Jouko Teeriaho part1

senior lecturer

Rovaniemi University of Applied Science

spring 2009

Lectures on

Waves and particles

501D3D

waves.nb 1

Contents:

1. Wave properies of light and electromagnetic waves

1.1 Question about the nature of light and matter

1.2 Longitudinal and transversal waves

1.3 Simple harmonic oscillations

1.4 Damped oscillations

1.5 Forced oscillations

1.6 Resonance

1.7 Basic parameters of waves

1.8 Wave equation

1.9 Superposition principle

1.10 Standing waves

1.11 Beat

1.12 Huygens principle

1.13 Young experiment with light waves

1.14 Diffraction of light waves

1.15 The laws of reflection and refraction

1.16 Total reflection

1.17 Polarisation

1.18 Doppler effect

2. Particle properties of light and electromagnetic waves

2.1 Planck's law for black body radiation

2.2 Stefan Boltzmann law

2.3 Wien' s law

2.4 The early history of atom theory

2.5 Bohr's model of atom 1913

2.6 DeBroglie's wave particle dualism 1924

2.7 Quantum mechanical atom model

2.8 Heisenberg's uncertainty principle

3. Phenomena explained by photon theory

3.1 Franck Hertz experiment 1914

3.2 Compton scattering

3.3 Röntgen waves ( X – rays)

3.4 Fotoelectric phenomenon

3.5 Laser

4. Radioactivity and nuclear energy

4.1 The law of radioactive decay

4.2 Mass defect and nuclear energy

5. Theory of relativity

Appendix:

Table of constants

waves.nb 2

Table of the types of electromagnetic waves

1.1 Question about the nature of light and matterAlready at the time of Isaac Newtonin (1600's) there has been two different opinions about the true nature of light.

Newton's opinion was that light consists of particles when his Dutch colleque Huygens claimed that light was

waves. They both could present evidence to support their views.

Today - due to the quantum theory - we understand that both Newton and Huygens were right. Yes - light appears

often in the form of particles: photons. And also - light appears also very often as electromagnetic waves. But light

never shows its both sides at a same experiment , which property Niels Bohr called the "complementary principle".

Light is "Janus - faced ". This strange wave - particle duality is difficult to understand even for the best scientists.

Best way to understand is mathematics - philosophical interpretations have been tried with more or less success.

The modern physics has changed profoundly our view of matter. We think that a table in the room is sure to be a

material. But when we take quantum mechanics seriously, we must say that also the table consists of waves.

In quantum mechanics the deterministic laws are replaced by probabilistic laws. Accoding to the classical theories

if we know the state of a particle at some time, we can predict its state in the future.

In quantum mechanics even if we should know the initial state of a particle, we can only make probabilistic predic-

tions about the future of the particle, but there will always be uncertainty about the exact behavior of the particle.

This uncertainty was stongly opposed by Einstein. He couldn't accept quantum mechanics because of it. He wrote

in his letter to Niels Bohr: "God doesn't play dice".

It seems that Einstein was wrong and quantum mechanics with all its uncertainty and probability properties is the

theory which best explains the nature of materia and radiation.

In the following presentation we shall talk about the waves and particles, famous experiments and their signifigance

and some results of modern physics of the 1900 's.

1.2 Longitudinal and transversal wavesWave motion results from the harmonic vibrations of individual oscillators.

Oscillators can be particles or atoms, like in the sound waves or surface waves of water.

Oscillators can be also fields like in the case of electromagnetic waves.

In transversal wave motion the insividual oscillators oscillate perpendicular to the direction of the wave propaga-

tion. This is the case in the surface waves of water.

In longitudinal wave motion oscillations happen in the direction of the wave propagation. In sound waves the

molecules of air oscillate forward and back in the direction of the propagation of the sound.

waves.nb 3

1.7 Basic parameters of the waves

1.9 Superposition principle

The displacement of an oscillator (atom, electric field,...) due to two different waves is the arithmetic sum of

the displacements due to the individual waves.

ü Example: In the picture below we see the sum wave (blue) of two sinusoidal waves (red).

wave1 = 2.2 Sin@2 xD;wave2 = 1.5 Sin@3 xD;sumwave = wave1 + wave2;

Plot@8wave1, wave2, sumwave<, 8x, 0, 5<, PlotStyle →

8RGBColor@1, 0, 0D, RGBColor@1, 0, 0D, RGBColor@0, 0, 1D<,PlotLabel −> "superposition of two waves"D;

1 2 3 4 5

-2

-1

1

2

3

superposition of two waves

1.10 Standing waves

When two waves with same wavelength and amplitude come from opposite directions the sum wave is s so

called standing wave: there are nodes , which do not oscillate and antinodes where oscillation is large.

