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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.
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.
However, many approximate methods work well: Expansion in a Fourier series may provide an equation of motion to arbitrary precision. The term, also called the Duffing term, can be approximated as small and the system treated as a perturbed simple harmonic oscillator. The Frobenius method yields a complex but workable solution.
The ladder operators of the quantum harmonic oscillator or the "number representation" of second quantization are just special cases of this fact. Ladder operators then become ubiquitous in quantum mechanics from the angular momentum operator , to coherent states and to discrete magnetic translation operators.
Simple harmonic motion can be considered the one-dimensional projection of uniform circular motion. If an object moves with angular speed ω around a circle of radius r centered at the origin of the xy-plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.
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 .
Relaxation oscillation in the Van der Pol oscillator without external forcing. The nonlinear damping parameter is equal to μ = 5. [12] When μ = 0, i.e. there is no damping function, the equation becomes + = This is a form of the simple harmonic oscillator, and there is always conservation of energy.
The Rabi problem concerns the response of an atom to an applied harmonic electric field, with an applied frequency very close to the atom's natural frequency. It provides a simple and generally solvable example of light–atom interactions and is named after Isidor Isaac Rabi.