# Harmonic Oscillator

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: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).

If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:

• Oscillate with a frequency smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator).
• Decay to the equilibrium position, without oscillations (overdamped oscillator).

The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called "critically damped."

If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator.

Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.

### Other articles related to "harmonic oscillator, harmonic":

Harmonic Oscillator - Examples - Spring/mass System - Energy Variation in The Springâ€“damping System
... When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring ... By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximum potential energy, the kinetic energy of the mass is zero ...
Landau Quantization - Derivation
... This is exactly the Hamiltonian for the quantum harmonic oscillator, except shifted in coordinate space by ... To find the energies, note that translating the harmonic oscillator potential left or right does not change the energies ... of this system are identical to those of the quantum harmonic oscillator The energy does not depend on the quantum number, so there will be degeneracies ...
Quantum Harmonic Oscillator
... The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator ... Because an arbitrary potential can 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 ...
Nondimensionalization - Nonlinear Differential Equation Example - Quantum Harmonic Oscillator
... for the one dimensional time independent quantum harmonic oscillator is The modulus square of the wavefunction

### Famous quotes containing the word harmonic:

For decades child development experts have erroneously directed parents to sing with one voice, a unison chorus of values, politics, disciplinary and loving styles. But duets have greater harmonic possibilities and are more interesting to listen to, so long as cacophony or dissonance remains at acceptable levels.
Kyle D. Pruett (20th century)