Energy Levels of Harmonic Oscillator in Classical-Like Formulation of Quantum Mechanics

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 oscillator in a variety of forms of classical and quantum mechanics

Petr Česnek

Jan 1, 20060 citations

Wave mechanics equivalent to classical mechanics

Dennis Shay

A nonlinear interaction term is added to the Schr\"odinger equation which makes it exactly equivalent to classical mechanics. In the hydrodynamic formulation of this equation, the usual classical theory is obtained. This theory is a Lagrangian density formulation of hydrodynamics and is used to calculate the density profile of superfluid helium in a line vortex. The theory can be canonically quantized, in the sense of quantum field theory, in terms of the density and velocity fields. The quantized theory is not equivalent to that which one would obtain if the quantization were done in the usual way in terms of the field operations ${\ensuremath{\psi}}^{*}$ and ($\ensuremath{\psi}$). The Schr\"odinger equation, with this interaction, has localized solutions with a definite energy and momentum and does not give quantized energy levels for a one-dimensional harmonic oscillator.A nonlinear interaction term is added to the Schr\"odinger equation which makes it exactly equivalent to classical mechanics. In the hydrodynamic formulation of this equation, the usual classical theory is obtained. This theory is a Lagrangian density formulation of hydrodynamics and is used to calculate the density profile of superfluid helium in a line vortex. The theory can be canonically quantized, in the sense of quantum field theory, in terms of the density and velocity fields. The quantized theory is not equivalent to that which one would obtain if the quantization were done in the usual way in terms of the field operations ${\ensuremath{\psi}}^{*}$ and ($\ensuremath{\psi}$). The Schr\"odinger equation, with this interaction, has localized solutions with a definite energy and momentum and does not give quantized energy levels for a one-dimensional harmonic oscillator.

Physical Review AJun 1, 19762 citations

Quantum Mechanics of the Harmonic Oscillator

WORLD SCIENTIFIC eBooksSep 1, 20121 citation

Symplectic structures and dynamical symmetry groups.

G. F. Torres del Castillo, Mercedes Paulina Velázquez-Quesada

Apart from the total energy, the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, with the standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2). We show that, by suitably redefining the symplectic structure, any of these three constants of motion can be used as a Hamiltonian, and that the remaining two, together with the total energy, generate a dynamical symmetry group isomorphic to SU(1,1). We also show that the standard energy levels of the quantum two-dimensional isotropic harmonic oscillator and their degeneracies are obtained making use of the appropriate representations of SU(1,1), provided that the canonical commutation relations are modified according to the new symplectic structure. Whereas in classical mechanics the different symplectic structures lead to equivalent formulations of the equations of motion, in quantum mechanics the modifications of the commutation relations should be accompanied by modifications in the interpretation of the formalism in order to obtain results equivalent to those found with the common relations.Apart from the total energy, the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, with the standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2). We show that, by suitably redefining the symplectic structure, any of these three constants of motion can be used as a Hamiltonian, and that the remaining two, together with the total energy, generate a dynamical symmetry group isomorphic to SU(1,1). We also show that the standard energy levels of the quantum two-dimensional isotropic harmonic oscillator and their degeneracies are obtained making use of the appropriate representations of SU(1,1), provided that the canonical commutation relations are modified according to the new symplectic structure. Whereas in classical mechanics the different symplectic structures lead to equivalent formulations of the equations of motion, in quantum mechanics the modifications of the commutation relations should be accompanied by modifications in the interpretation of the formalism in order to obtain results equivalent to those found with the common relations.

Revista Mexicana de FísicaDec 1, 20048 citations

Quantum Mechanics of Harmonic Oscillators

Thomas A. Garrity

Cambridge University Press eBooksJan 19, 20150 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.