# Harmonic Oscillator: Definition, Applications, and Quantum Mechanics

Classified in Physics

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## Harmonic Oscillator

In classical mechanics, a harmonic oscillator (also known as a linear oscillator or simple oscillator) is a physical system bound to a position of stable equilibrium by a restoring force proportional to the displacement from this position. A typical example of a harmonic oscillator is a mass attached to a spring. The restoring force is the elastic force F given by Hooke’s law:

*F* = −*kx*,

where *x* is the displacement and *k* is the spring constant. The motion of a body of mass *m* attached to the spring is governed by Newton’s second law:

[*m**(d^{2}/dt^{2})] * *x*(*t*) = −*kx*

whose general solution is:

*x*(*t*) = *A*cos(*ωt* + *φ*).

Here, *ω* = rad(*k*/*m*) is the natural oscillating frequency, *A* is the amplitude of the oscillation, and *φ* is the phase constant; both *A* and *φ* are constants determined by the initial condition (initial displacement and velocity).

### Importance of the Harmonic Oscillator

The harmonic oscillator is one of the most important models in mechanics because any potential *V* (*x*) can be approximated as a harmonic potential in the vicinity of a stable equilibrium point.

### Applications of the Quantum Oscillator

- Study of the oscillations of the atoms of a molecule about their equilibrium position
- Study of the oscillations of atoms of a crystalline lattice
- Study of the electromagnetic field of radiation

## Potential Well

The region surrounding the local minimum of potential energy is called a potential well.

Classically, a particle having the potential energy *V* (*x*) is subjected to the force *F _{x}*(

*x*) = −(d/dx)*

*V*(

*x*).

The force is zero for the local minimum position, *x*_{0}; this position is an equilibrium one. If the particle is displaced from *x*_{0}, the force tends to bring the particle back toward this position. The energy of the particle is *E* ≥ *V*_{min}, and the motion takes place between the abscissas *x*_{min} and *x*_{max} determined as solutions of the equation *V* (*x*) = *E*. It is said that the particle is in a **bound state**.

Suppose now a potential that is bounded above and a particle of energy *E* such that *V* (*x*) ≤ *E* for all *x*-values. The classical mechanics treatment of the motion allows the particle to be found in whatever position on the axis; the state of the particle is called an **unbound state**.