Try this animation:

v = 0.5;

Animate@Plot@Sin@x + v tD + Sin@x − v tD, 8x, 0, 8 π<,PlotRange → 880, 8 π<, 8−4, 4<<, Filling → Axis,

PlotLabel → "Standing Wave"D, 8t, 0, 8 π, π ê 8<D

waves.nb 4

1.11 Beat

When two sinusoidal waves with frequencies f1 and f2 which are very near each other, interfere the sum wave

has a frequency f1+f2

2 and the amplitude is oscillating with frequency

f1- f2

2.

Phenomenon is based on the mathematical formula:

sin(ax) + sin(bx) = 2 cos(a-b

2x) sin(

a+b

2x)

Example: Calculation and Plot of the sum wave of 10 Hz and 9 Hz waves

f1 = 10.0; f2 = 9.0;

wave1 = Sin@2 π f1 tD; wave2 = Sin@2 π f2 tD;sumwave = wave1 + wave2;

Plot@sumwave, 8t, 0, 3<D

0.5 1.0 1.5 2.0 2.5 3.0

-2

-1

1

2

ü Beat

A human ear senses the intensity, which is the square of the amplitude:

Intensity is proportional to the square of amplotude 4 cos I2 p f1-f2

2M2

The intensity has a beat frequency of | f1 - f2|

waves.nb 5

PlotB4 CosB2 π H11 − 10L2

tF2

, 8t, 0, 3<,

PlotLabel −> "Intensity", Filling → AxisF

In our example the intensity oscillates with period of 1 second : frequency is 1 Hz.

"Beat" frequency f = | f1 – f2 |

With Mathematica you can hear the beat using command Play instead of Plot

Play@Sin@2 π 440 tD + Sin@2 π 441 tD, 8t, 0, 5<D;

1.12 Huygens' principleA Dutch physicist Huygens in 1600's found the following law from observations:

The law is called "Huygens principle".

Huygens principle:

Wavefronts porpagate so that every single oscillator at the wave front send spherical (circular) waves. The

wavefronts are formed by interference of those spherical waves. The wavefronts are formed from the common

tangent lines of the spherical waves

The picture below descibes hoe wave fronts a build from circular waves

waves.nb 6

1.13 Young's double slit experiment in 1803It is a known fact the if the surface wavefronts on the water come through two slits, on the other side the circular

waves interfere and there will be maxima, where amplitude is high and minima, where amplitude is zero.

Young tried if this phenomenon could be seen if instead of water waves monochrome ligth was used.

The experiment was made in 1803 with light. The result showed without doubt that light is waves, too.

Below is the picture seen on the screen.

waves.nb 7

The maxima are seen on points, where the difference of the distances from the slits is a multiple of wave length.

In those points there is a constructive interference:

Those curves where interference is constructive are in fact hyperbolas, but later we present a good

approximation for the angles, where maximal amplitudes are observed.

In the picture the curves represet curves, where maximal constructive interference is observed

1 2 3 4 5

-1

1

2

ü Young's formula

A good approximation for the angles a where intensity has maxima is

Intensity maxima are observed in angles a, where

d sina = n l , where n = 0, 1, 2, ...

d = the distance of the gaps, l = wave length

Explanation of the formula.

When the path length difference of two waves BP - PA is a multiple of wave length l , the waves come to point P

havong same phase angle, and the waves have a constructive interference: they strengthen each other. From the

picture: BP - PA = BC = d sin a

waves.nb 8

Plot@Sin@4 xD, 8x, 0, 5<, Axes → NoneD

When the path length difference of two rays, BP - PA = BC is Å1

2l + n l, the rays from slits will have an opposite

phase and the sum wave will have a zero amplitude. (desctructive interference)

Minima ( dark areas ) are in directions

d sina = (n+ Å1

2) l , where n = 0, 1, 2, ...

We talk about destructive interference in directions where the waves come in opposite phases.

Below Picture A describes constructive and B destructive interference: Lowest wave is the sum wave of the two

waves above.

ü Greating and spectrum

If we use instead of two slits a grating , which has many slits with constant intervals, we get very sharp intensity

maxima.

