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AMRUTA INSTITUTE OF ENGINEERING & MANAGEMENT SCIENCES [Approved by AICTE NEW DELHI, Affiliated to VTU BELGAUM]

DEPARTMENT OF PHYSICS

COURSE MATERIAL SUBJECT:- 06 PHY 12 Name of the Faculty Member:- Daruka Prasad B

Email:- [email protected]

MODERN PHYSICS

VTU University Syllabus (2007-08)

Introduction to Blackbody radiation spectrum, photo-electric effect, Compton effect. Wave-particle Dualism.

deBroglie hypothesis- deBroglie wavelength, extension to electron particle, - Davisson and Germer Experiment.

Matter waves and their characteristic properties. Phase velocity, group velocity and Particle velocity. Relation

between group velocity and particle velocity. Expression for deBroglie wavelength using group velocity. 8 Hrs.

Reference Material

Arthur Beiser, “Concepts of Modern Physics”, S.Chand Publications, 3rd Edition, 2004.

Arthur Beiser, “Perspectives of Modern Physics”, S.Chand Publications, 3rd Edition, 2004.

R.K. Gaur & S.L Gupta “ Engineering Physics” Dhanpat Rai Publications-2008,

Pages 56.1-57.21

S.O.Pillai & Sivakami “ A Text book of Engineering Physics”, New Age International (p) Ltd.,

Prabir. K. Basu & Hrishikesh Dhasmana, “ Solid State Engineering Physics”, Ane Books India

Publications 2008, 34-52.

S.O. Pillai “ Solid State Physics” 6th edition, New Age International (p) Ltd., publications, 2007, Pages 157-

178.

Feynman, Leighton, Sauds, “ The Feynman Lectures on Physics, Volume 3, Narosa Publications, 2007,

pages17-20.

M.N.Avadanulu & P.G. Kshirsagar, “ A Textbook of Engineering Physics” S.Chand Publications,2006

S.P. Basavaraju, “ Engineering Physics”, Subhash Stores Publications, 2007-08.

G.K. Shivakumar “Engineering Physics”, Prism Publications, 2007-08.

INTRODUCTION

Classical Physics:

Newtonian Mechanics, Maxwell’s Electromagnetic theory and Thermodynamics explained almost

all the scientific results, which are successful in the realms of macroscopic world, are regarded as Classical Physics.

In the Classical Physics, matter and fields are treated as entirely independent entities.

The physical quantities such as particle energy are continuously variable and take any

possible value.

A particle can be isolated from its environment and can be treated as an independent entity for the investigation. Thus, the particle under investigation and the instrument

measuring any of its parameters are mutually independent.

Electromagnetic waves are generated by accelerated charges; if a charge oscillates with a

constant frequency ‘’, it produces an electromagnetic wave of same frequency ‘’. The waves spread out continuously through space and the wave energy is not localized;

but it is distributed over the volume of the wave.

Energy of the wave is not related to the frequency of the wave, but it is proportional to the

square of the amplitude of the wave2

AI .

The thermodynamic equilibrium of an assembly of neutral particles is governed by

Maxwell-Boltzmann statistics. A large number of particles can occupy a given energy

state and the particles can have a continuous range of energies.

In 19th century, a number of new phenomena such as photoelectric effect, X-rays, Line spectra,

Ultraviolet catastrophe and radioactivity were discovered which defied explanation on the basis of classical physics. The classical laws are not valid in the microscopic world. The micro world of

atoms obeys different laws. The new laws applicable for micro particles constitute Quantum

Mechanics.

INTRODUCTION BLACK BODY RADIATION

A body which absorbs all the radiations incident on its surface whatever may be their wavelengths is called Black Body. An isothermal cavity in which radiations from outside can enter in

through small aperture in it can be considered black body. When radiation inside the cavity are in equilibrium with

its walls, the emissivity in a unit wavelength be E. In 1899 Lummer and Pringshein experimentally studied the

distribution of radiant energy emitted from a black body in different wavelengths by keeping the body at different

temperatures.

