantum mechanics into a single frame, if we assume that time is discrete.We describe quantum behaviors of a simple harmonic oscillator, starting from the classical mechanics. By imposing two conditions on the phase points generated from a symplectic algorithm, we obtain discrete energy levels, satisfying $E_n τ_n = h$ where $E_n$ is the total energy of the oscillator and $τ_n$ is the time step for the closed orbit of $n$-polygon in phase space. We can thus successfully integrate classical and quantum mechanics into a single frame, if we assume that time is discrete.

arXiv:1307.0163Jun 30, 20130 citations

Harmonic Oscillators

Julian Schwinger

Springer eBooksJan 1, 20010 citations

E_n τ_n = h

where

E_n

is the total energy of the oscillator and

τ_n

is the time step for the closed orbit of

n

-polygon in phase space. We can thus successfully integrate classical and quantum mechanics into a single frame, if we assume that time is discrete.