In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional...

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually...

displacement from the fixed point is called simple harmonic motion. In the diagram, a simple harmonic oscillator, consisting of a weight attached to one end...

three dimensions, harmonic oscillators behave similarly to one dimension. The simplest example of this is an isotropic oscillator, where the restoring...

two general types of electronic oscillators: the linear or harmonic oscillator, and the nonlinear or relaxation oscillator. The two types are fundamentally...

by the uncertainty principle. Consider a one-dimensional quantum harmonic oscillator. It is possible to express the position and momentum operators in...

the zero-point energy, and the wave function is a Gaussian. The harmonic oscillator, like the particle in a box, illustrates the generic feature of the...

model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum...

applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually denoted...

system from being a harmonic oscillator. An oscillator that is not oscillating in harmonic motion is known as an anharmonic oscillator where the system can...

Fourier coefficients of the classical orbits, the simplest case is the harmonic oscillator, where the classical position and momentum, X(t) and P(t), are sinusoidal...

operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum. There is a relationship between the raising...

are modeled as harmonic oscillators near their equilibria, a derivation of the resonant frequency for a driven, damped harmonic oscillator is shown. An...

semiconductor devices. The timekeeping element in every modern clock is a harmonic oscillator, a physical object (resonator) that vibrates or oscillates at a particular...

potential with a shape somewhere between the square well and the harmonic oscillator. To this potential, a spin-orbit term is added. Even so, the total...

) determines in which basis the description is easiest. For the harmonic oscillator, x and p enter symmetrically, so there it does not matter which description...

-dimensional isotropic harmonic oscillators. That is, each particle in our ensemble can be treated as a simple harmonic oscillator. The Hamiltonian for...

equation for a simple harmonic oscillator with frequency ω = g / L {\displaystyle \omega ={\sqrt {g/L}}} . A harmonic oscillator can be damped, often by...

any type of harmonic oscillator, such as a pendulum, a vibrating molecule or a passive electronic oscillator. Systems of such oscillators arise in many...

promotion of a classical harmonic oscillator to a quantum harmonic oscillator. The displacement of a classical harmonic oscillator is described by x ( t...

reaching exactly 0 velocity. Rather, the initial ensemble of stochastic oscillators approaches a steady state in which the velocity and position are distributed...

A parametric oscillator is a driven harmonic oscillator in which the oscillations are driven by varying some parameters of the system at some frequencies...

freedom of molecules towards its thermodynamic variables. A quantum harmonic oscillator has an energy spectrum characterized by: E j , n = ℏ ω j ( n j +...

be adiabatic invariants. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize...

study and the classical limit. Consider the example of a simple harmonic oscillator initially at rest with amplitude A. Suppose that this system was...

f(r)=-{\frac {dV}{dr}}=-{\frac {k}{r^{2}}}.} The second is the radial harmonic oscillator potential: V ( r ) = 1 2 k r 2 {\displaystyle V(r)={\frac {1}{2}}kr^{2}}...

Their behavior is analogous to translational spring-mass oscillators (see Harmonic oscillator Equivalent systems). The general differential equation of...

to call a  partial  a  harmonic,  the first being actual and the second being theoretical). Oscillators that produce harmonic partials behave somewhat...

described by its Hamiltonian which also describes the system as a harmonic oscillator, or wave function, that fluctuates between various energy states...

_{t}=-{\frac {\nabla ^{2}}{2}}\psi _{t}.} The Lagrangian for the simple harmonic oscillator is L = 1 2 m x ˙ 2 − 1 2 m ω 2 x 2 . {\displaystyle {\mathcal {L}}={\tfrac...