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Fig.1:- Distribution of radiant energy emitted from a black body in

different wavelengths by keeping the body at different temperatures.

Conclusions of the Lummer and Priengshein Experiment:

At each temperature, wavelength is distributed in all wavelengths from 0 to. The value

of E increases with increase in and at particular m the vale of E becomes maximum.

As the wavelength increases beyond m, E decreases and at = the value of E becomes

zero. Thus E - curve is continuous. In other words, black body spectrum is continuous.

Area enclosed by the curve represents the total energy emitted by the black body. Higher the temperature of the body greater is the total radiant energy. Energy radiated by black

body is directly proportional to fourth power of the absolute temperature is the statement

of Stefan’s law. 4TE

As the temperature increases, the wavelength m corresponding to the maximum radiant energy decreases, i.e., the peak of the curve shifts towards the shorter wavelength region.

The wavelength m corresponding to the maximum emissivity is inversely proportional to

the absolute temperature T of the body T

m

1 ; )(tanbtconsTm

b= 2.88 x 10-3 mK at lower temperature (Wien’s displacement law). At any temperature

the energy radiations of wavelength emitted by the blackbody is directly proportional to

the fifth power of Temperature 5TE

Wien’s Law:- Wien’s law of energy distribution in the blackbody radiation spectrum varies as below

decdE

TC )/(5

12

where E d is the energy per unit volume for wavelengths in the range and +d and C1 and C2 are constants. Wien’s law suits only for shorter wavelength region and high

temperature value of the source.

Rayleigh and Jean’s law: - To explain the spectral distribution of the black body radiation, Rayleigh and Jean assumed that inside the black body cavity, due to

superposition of radiation waves incident on the wall the waves are reflected from it, and

they form stationary waves. Hence, the energy density in the wavelength range and

+d is equal to the number of modes of vibrations in unit volume. According to classical theory, mean energy per mode of vibrations at an absolute

temperature T is kT, where k is Boltzmann’s constant (k=1.38x10-23 J/K). Hence, according to Rayleigh-Jean’s law

dkT

dE4

8

This law explains only the higher wavelength part of the experimental energy distribution

curve as shown in Fig.1. Ultraviolet Catastrophe (Drawback of Rayleigh-Jean’s law)

When a body is heated, it emits radiation called thermal radiation. The relative brightness of

different frequencies depends on the temperature of the body. As the temperature of the body is

increased, the component of maximum intensity shifts to a higher and higher frequency. The experimental results, illustrated in the Fig.1 showed that at a given temperature, the radiation

energy density initially increases with frequency, then peaks at around a particular frequency

and after that decreases finally to zero at very high frequency. According to the classical theory,

the intensity of thermal radiation should increase with square of frequency2U .

The theory agrees with experimental result at lower frequencies but leads to absurd result at

higher frequencies. The theory implies that the radiation emitted by a hot body should have a

large portion of UV rays. This is contrary and fails the law of conservation of energy. This

1646K

1259K

998K

E

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contradiction is called the ultraviolet catastrophe. Thus, classical theories of physics failed to

explain the distribution of thermal radiation emitted by solid bodies.

PLANCK’S QUANTUM HYPOTHESIS: In 1900 Max Planck succeeded in developing a theory that correctly predicted the frequency

distribution of thermal radiation for black body.

Postulates:

The atomic oscillators in a body cannot have any arbitrary amount of energy. They

could have only discrete units of energy given by nhEn

where n is any positive integer (n=1,2,3…..); is the frequency of oscillation, and h is a constant known as planck’s constant (h= 6.62x10-34 Js).

