SIMPLE HARMONIC OSCILLATOR

(1)

SIMPLE HARMONIC OSCILLATOR

Equation of Motion

The simple harmonic oscillator is a model system that represents a mass suspended from a stationary object by a spring as shown in Figure. There are two forces acting on the mass

Stationary object

Spring

Mass

Restoring Force Weight Force

x =- maximum at t= another time

x = maximum at t=time x = 0 at t=0,….

(Equilibrium Position)

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1-Weight force (Newton's 2

^nd

Law) Force = mass acceleration

2-Restoring force (Hook's Law)

Force = force constant displacement

( )

+ = 0

The above equation is called second order ordinary differential equation and has a general solution (the equation of motion of the system) can be written as

= +

Note that at t=0 x=0 so

0 = 0 + 0

0 = 0 & 0 = 1 0 = 0 +

=

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The simple harmonic oscillator can also be modeled as

SIMPLE HARMONIC OSCILLATOR

The Figure 2 shows the position as a function of a time, this is uniform harmonic motion, it is a periodic motion. The constant (A) is the largest magnitude of oscillation and called the maximum

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= 2

2 =

= 1 2

If the model oscillator consists of two movable masses connected by a spring, as depicted in Figure 3. so, it is necessary to replace the mass in the harmonic oscillator formulas by the reduced mass

= +

= 1 2

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SIMPLE HARMONIC OSCILLATOR

= 1 2

That for a diatomic chemical compound if we consider the bond between them is a spring and the two atoms oscillates, the frequency of that oscillation can be calculated from above equation.

The Force constant (k ) can give an indication of the stiffness of the spring, the larger k the stiffer spring

= 1 2

Molecular Vibrations

A covalent bond between two atoms can be envisaged as a spring holding them together. If the bond is compressed, there is a restoring force which pushes the atoms apart, back to the equilibrium bond length. Similarly, if the bond is stretched, there is a restoring force that forces the atoms back closer together, again restoring the equilibrium bond length

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SIMPLE HARMONIC OSCILLATOR

Molecular Vibrations

The above equation show that the frequency of the stretching vibrations have discrete values and depends on two factors:

(1) The mass of the atoms

Heavier atoms vibrate more slowly than lighter ones, so a C-D bond will vibrate at a lower frequency than a C-H bond. (HF < HI)

= 1 2

SIMPLE HARMONIC OSCILLATOR

Molecular Vibrations

(2) The stiffness of the bond

A-Stronger bonds are stiffer than weaker bonds, and therefore require more force (more energy) to stretch or compress them. Thus, stronger bonds generally vibrate faster (why?) (at higher energy) than weaker bonds. So O-H bonds which are stronger than C-H bonds vibrate at higher frequencies.

B-For multiple bonds, the more order the stiffer the bond and therefore the higher energy of vibration

= 1

2

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SIMPLE HARMONIC OSCILLATOR

Molecular Vibrations

Single C-C bonds absorb around 1200 cm

^-1

Double C=C bonds absorb around 1660 cm

^-1

Triple C C bonds absorb around 2200 cm

^-1

.

= 1 2

Figure 7 shows a vibrational spectrum of

benzoic acid indicating that the vibration have

a discrete values

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Energy of Motion

The kinetic energy of the harmonic oscillator is

= 1 Where 2 v

v = = So

= 1 2

The potential energy of the harmonic oscillator relates to the force by

= = =1

2

SIMPLE HARMONIC OSCILLATOR

Energy of Motion

Due to A can take any positive or negative values so

•The energy can take any positive value

•The energy can be absorbed or emitted continuously

•This equation gives continuum energy level diagram and this is one of the failure of classical mechanics.

The total energy will be

= + =1

2 + = 1

2