Harmonic Oscillator

1. THE HARMONIC
OSCILLATOR
By-
Manish Sahu
M.Sc. Chemistry (Final)
Sp.- Physical Chemistry

2.  Introduction
 Definition
 History
 Force
 Frequency
 Energy
 Application
 Limitations
 Conclusion
 Reference

3. A Physical system in which some value oscillates
above and below a mean value at one or more
characteristic frequencies . Such system often arise
when a contrary force result from displacement from
a force neutral position and gets stronger in
proportional to the amount of displacement , as in
the force exerted by a spring that is stretched or
compressed or by a vibrating string on a musical
instrument .

4. A system executing
harmonic motion is called a HARMONIC
OSCILLATOR.
It is a basic model for a
vibrating diatomic molecule.

5. The Harmonic Oscillator played a leading role
in the development of quantum mechanics.
In 1900 , PLANCK made the bold assumption
that atoms acted like oscillators with quantized energy when
they emitted and absorbed radiation .
In 1905 , EINSTEIN assumed that
electromagnetic radiation acted like electromagnetic
harmonic oscillators with quantized energy.

6. In SIMPLE HARMONIC MOTION, the force acting on
system at any instant , is directly proportional to the
displacement . If the displacement of the system from a
fixed point x,
The linear restoring force is F ,
F α x
F = - k x ………(1)
Where ,
F = Restoring Force
x = Displacement
k = Force Constant

8. Now according Newton's 2nd law of motion,If ‘m’ is the mass of a particle,than
F = ma
a = d2
x /dt2
F = m(d2
x/dt2
) ………(2)
Now equaling equation (1) & (2),then
d2
x/dt2
+ kx/m = 0 ……....(3)
If x is the displacement Harmonic ,then
x = Asin2πʋt ………..(4)
where,
A= amplitude of vibration , ʋ = frequency of vibration

9. If we take differential form of eq
n
(4)
dx/dt = A (2πʋ) cos2πʋt
Again differential…
d
2
x/dt
2
= - A ( 4π
2
ʋ
2
) sin2πʋt
d
2
x/dt2 = - 4π2ʋ2x
Or
d2x/dt2 + 4π2ʋ2x = 0 ………(5)
Now equaling then eqn (3) & (5) ,
then,
4π2ʋ2 x = kx/m
or in other words the frequency of the vibration in simple harmonic oscillator i.e.
……..(6)
This equation is the frequency of LINEAR HARMONIC OSCILLATOR.
ʋ =1/2π√k/m

10. If it is in classical treatment force is related to potential energy by the expression :-
F = - ∂V/∂x
where V = P.E.
F = - kx
- ∂V/∂x = - kx
∂V/∂x = kx
∂V = kx ∂x
Now If we integrate these eqn
,then
0∫v dV = k 0∫x x dx
V = k x2
/ 2
This equation is potential energy of LINEAR HARMONIC OSCILLATION.
V = ½ kx2
Fig:- P.E. Diagram from Linear Harmonic Oscillation

11.  The problem of simple harmonic oscillator occurs
frequency in physics because a mass at
equilibrium under the influence of any
conservative force,in the limit of small motions
behaves as a SIMPLE HARMONIC OSCILLATOR.
 A conservative force is one i.e.,associated with a
potential energy .The potential energy function of
a HARMONIC OSCILLATOR is
V(x) = ½ kx2

12. The HARMONIC OSCILLATOR is a great
approximation of a molecular vibration,but
has key limitations :-
 Due to equal spacing of energy ,all
transitions occure at the same
frequency.However experimentally many
lines are often observed.
 The HARMONIC OSCILLATOR does not
predict bond dissociation, you cannot break
it no matter how much energy is introduced.

13. The HARMONIC OSCILLATOR
is among the most important examples of
explicitly solvable problems, whether in
classical or Quantum Mechanics .It appears in
every textbook in order to demonstrate some
general principle by explicit calculation.

14.  Advanced physical chemistry by :-
“Dr. J.N. Gurtu & A. Gurtu”
 Theoretical chemistry by :-
“Samuel Glasstone”