The atomic oscillators cannot absorb or emit energy of any arbitrary amount. They

absorb or emit energy in indivisible discrete units. The amount of radiant energy in

each unit is called a quantum of energy. Each quantum carries an energy hE . The energy of each quantum is a minimum amount of energy that can be found

separately and that cannot be further subdivided. It represents the smallest quantity

of energy of radiation of that frequency. The hypothesis that radiation energy is emitted or absorbed in a discontinuous manner and in

the form of quantum is called the planck’s quantum hypothesis. The integer ‘n’ is called

quantum number. The energies of the atoms are said to be quantized and the allowed energy

states are called quantum states.

Fig.2: Discrete Energy levels

The atom absorbs or emits energy by jumping from one quantum state to another quantum state, indicates that the energy content of a wave is related to the frequency of wave rather than to the square of its amplitude as postulated in classical wave theory. On the basis of above hypothesis, following relation was given by Planck for the energy distribution in the black body radiation, which could successfully explain the experimental curves of Lummer and Pringsheim

or

e

dhcdE

kThc 1

8/5

1

8)(

3

3

kT

h

e

d

c

hdu

(1)

where k is the Boltzmann constant (k=1.38x10-23 J/K), T is the absolute temperature, c is the velocity of light in vacuum. Equation (1) gives the radiation energy density per

unit frequency of d. If h >>kT, kT

h

e

then 0)( du

kT

hekT

h

1 (2)

since

x

xxxex

1

................!3!2

132

(3)

because x is very small

h

kT

ekT

h

1

1

(4)

Planck’s law changes to dc

kTdu

3

28)(

(5) Rayleigh Jean’s law.

PHOTOELECTRIC EFFECT:

The ejection of electrons from the surface of a metal under the action of light is called the photoelectric effect. The materials that exhibit photoelectric effect are called

hE

Energy

Discrete energy levels

1

2

3

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photosensitive materials. The emitted electrons are called photoelectrons. The

photoelectric effect was first observed by Henrich Hertz in 1887.

In order to study the photoelectric effect, the photosensitive material is made as the cathode of a vacuum tube and the tube is connected` to the circuit as shown in Fig.2.

The vacuum tube G is made of glass or quartz. The cathode K is coated with the photosensitive material and is connected to the negative terminal of a battery. A metal plate p is maintained at a positive potential by the battery. When the tube is kept in the dark then there will be no current in the circuit. When it is exposed to light, photoelectrons are liberated from the cathode k. They are attracted to the anode p and flow through the circuit. The strength of the current read by the meter A is a measure of the number of photoelectrons. The potential difference across K and p is varied with the help of a potentiometer. The polarity of the voltage can be reversed through suitable arrangement. The experiments showed that the photocurrent and energy of photoelectrons depend on the intensity and frequency of the incident light. The experimental observations on the photoelectric effect can be summed up as follows:

Electrons are emitted from photosensitive surfaces. Each material has a definite

minimum frequency 0 below which the photoelectric emission doesn’t occur. The minimum frequency is called the threshold frequency.

For frequencies 0, the number of electrons emitted per second is proportional to the intensity of light.

For frequencies 0, photoelectrons are emitted with kinetic energies ranging up to a maximum value. The maximum kinetic energy can be determined in terms of

stopping potential (Vs)

seVm 2

max2/1 (1)

Stopping potential is the potential, which is required to halt the most energetic

electrons. The stopping potential doesn’t depend on the intensity of light.

The stopping potential Vs is a linear function of the frequency of the incident light.

Emission of photoelectrons is instantaneous no matter how feeble the incident light may be.

Failure of classical theory to explain photoelectric effect: As the frequency and wave intensity are not related in classical theory, the frequency

is not expected to influence the emission of photoelectrons. Therefore, a threshold

frequency should not be exhibited. However, these conclusions are contrary to the

experimental observations. It means that classical physics cannot account for the

existence of threshold frequency.

