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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.
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 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.
In order to calculate the internal energy and the specific heat, we must know the number of normal vibrational modes a frequency between the values ν and ν + dν. Allow this number to be f(ν)dν. Since the total number of normal modes is 3N, the function f(ν) is given by: =
In the context of the quantum harmonic oscillator, one reinterprets the ladder operators as creation and annihilation operators, adding or subtracting fixed quanta of energy to the oscillator system. Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin).
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 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.
The zero-point energy E = ħω / 2 causes the ground-state of a harmonic oscillator to advance its phase (color). This has measurable effects when several eigenstates are superimposed. The idea of a quantum harmonic oscillator and its associated energy can apply to either an atom or a subatomic particle.