Harmonic Oscillator: Definition, Applications, and Quantum Mechanics

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*(d²/dt²)] * x(t) = −kx

whose general solution is:

x(t) = Acos(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, x₀; this position is an equilibrium one. If the particle is displaced from x₀, 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.

Fig. 2.2 The infinitely deep one-dimensional square potential well of width a.