The wave theory expects that the electrons would be emitted with higher velocities

when the intensity of light is large. The velocity, and therefore the Kinetic Energy (K.E) of photoelectrons would be required to be proportional to the incident wave

intensity. As according to wave theory, the frequency is not related to the energy of

the wave, the kinetic energy of photoelectrons and hence stopping potential should be

independent of the frequency. This is in contradiction with the observation. A feeble

light of a higher frequency produces a smaller number of photoelectrons. However, they have much greater kinetic energies. It implies that classical theory fails to trace

the connection between the K.E of photoelectrons and the frequency of the incident

light.

e’s

Zinc Plate

V

A

UV Radiation

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According to the classical theory the electrons at the surface of a metal would require

a time of the order of an hour to absorb enough energy from an incident weak radiation so that they can escape from the metal. In reality, the electrons are emitted almost instantaneously taking a time of the order of 10-9 sec.

Einstein’s theory of Photoelectric Effect: Albert Einstein in 1905 showed that all the features of photoelectric effect with the extension of planck’s quantum hypothesis. Einstein assumed that the light of frequency illuminating the photosensitive material might be regarded as a stream of

photons, each photon carrying an energy h. When photon encounters an electron in the material, it gives up all its energy to the electron, and the electron acquires the energy. Each free electron in the metal is prevented from leaving it because of a potential barrier. The amount of energy required by an electron to surmount the potential barrier is equal to the work function W0 of that metal. Therefore, when an

electron absorbs the photon energy h, part of the energy is spent in overcoming the potential barrier and the remaining part of energy goes into its kinetic energy. According to the law of conservation of energy Energy of photon = (Energy needed to liberate the electron) + (Maximum K.E of the

liberated electron)

max0 .EKWh or 2

max02

1 mWh (2)

However, using the relation that connects K.E with stopping potential, we get

seVWh 0 (3) The above equation is called photoelectric equation. This equation derived based on

quantum theory of light can explain all the features of the photoelectric effect.

According to Photoelectric Equation, the kinetic energy of the photoelectron is

given by 0max. WhEK . If the energy of the incoming photon is equal to or greater that W0, the electron will be ejected from the material. On the other

hand, if 0Wh , the photons will not have enough energy and photoemission does not take place, however intense the incident light may be. The minimum

energy that a photon should have to cause photoelectric effect is 00 Wh . Thus, the quantum theory accounts for the existence of the threshold frequency for photoelectric effect.

According the quantum theory, an individual photon ejects an electron. The intensity of light is proportional to the number of photons in the incident light

.NI therefore; the number of photoelectrons will be proportional to the number of incident photons. As a result, the photocurrent will be proportional to the intensity of the incident light.

In case of a monochromatic light of frequency , the photon energy h is fixed quantity. Therefore, by increasing the intensity of the light, only the number of photons and hence number of ejected electrons increases but the K.E doesn’t vary. Thus, the quantum theory explains the independence of K.E and hence of stopping potential on intensity of the incident light.

The greater the frequency, the greater is the photon energy. The maximum K.E of the photoelectrons will increase with increasing frequency of the incident light. Therefore, the stopping potential will be proportional to the frequency.

As the energy is transferred from the photon to the electron instantaneously, there will be no time delay in the emission of photoelectrons.

Experimental Verification of Einstein’s theory: We Know that

0WheVs (4)

Therefore, Vs = e

W

e

h 0

(5)

the above relationship between the stopping potential Vs and the incident frequency is linear as shown in Figure. The slope of the line gives the value of (h/e) from which the planck’s constant can be calculated and compared with the values obtained from other

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experiments. In 1916, Robert A Millikan established the experimental verification for the correctness of the Einstein’s equation.

Compton Effect

When X- rays of sharply defined frequency are allowed to incident on a material of low atomic number, there is a change in frequency on scattering. The scattered beam has two wavelengths. One is having incident wavelength and the second one is having longer wavelength. This change in wavelength is due to the energy loss of incident x-rays. This interaction is known as Compton effect, which is the scattering of a photon by an electron is called Compton scattering. Compton explained the effect based on the quantum theory of radiation. Considering radiation to be made up of photons, he applied the laws of conservation of energy and momentum for the interaction of photon with electron. Consider an x-ray photon of

energy h incident on an electron at rest as shown in the figure below.

