What is a Harmonic Oscillator : Block Diagram and Its TypesThe 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 = -KxWhere,F = Restoring forceK = Spring constantX = Distance from equilibriumblock-diagram-of-harmonic-oscillatorThere 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 DiagramThe 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.ExampleMass on a Spring: The spring provides restoring force that accelerates the mass and the restoring force is expressed asF = maWhere ‘m’ is the mass and a is an acceleration.mass-on-a-springSpring 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 OscillatorTypes of this oscillator mainly include the following.Forced Harmonic OscillatorWhen we apply external force to the motion of the system, then the motion is said to be a forced harmonic oscillator.Damped Harmonic OscillatorThis 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 aredamping-waveformsOver DampedWhen the system moves slowly towards the equilibrium point then it is said to be an overdamped harmonic oscillator.Under DampedWhen the system moves quickly towards the equilibrium point then it is said to be an overdamped harmonic oscillator.Critical DampedWhen the system moves quickly as possible without oscillating about the equilibrium point then it is said to be an overdamped harmonic oscillator.QuantumIt 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 EnergyThe 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 EquationThere 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 energyK = Kinetic energyU = Potential energyWhere k = k = 1/2 mv2…………eq(2)U = 1/2 kx2………… eq(3)oscillation-cycle- for- average- valuesThe average values of kinetic and potential energy per oscillation cycle is equal toWhere v2 = w2(A2-x2) ……. eq(4)Substitute eq(4) in eq(2) and eq(3) will getk = 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 valueE = 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 A2Total energy (E) = 1/2 K A2Average energies for one time period is expressed asKavg = Uavg = 1/2 (1/2 K A2)Harmonic Oscillator Wave FunctionThe Hamiltonian operator is expressed as a sum of kinetic energy and potential energy and it is expressed asђ (Q) = T + V……………….eq(1)Where ђ =Hamitonian operatorT= Kinetic energyV = Potential energyTo generate the wave function, we have to know the Schrodinger equation and the equation is expressed as-ђ2/2μ*d2ѱυ(Q)/dQ2+ 1/2KQ2ѱυ(Q) = Eυѱυ(Q)…………. eq(2)Where Q = Length of the normal coordinateΜ = Effective massK = Force constantSchrodinger equation boundary conditions are:Ѱ(-∞) = øѰ(+∞) = 0We can also write the eq (2) as d2ѱυ (Q)/dQ2 + 2μ/ђ2(Eυ-K/2 *Q2) ѱυ(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 becomesd2ѱυ (Q)/dx2 + (2μβ2 Eυ/ ђ2 – x2) ѱυ(x) = 0 ……….. eq(6)The general expression for power series isΣC¬nx2 …………. eq(7)An exponential function is expressed asexp (-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 equationђυ(x) = (-1)υ*exp(x2)d/dxυ*exp(-x2) …………….. eq(10)The normalizing constant is expressed as Nυ= (1/2υυ!√Π)1/2 …………….eq(11)The simple harmonic oscillator solution is expressed asѰυ(x) = NυHυ(y)e-x2/2………………eq(12)Where Nυ is the Normalization constantHυ is the Hermite e-x2/2 is the GaussianAn equation (12) is the wave function of the harmonic oscillator.This table shows the first term Hermite polynomials for the lowest energy statesυ0123Hυ(y)12y4y2-28y3-12yThe wave functions of the simple harmonic oscillator graph for four lowest energy states are shown in the below figures.wave-functions-of-harmonic-oscillatorThe probability densities of this oscillator for the four lowest energy states are shown in the below figures.probability-densities-of -waveformsApplicationsThe simple harmonic oscillator applications mainly include the followingAudio and Video systemsRadio and other communication devicesInverters, AlarmsBuzzersDecorative lightsAdvantagesThe advantages of the harmonic oscillator areCheapHigh-frequency generationHigh efficiencyCheapPortableEconomicalExamplesThe example of this oscillator includes the following.Musical instrumentsSimple pendulumMass spring systemSwingThe motion of the hands of the clockThe motion of the wheels of car, lorry, buses, etcIt 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? Share This Post: Facebook Twitter Google+ LinkedIn Pinterest Post navigation ‹ Previous What is a SSB Modulation and Its ApplicationsNext › What is an LM380 Audio Amplifier Working & Its Applications Related Content What is Static VAR Compensator : Design & Its Working What is a Carey Foster Bridge & Its Working What is a Spectrum Analyzer : Working & Its Applications What is an Inductive Reactance : Definition, Unit and FormulaAdd Comment Cancel replyComment:Name * Email * Website