Harmonic Oscillator

A harmonic oscillator is a type of oscillator, which has several significant applications in classical and quantum mechanics. It functions as a model in the mathematical treatment of diverse phenomena, such as acoustics, molecular-crystal vibrations, AC circuits, elasticity, optical properties, and electromagnetic fields.

What is a Harmonic Oscillator?

When a body oscillates about its location along a linear straight line under the influence of a force that is pointed towards the mean location, and is proportional to the displacement at any moment from this location, the motion of the body is considered to be simple harmonic, and the swinging body is known as a linear harmonic oscillator or simple harmonic oscillator. This form of oscillation is the best example of periodic motion.

At the molecular level, above 0K temperature, the atoms in a crystal are temporarily displaced from their normal locations due to thermal energy intake. Interatomic forces act on the displaced atoms. Under the influence of such restoring forces, individual atoms vibrate about their normal location, which is the correct location in the ideal structure. Therefore, vibrations of the individual atoms are similar to a simple harmonic oscillator.

A harmonic oscillator in classical physics is a body that is being exerted by a restoring force proportional to its displacement from its equilibrium location.

In the case of motion in one dimension,

F=−kx

Hooke’s law is generally applied to real springs for small displacements; the restoring force is usually proportional to the displacement (compression or stretching) from the equilibrium position.

The constant ‘K’ is a measure of the spring’s stiffness. The variable ‘x’ is selected equal to zero at the equilibrium location (negative for compression and positive for stretching). The negative sign shows the fact that ‘F’ is a type of restoring force, opposite to the direction of displacement ‘x’. If ‘F’ is the lone force exerting on the system, it is known as a simple harmonic oscillator. It possesses sinusoidal oscillations about the equilibrium position with a fixed amplitude and a fixed frequency (independent of the amplitude). If a frictional force is present in the system, the harmonic oscillator is called a damped oscillator. Based on the friction coefficient, a body (underdamped oscillator) can oscillate with a frequency lesser than in the undamped scenario, and amplitude descends with time. Here, a body can decay to the equilibrium location in the absence of oscillations (overdamped oscillator). In fact, the boundary solution between an overdamped oscillator and an underdamped oscillator happens at a distinct value of the friction coefficient. It is called critically damped. The harmonic oscillator is termed a driven oscillator if an outside time-dependent force exists.

The video explains the fundamentals of simple harmonic motion.

Harmonic Oscillator Examples

Important mechanical examples include acoustic systems, pendulums with low displacement angles, and springs with weights. There are analogous systems, such as electrical harmonic oscillators (RLC circuits). The model of the harmonic oscillator is very important in physical science. Any body that is subject to a force in steady equilibrium functions as a harmonic oscillator (small vibrations). Harmonic oscillators exist extensively in nature. They have been reverse-engineered into many man-made devices. They are the fundamental sources of almost all sinusoidal waves and vibrations.

Simple Harmonic Oscillator

A simple harmonic oscillator is a type of oscillator that is either damped or driven. It generally consists of a mass’ m’, where a lone force ‘F’ pulls the mass in the trajectory of the point x = 0, and relies only on the position ‘x’ of the body and a constant k. The Balance of forces is,

Solving the differential equation, the function that describes the motion is,

Where

The motion is periodic, recurring itself in a sinusoidal fashion with fixed amplitude ‘A’. The movement of simple harmonic oscillators is characterised by its period

Quantum Model of the Harmonic Oscillator

The quantum harmonic oscillator is the subatomic analogue version of the conventional harmonic oscillator. It is one of the most relevant model systems in quantum physics. A random smooth potential can generally be estimated as a harmonic potential at the locale of a stable equilibrium point. It is one of the rare quantum-mechanical systems, which has an exact known analytical solution. This class of harmonic oscillators is characterised by its Schrödinger Equation. The harmonic oscillator only possesses discrete energy states as is valid of the one-dimensional body in a box problem. It is one of the basic applications of quantum physics that opens up the vast quantum world. Systems with unsolved equations are usually broken into small systems.

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