After the interaction, the x-ray photon gets scattered at an angle with its energy

changed to a value h’ and the electron which was initially at rest recoils at an angle . It can be shown that the increase in wavelength is given by

cos10

cm

h (1)

where m0 is the rest mass of the electron.

When = 900, 0

0

0242.0 Acm

h .

This constant value is called Compton wavelength. When =1800, cm

h

0

2

Experimental observation indicates that the change in the wavelength of the scattered x-rays is in agreement with Eq. (1), thus providing further confirmation to the photon model.

de Broglie’s Concept of Matter Waves

Upto 1924 Physicists were of the view that only light or electromagnetic radiation possesses dual character, sometimes they behave as a wave and at the other as a particle. The matter was considered exclusively corpuscular in nature. This ideology was broken by Louis de-Broglie in 1924 by proposing that wave-particle duality may not be a monopoly of radiatin, but a universal characteristic of nature. In his doctoral dissertation, he suggested a new idea of the existence of matter waves purely on theoretical ground. According to de-Broglie’s concept, a moving particle always has a wave associated with it and

the motion of the particle is guided by that wave in a similar manner as photon is controlled

by a wave.

The expression for the wavelength of a matter wave or de-Broglie’s wave can be obtained by analogy of the behavior of radiation with matter. Accordingly, the wavelength of matter wave or de-Broglie’s wave should be the same as that for electromagnetic

radiation. E=hAccording to Einstein’s theory of relativity, a mass m of a particle is equivalent to an amount of energy as E= mc2.

De Broglie assumed m to be the mass of the photon of energy h which moves with the velocity of light c. Comparing the above relations of energy, we get

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For photon’s, h = mc2

If p is the momentum of photon then p= mc

Or p

h c

This is the de-Broglie’s wavelength for photons. In case of particle of mass m moving with velocity v and has a momentum p=mv, the expression for the wavelength of a wave that guided the motion of the material particles or matter waves, is obtained by

substituting p=mv in wavelength expression i.e., mv

h

p

h

If Ek is the kinetic energy of the material particle, then

Kk mEporm

pmvE 222

122

Therefore, the de-Broglie wavelength may also be written as

kmE

h

2

If the charged particle carrying charge q is accelerated through a potential difference of V Volts, then the wavelength of the wave associated with this moving charged particle is obtained as follows. The kinetic energy of the charged particle, E=qV

mqV

hwavelengthBrogliede

2,

For a neutral material particles like neutrons, which possess Maxwellian distribution of

velocities at absolute temperature T in thermal equilibrium, the average kinetic energy is

given by kTmvE rmsk2

3

2

1 2

where k is the Boltzmann’s constant. The wavelength of a wave associated with such a particle is given by

mkT

h

3

Properties of Matter Waves

1. The de-Broglie’s wavelength of a wave associated with moving lighter particle is

greater than the wavelength associated with heavier particle.

2. The de-Broglie’s wavelength of a wave associated with a slow moving particle is

greater than the wavelength associated with fast moving particle.

3. For a particle at rest, that is, if v=0, the de-Broglie’s wavelength becomes infinite, that

is, wave becomes indeterminate and if v=, then =0. This indicates that the matter

waves are generated only when the material particles are in motion.

4. The expression for de-Broglie wavelength is independent of charge of the particle,

therefore the matter waves are generated by moving charged particles as well as by

moving neutral particles.

5. The velocity v of a matter wave is greater than the velocity of electromagnetic wave,

that is, the velocity of light.

It can be shown as below:

The energy of a wave of frequency is given by E= h

The energy of a particle of mass m is given by E=mc2 where c is the velocity of light.

h

mc2 (1)

The wave velocity(or Phase velocity) u is related with as u= or

mv

hu

(2)

Substituting the value of from Eq.1 to Eq.2, we get

v

cu

2

(3)

As the particle velocity v is always less than the velocity of light, it follows that the

velocity of propagation of the associated matter wave (or phase velocity) is greater

than c, the velocity of light.

