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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 ...
Simple harmonic motion can serve as a mathematical model for a variety of motions, but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. The motion is sinusoidal in time and demonstrates a single resonant frequency. Other phenomena can be modeled by simple ...
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 ...
Oscillation. An undamped spring–mass system is an oscillatory system. Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current.
The Hooke's atom is a simple model of the helium atom using the quantum harmonic oscillator. Modelling phonons, as discussed above. A charge q {\displaystyle q} with mass m {\displaystyle m} in a uniform magnetic field B {\displaystyle \mathbf {B} } is an example of a one-dimensional quantum harmonic oscillator: Landau quantization .
Starting with the example used in the derivation above, the simple harmonic oscillator has the potential energy function = =, where k is the spring constant of the oscillator and ω = 2π/T is the natural angular frequency of the oscillator.
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 motion is simple harmonic motion where θ 0 is the amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). The corresponding approximate period of the motion is then