If the light is not monochrome but consists of many colors, we see from the formula for maxima that different

wavelength have maxima in different angles.

In other words we see a spectrum in directions corresponding the side maxima ( n = 1 , 2, ....).

Grating is often used to produce spectrum. Another way is to use a prism.

waves.nb 9

1.14 Diffraction of wavesBy diffraction we mean the phenomenon, where electromagnetic waves bend around the edges forming interference

patterns.

Diffraction can be explained by Huygens principle:

"The oscillators of the wave front act as a centre of spherical waves"

ü Single slit diffraction

Even when light goes through one narrow slit (size equal or less than l), it bends and forms a similar pattern on the

screen as in the double slit experiment.

In this case the intensity minima are found in the angles

Maxima d sina = n l , n = 1, 2, ... d = width of the slit

Explanation of the formula.

In the direction a, where the path length difference AP-BP = AC = d sina is a multiple of wave length l,

we find for all the rays passing the slit a pair with opposite phase. For example, if the path length difference of rays

through B and A: d sina = l, the ray going through B and the ray which goes through the centre M of the slit have

opposite phases and therefore destroy each other. Similarly the rays just below B and M form a pair, which destroy

each other. Consequently in point P in the direction defined by

d sina = n l , is completely dark, all the rays have pirwise destroyed each other. Intensity minimum is found in

that direction.

In single slit diffraction the central maximum is bright and the side maxima are quite weak - hardly noticeable.

waves.nb 10

There is a rule: the narrower is the slit, the wider is the diffraction pattern.

If the width of the slit is near wave length , then the light rays spread strongly and they form a wide diffraction

pattern.

ü Diffraction of light passing through a circular aperture (lens)

When light rays pass a narrow circular hole, it bends and you can see on the screen

the following diffraction pattern.

ü Edge diffraction

Radio waves bend also around the wall edges because of the diffraction (see picture). In the bended waves there are

also minima and maxima, which are not clearly seen in the picture.

1.15 The laws of reflection and refraction

ü The law of reflection

waves.nb 11

The angle of incident ray and the angle of reflected ray are indentical

ü The law of refraction (Snell's law)

Example: When a ray of light comes to the surface between air and glass, the ray bends to the direction of the

normal of the surface. That is because the speed of light is smaller in glass than in air.

If light would come from a slower material to faster, it would bend away from the normal.

Definition: The refractive index ot the material n = Åc

v

c = speed of the light in vacuum = 300 Mm/s , v = speed of the light in the material

The law of Snell: sin q1

sin q2 =

v1

v2=

n2

n1

where v1 and v2 are speeds of the light and n1 and n2 are refractive indexes

table: medium speed of light refractive index

vacuum 300 Mm/s 1.0

air 300 mm/s 1.0

glass 200 Mm/s 1.5

water 225 Mm/s 1.33

1.16 Total internal reflectionWhen the light comes from a slower medium to a faster medium and the incident angle

exceeds a certain limit, which is is called the critical angle, the ray is 100% reflected.

The phenomenon is called total internal reflection.

waves.nb 12

Example: The speed of light in water is 225 Mm/s. Calculate a) the refractive index of water

b) the critical angle when light comes from water to air.

Solution: a) refractive index n = Åc

v =

300

225 = 1.33

b) critical angle is obtained from sin 90 ±

sin qc=

300

225

=> sin qc = 225

300= 0.75

=> critical angle qc = 48.6 °

ü Glass fiber cable technology

In a reflection from a mirror about 5% of the intensity of light is absorbed. But in the total internal reflection energy

loss does not exist. That is why glass cables are a very good way to transmit light even hundreds of kilometers.

Glass cables are used in data transfer between the countries and even between continents.

Glass cables are also used in optics : traffic lights, medical instruments, ...

Total internal refection confines light within optical fibers (similar to looking down a mirror made in the

shape of a long paper towel tube). Because the cladding has a lower refractive index, light rays reflect

back into the core if they encounter the cladding at a shallow angle (red lines). A ray that exceeds a

certain "critical" angle escapes from the fiber (yellow line).

1.17 PolarizationNormal light light sun light is not polarized, which means that the electric field vibrates in all possible

directions.

waves.nb 13

The most common method of polarization involves the use of a Polaroid filter. Polaroid filters are made of a

special material which is capable of blocking one of the two planes of vibration of an electromagnetic wave.

Only the component of vibration which is parallel with the polarization axis of the filter can pass the filter.