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6. The velocity of matter wave is not constant as radiation which move with constant

velocity equal to the velocity of light. The velocity of matter wave depends upon the

velocity of material particle.

7. The wave and particle aspects of matter wave never appears simultaneously in the

same experiment. In some phenomena particle nature predominate while in others

wave nature predominate.

Davission and Germer Experiment

Davission and Germer succeded in confirming de-Broglie wavelength of matter wave associated with electron beam. The schematic of the apparatus used by Davisson and Germer is as shown in Figure below:

45°

D S

F G C

+-50V

The apparatus consists of a filament heated with a small a.c power supply to produce thermionic emission of electrons. These electrons are attracted towards an anode in the form of a cylinder with a small aperture maintained at a finite positive potential with respect to the filament. They pass through the narrow aperture forming a fine beam of accelerated electrons. This electron beam was made to incident on the face of (1 1 1) a single crystalline sample of nickel. The electrons scattered at different angles were counted using a Faraday’s ionization counter as a detector. This counter detects only the scattered electrons generated from the filament. The weaker electrons eject from the nickel surface in the process will not be counted in the counter. The experiment was repeated by recording the scattered electron intensities at various positions of the detector for different accelerating potentials as shown below.

20°

I I I I

I

40 volt 44 volt 48 volt 54 volt

60 volt

50°

When a beam of electrons accelerated with a potential of 54 V was directed perpendicular to the nickel target, a sharp maximum occurred in the electron density at

an angle of 500 with the incident beam. When the angle between the direction of the incident beam and the direction of the scattered beam is 500, the angle of incident will

be 250 and the corresponding angle of diffraction will be 650. The spacing of the planes responsible for diffraction was found to be 0.91A0 from x-ray diffraction experiment. Assuming first order diffraction, the wavelength of the electron beam can be calculated as

00 65.165sin91.02sin2 Ad The wavelength of the electrons can also be calculated using the de Broglie’s relation as

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0

1931

34

66.1

54106.1101.92

1063.6

2

A

meV

h

p

h

Thus, the Davisson-Germer experiment directly verifies the de Broglie’s hypothesis.

Interpretation of Bohr’s Quantisation Rule on the Basis of de-Broglie’s Concept of Matter wave.

Bohr’s Quantisation rule can be explained on the basis of de-Broglie concept of matter wave. According to wave mechanical model of an atom, an electron behaves as a stationary wave, which goes round the nucleus in a circular orbit. Since the electron does not radiate energy while moving in its orbit, the wave associated with it must be stationary wave in which there is no transfer of energy. According to de-Broglie’s concept of stationary orbit, only those orbits are allowed as stationary orbits whose

circumference or perimeter is an integral multiple of wavelength associated with

electron. That is, 2r= n

Where r is the radius of orbit and the de-Broglie wavelength of electron. According the de-Broglie concept, an electron of mass m moving with velocity v has

associated with it a wave whose wavelength is =h/mv. There fore,

,........3,2,1,2

2 nwherenh

mvrormv

nhr

Since mvr is the angular momentum of an electron as particle, the wave mechanical picture leads automatically to Bohr’s Postulate.

Wave Packet The matter has the dual nature; particle and wave. According to de-Broglie hypothesis, a wave is associated with each moving material particle whose wavelength is given as

p

h

where h is Planck’s constant, m is the mass of particle and is the velocity of particle.

