What is a Harmonic Oscillator : Block Diagram and Its Types

The simple harmonic motion is invented by French Mathematician Baron Jean Baptiste Joseph Fourier in 1822. Edwin Armstrong (18th DEC 1890 to 1st FEB 1954) observed oscillations in 1992 in their experiments and Alexander Meissner (14th SEP 1883 to 3rd JAN 1958) invented oscillators in March 1993. The term harmonic is a Latin word. This article discusses an overview of the harmonic oscillator which includes its definition, type, and its applications.

What is Harmonic Oscillator?

Harmonic Oscillator is defined as a motion in which force is directly proportional to the particle from the equilibrium point and it produces output in a sinusoidal waveform. The force which causes harmonic motion can be mathematically expressed as

F = -Kx

Where,

F = Restoring force

K = Spring constant

X = Distance from equilibrium

There is a point in harmonic motion in which the system oscillates, and the force which brings the mass again and again at the same point from where it starts, the force is called restoring force and the point is called equilibrium point or mean position. This oscillator is also known as a linear harmonic oscillator. The energy flows from active components to passive components in the oscillator.

Block Diagram

The block diagram of the harmonic oscillator consists of an amplifier and a feedback network. The amplifier is used to amplify the signals and that amplified signals are passed through a feedback network and generates the output. Where Vi is the input voltage, Vo is the output voltage and Vf is the feedback voltage.

Example

Mass on a Spring: The spring provides restoring force that accelerates the mass and the restoring force is expressed as

F = ma

Where ‘m’ is the mass and a is an acceleration.

Spring consists of a mass (m) and force (F). When the force pulls the mass at a point x=0 and depends only on x – position of the mass and the spring constant is represented by a letter k.

Types of Harmonic Oscillator

Types of this oscillator mainly include the following.

Forced Harmonic Oscillator

When we apply external force to the motion of the system, then the motion is said to be a forced harmonic oscillator.

Damped Harmonic Oscillator

This oscillator is defined as, when we apply external force to the system, then the motion of the oscillator reduces and its motion is said to be damped harmonic motion. There are three types of damped harmonic oscillators they are

Over Damped

When the system moves slowly towards the equilibrium point then it is said to be an overdamped harmonic oscillator.

Under Damped

When the system moves quickly towards the equilibrium point then it is said to be an overdamped harmonic oscillator.

Critical Damped

When the system moves quickly as possible without oscillating about the equilibrium point then it is said to be an overdamped harmonic oscillator.

Quantum

It is invented by Max Born, Werner Heisenberg, and Wolfgang Pauli at “University of Gottingen”. The word quantum is the Latin word and the meaning of quantum is a small amount of energy.

Zero Point Energy

The zero-point energy is also known as ground state energy. It is defined when ground state energy is always greater than zero and this concept is discovered by Max Planck in Germany and the formula developed in 1990.

Average Energy of Damped Simple Harmonic Oscillator Equation

There are two types of energies they are kinetic energy and potential energy. The sum of kinetic energy and potential energy is equal to the total energy.

E = K+U ………………. Eq (1)

Where E = Total energy

K = Kinetic energy

U = Potential energy

Where k = k = 1/2 mv²…………eq(2)

U = 1/2 kx²………… eq(3)

The average values of kinetic and potential energy per oscillation cycle is equal to

Where v² = w²(A²-x²) ……. eq(4)

Substitute eq(4) in eq(2) and eq(3) will get

k = 1/2 m [w²(A²-x²)]

 = 1/2 m [Aw cos(wt+ø₀)]²…….  eq(5)

U = 1/2 kx²

= 1/2 k [A sin(wt+ø₀)]²…….  eq(6)

Substitute eq(5) and eq(6) in eq(1) will get the total energy value

E = 1/2 m [w² (A²-x²)]+ 1/2 kx²

= 1/2 m w²-1/2 m w²A²+ 1/2 kx²

                = 1/2 m w²A²+1/2 x²(K-mw²)…….  eq(7)

Where mw² = K, substitute this value in eq(7)

            E = 1/2 K A²­- 1/2 Kx² + 1/2 x² = 1/2 K A²

Total energy (E) = 1/2 K A²

Average energies for one time period is expressed as

K_avg = U_avg = 1/2 (1/2 K A²)

Harmonic Oscillator Wave Function

The Hamiltonian operator is expressed as a sum of kinetic energy and potential energy and it is expressed as

ђ (Q) = T + V……………….eq(1)

Where ђ =Hamitonian operator

T= Kinetic energy

V = Potential energy

To generate the wave function, we have to know the Schrodinger equation and the equation is expressed as

-ђ²/2μ*d²ѱ_υ(Q)/dQ²+ 1/2KQ²ѱ_υ(Q) = E­_υѱ_υ(Q)………….  eq(2)

Where Q = Length of the normal coordinate

Μ = Effective mass

K = Force constant

Schrodinger equation boundary conditions are:

Ѱ(-∞) = ø

Ѱ(+∞) = 0

We can also write the eq (2) as

    d²ѱ_υ (Q)/dQ² + 2μ/ђ²(E­_υ-K/2 *Q²) ѱ_υ(Q) = 0        …………   eq(3)

Parameters used to solve an equation is

 β= ђ/√μk            ………..   eq (4)

d²/dQ² = 1/β² d²/dx²  …………..   eq(5)

Substitute eq (4) and eq (5) in eq (3), then the differential equation for this oscillator becomes

d²ѱ_υ (Q)/dx² + (2μβ² E­_υ/ ђ² – x²) ѱ_υ(x) = 0      ………..   eq(6)

The general expression for power series is

ΣC¬nx2 …………. eq(7)

An exponential function is expressed as

exp (-x²/2)       …………  eq(8)

eq(7) is multiplied with eq(8)

ѱυ(x) = ΣC¬nx2exp (-x2/2) ……………..eq(9)

Hermite polynomials are obtained by using the below equation

ђ­_υ^­­­(x) = (-1)^υ*exp(x²)d/dx^υ*exp(-x²)   ……………..  eq(10)

The normalizing constant is expressed as

                                                          N_υ= (1/2^υυ!√Π)^1/2   …………….eq(11)

The simple harmonic oscillator solution is expressed as

Ѱ_υ(x) = N_υH_υ(y)e^-x2/2………………eq(12)

Where N_υ­ is the Normalization constant

H_υ­­­ is the Hermite

 e^-x2/2 is the Gaussian

An equation (12) is the wave function of the harmonic oscillator.

This table shows the first term Hermite polynomials for the lowest energy states

The wave functions of the simple harmonic oscillator graph for four lowest energy states are shown in the below figures.

The probability densities of this oscillator for the four lowest energy states are shown in the below figures.

Applications

The simple harmonic oscillator applications mainly include the following

• Audio and Video systems
• Radio and other communication devices
• Inverters, Alarms
• Buzzers
• Decorative lights

Advantages

The advantages of the harmonic oscillator  are

• Cheap
• High-frequency generation
• High efficiency
• Cheap
• Portable
• Economical

Examples

The example of this oscillator includes the following.

• Musical instruments
• Simple pendulum
• Mass spring system
• Swing
• The motion of the hands of the clock
• The motion of the wheels of car, lorry, buses, etc

It is one type of motion, that we can observe on our daily bases. Harmonic oscillator wave function using Schrodinger and equations of the harmonic oscillator are derived. Here is a question, what type of motion performed by bungee jumping?