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The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases. The velocity is maximal for zero displacement, while the acceleration is in the direction opposite to the displacement.
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.
Harmonic oscillator; Inertial / Non-inertial reference frame; Motion Newton's law of universal gravitation; ... where g is the acceleration due to gravity.
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).
English: In harmonic oscillatory motion, the acceleration is proportional to the restoring force. This is demonstrated here using Pasco's Visual Accelerometer; full scale is set at 1 m/s². Of course, in this damped oscillator, there is also friction.
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.
For a simple harmonic oscillator released from rest, regardless of its initial displacement, the time it takes to reach the lowest potential energy point is always a quarter of its period, which is independent of its amplitude. Therefore, the Lagrangian of a simple harmonic oscillator is isochronous.