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A simple harmonic oscillator is an oscillator that is neither driven nor damped.It consists of a mass m, which experiences a single force F, which pulls the mass in the direction of the point x = 0 and depends only on the position x of the mass and a constant k.
In mechanics and physics, simple harmonic motion (sometimes abbreviated as 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.
Therefore, the Lagrangian of a simple harmonic oscillator is isochronous. In the tautochrone problem, if the particle's position is parametrized by the arclength s ( t ) from the lowest point, the kinetic energy is then proportional to s ˙ 2 {\displaystyle {\dot {s}}^{2}} , and the potential energy is proportional to the height h ( s ) .
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 systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the spring-mass system, are described mathematically by the simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion.
In the semiclassical picture of vibrations (oscillations) of a simple harmonic oscillator, the necessary conditions can occur at the turning points, where the momentum is zero. Classically, the Franck–Condon principle is the approximation that an electronic transition is most likely to occur without changes in the positions of the nuclei in ...
Increase of amplitude as damping decreases and frequency approaches resonant frequency of a driven damped simple harmonic oscillator. [1] [2]Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration that matches its natural frequency.
Simple pendulum, see picture (right). Simple harmonic oscillator where the phase portrait is made up of ellipses centred at the origin, which is a fixed point. Damped harmonic motion, see animation (right). Van der Pol oscillator see picture (bottom right).