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In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x: where k is a positive constant. If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion ...
In mechanics and physics, simple harmonic motion (sometimes abbreviated SHM) is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an oscillation that is ...
To see an example where Liouville's theorem does not apply, we can modify the equations of motion for the simple harmonic oscillator to account for the effects of friction or damping. Consider again the system of N {\displaystyle N} particles each in a 3 {\displaystyle 3} -dimensional isotropic harmonic potential, the Hamiltonian for which is ...
The Wigner function of a simple harmonic oscillator at different levels of excitations. The ( q , p ) {\displaystyle (q,p)} are rescaled by n + 1 {\displaystyle {\sqrt {n+1}}} in order to show that the Wigner function oscillates within that radius, and decays rapidly outside of that radius.
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point , it is one of the most important model systems in quantum mechanics.
The harmonic oscillator is an important case. Finding the matrices is easier than determining the general conditions from these special forms. For this reason, Heisenberg investigated the anharmonic oscillator , with Hamiltonian H = 1 2 P 2 + 1 2 X 2 + ε X 3 . {\displaystyle H={1 \over 2}P^{2}+{1 \over 2}X^{2}+\varepsilon X^{3}~.}
Here we show that a necessary condition for stable, exactly closed non-circular orbits is an inverse-square force or radial harmonic oscillator potential. In the following sections, we show that those two force laws produce stable, exactly closed orbits (a sufficient condition ) [it is unclear to the reader exactly what is the sufficient ...
Quantum field theory. In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum ...