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 mv^{2}…………eq(2)
U = 1/2 kx^{2}………… eq(3)
The average values of kinetic and potential energy per oscillation cycle is equal to
Where v^{2} = w^{2}(A^{2}-x^{2}) ……. eq(4)
Substitute eq(4) in eq(2) and eq(3) will get
k = 1/2 m [w^{2}(A^{2}-x^{2})]
= 1/2 m [Aw cos(wt+ø_{0})]^{2}……. eq(5)
U = 1/2 kx^{2}
= 1/2 k [A sin(wt+ø_{0})]^{2}……. eq(6)
Substitute eq(5) and eq(6) in eq(1) will get the total energy value
E = 1/2 m [w^{2} (A^{2}-x^{2})]+ 1/2 kx^{2}
= 1/2 m w^{2}-1/2 m w^{2}A^{2}+ 1/2 kx^{2}
^{ }= 1/2 m w^{2}A^{2}+1/2 x^{2}(K-mw^{2})……. eq(7)
Where mw^{2 }= K, substitute this value in eq(7)
E = 1/2 K A^{2}- 1/2 Kx^{2 }+ 1/2 x^{2 }= 1/2 K A^{2}
Total energy (E) = 1/2 K A^{2}
Average energies for one time period is expressed as
K_{avg} = U_{avg} = 1/2 (1/2 K A^{2})
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
-ђ^{2}/2μ*d^{2}ѱ_{υ}(Q)/dQ^{2}+ 1/2KQ^{2}ѱ_{υ}(Q) = E_{υ}ѱ_{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
d^{2}ѱ_{υ} (Q)/dQ^{2} + 2μ/ђ^{2}(E_{υ}-K/2 *Q^{2}) ѱ_{υ}(Q) = 0 ………… eq(3)
Parameters used to solve an equation is
β= ђ/√μk ……….. eq (4)
d^{2}/dQ^{2} = 1/β^{2 }d^{2}/dx^{2 }………….. eq(5)
Substitute eq (4) and eq (5) in eq (3), then the differential equation for this oscillator becomes
d^{2}ѱ_{υ} (Q)/dx^{2} + (2μβ^{2} E_{υ}/ ђ^{2} – x^{2}) ѱ_{υ}(x) = 0 ……….. eq(6)
The general expression for power series is
ΣC¬nx2 …………. eq(7)
An exponential function is expressed as
exp (-x^{2}/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(x^{2})d/dx^{υ}*exp(-x^{2}) …………….. eq(10)
The normalizing constant is expressed as
N_{υ}= (1/2^{υ}υ!√Π)^{1/2 }…………….eq(11)
The simple harmonic oscillator solution is expressed as
Ѱ_{υ}(x) = N_{υ}H_{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
^{υ} | 0 | 1 | 2 | 3 |
H_{υ}(y) | 1 | 2y | 4y^{2}-2 | 8y^{3}-12y |
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?