If two filters are placed polarization axises parallel, the result is the same as in the case of one filter.

If two filters are placed polarization axises perpendicular, no light passes through the system.

ü Malus' law

If EM radiation (light) is already linearly polarized, and another polarizing filter with polarizing axis,

which forms an angle a with respect to the light, the second polarizer lets through a component

E = E0 cos a of the original electric field E0

and intensity I = I0 HcosaL2

Sun light consist of all possible angles => polaroid glasses let through intensity I = Å1

2I0 , because the average

value over all angles of HcosaL2 = Å1

2

ü Polarization in reflection

When the sunlight comes to the water, both the reflected light and the refracted lights are partially polarized.

Reflected light is polarized parallel to the surface.

Rafracted light is polarized poerpendicular to the surface.

waves.nb 14

Brewster's law

The polarization in reflection is not 100%.

Brewster has shown that the degree of polarization in the reflected light is at its maximum, when the angle between

reflected ray and refracted ray is 90°. In this situation a + b = 90° and

the angle of the incident ray a satisfy the following condition:

Brewster's formula tan a = n

The condition can be also said in the form: "Polarization is greatest, when the reflected ray and the refracted ray are

perpendicular to each other"

Example: When sunlight reflects from water, the polarization of the reflection is greatest, when

tan q = 300

225= 1.33 => q = 53.1° .

This angle is called Brewster's angle for water.

ü Polaroid sunglasses

Polaroid sunglasses are made of polarizing filter material with vertical axis of polarization.

They filter out the reflections from water.

1.18 Doppler effect

ü Definition of Doppler effect

Doppler effect is the apparent change in frequency and wavelength of a wavethat is perceived by an observer

moving relative to the source of the waves

Everybody has experienced Doppler effect when an ambulance has passed with sirens on.

The pitch of the siren has seemed high and then suddenly it has become lower.

waves.nb 15

For waves that travel through a medium (sound, ultrasound, etc...) the relationship between observed frequency f'

and emitted frequency f is given by:

Doppler's formula f ' = c+ v0

c - vs f

c = the velocity of the waves (340 m/s for sound, 300 Mm/s for light)

vo = is the velocity of the observer

vs = the velocity of the wave source

For sign convention on velocity: a positive value is used if the motion is towards the other, and a negative value if

the motion is away from the other

ü Police radars

Police radar is usually a Doppler -radar. The radar send a pulse at 3 GHz frequency.

The signal is reflected from the target, and the observed frequncy is due to Doppler effect greater that the sent .

The radar registers the difference of the sent and observed frequencies and calculates

the speed of the target using the following formula

Speed of the target v = Df2 f

c

f = is the frequency sent by the radar ( about 3 GHz)

Df = is the difference of the sent and received frequency

c = speed of light

Note: Factor 2 in the denominator is due to the fact that Doppler effect takes place twice: First when the pulse

reaches the target car, second time when it reaches the radar.

waves.nb 16

ü Red shift of light

Doppler effect is also observed in astronomy. Universe is expanding and faraway stars are going away from the

Earth with high speeds.

Consequently the typical spectral lines of the light which they send originated from hydrogen and helium) are

shifted towards red.

The changes of observed and original frequncies can be used to calculate the relative speed of the stars with respect

to the Earth..

Just like the police radar.

That is how we know that the Universe is expanding and we know also the speed of the expansion.

Speed of the star v = Dff

c

f = is the frequency of the spectral line

Df = is the frequncy shift

c = speed of light

1.19 Vibrations in a stringIn a quitar string is plucked , vibration waves start to travel back and forth in the string.

Standing waves are formed in the string.

Only standing waves with nodes at the end of the string survive.

Because the ends of the string can not vibrate, only those waves survive, where there are nodes at the end of the

string. Mathematically this means that 2L = n l

resonance wavelengths: l = 2 L

n , where n = 1,2, ...

resonance frequencies: f = Åv

l =

n v

2 L

l = wavelength

f = frequency

v = speed of the wave

The first frequency ( n = 1) is called the fundamental and the others are its multiples, called harmonics.

waves.nb 17

It can be shown that the speed of the wave depends on the tension of the string T and the mass per unit length r

according to the following formula:

v = ÅT

r

When one tunes a quitar, one changes the tension T to gain the desired pitch ( = frequency).

Combining previous formulas we get a formula for the fundamental frequncy of a string:

f = n

2 LÅT

r

waves.nb 18