If E is the energy of the particle, from quantum condition E=h, the frequency of the

wave associated with it is h

E

But from Einstein’s mass-energy relation E=mc2

h

mc2

Hence, velocity of de-Broglie wave

22 c

mv

h

h

mcu

But according to Einstein’s theogy of relativity, no material particle can have speed greater than the speed of light c ( i.e., the speed of light c is the maximum possible speed). Hence, from above equation it is concluded that the speed of de-Broglie wave u will be greater than the speed of light c, which is impossible. Apart from this, if the speed of de-Broglie wave u associated with the particle is greater than the speed of particle, the particle will be left behind. Hence, we conclude that only one wave is not

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associated with the material particle. Schrödinger assumed that a moving material particle is equivalent to a wave packet, instead to a single wave. A wave packet is a group of several waves of slightly different velocities ( v and v+dv) and

different wavelengths( and +d). The amplitudes and phases of the component waves are such that they interfere constructively in a limited region where the particle is found and outside the region they interfere destructively (i.e., outside the region where the particle is not found, resultant amplitude abruptly falls to zero). Thus, the wave behavior of a moving material particle can be represented by the resultant wave obtained due to the superposition of component waves of slightly different velocities and different wavelengths. The spread of amplitude of resultant wave with distance determines the size of the wave packet. If the velocity of all the superposing component waves is the same, the wave packet also moves with the same velocity. But if the velocities of the component waves differ, the velocity of the wave packet differs from the velocities of the component waves. The mean velocity of

component waves of a wave packet is called the phase velocity vp and the velocity of the wave packet is called the group velocity vg.

Phase velocity and Group Velocity The average velocity of component waves of a wave packet is called the phase velocity of those waves.

Let a plane harmonic wave of frequency ( or angular frequency = 2) and wavelength

( or propagation constant k=2) traveling in X-direction is represented as )( tkxiey

The speed of propagation of this wave will be the speed associated with a point for

which the phase (kx-t) is constant, i.e., (kx-t)= constant

Or x= constant + (/k) t

/2

2

kdt

dxvvelocityPhasep

Group Velocity

The velocity with which the wave packet (or the point of reinforcement of waves) obtained due to the

superposition of waves traveling in a group, advances is called the group velocity.

dk

dvg

Derivation

Consider two waves of same amplitudes, but of slightly different frequencies 1 and 2 and slightly

different wavelengths 1 and 2 superpose and form a wave packet. If A the amplitude,

the angular frequencies of these waves are 1=21 and 2= 22 and their propagation constants are k1 and k2 respectively. Then their separate displacement at any instant t can be written as

txkA 111 sin (1)

txkA 222 sin (2)

According to Young’s principle of superposition, the resultant displacement at any instant t is given

by txktxkA 221121 sinsin

(3)

Using trigonometric relation

2cos

2sin2sinsin

BABABA

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22

cos22

sin2 21212121 txkktxkkA

(4)

or )(sin tkxA (5)

where 2

;2

;;,22

cos 21212121

kkkkkktx

kAA

A is the modified amplitude of the wave packet which is modulated both in space and time by a very slowly varying envelop of frequency and propagation constant, and has maximum value of 2A.

The velocity with which this wave packet moves or group velocity vg of the wave packet is

given by kkkvg

21

21

This is the required expression for group velocity. If a bunch of waves contains a number of frequency components in an infinitely small frequency interval.

Relation between Group Velocity and Wave (or Phase) Velocity If the individual waves with which a wave packet is constructed have an average velocity or the wave

velocity vp , then

pp kvork

v

According to the expression of group velocity

d

dvvv

ddBut

d

dvvvor

d

dvvv

Hencewavelengththeiswherekhere

dk

dvkv

dk

vkd

dk

dv

p

pg

p

pg

p

pg

p

p

p

g

,11

1

1

2

2

,.,2

2

This is the required relation between group velocity vg and wave velocity vp in a dispersive medium in which the wave velocity is frequency dependent.

In a medium in which the wave velocity is independent of frequency, that is, vp is constant, therefore,

,0d

dvp

Hence, above equation becomes vg = vp

that is,. In a non-dispersive medium( where vp is constant) or in free space, the group velocity of a wave packet is equal to wave (or phase) velocity.

Relation Between Group Velocity of de-Broglie’s waves and Particle Velocity

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Suppose a particle of rest mass m0 is moving with velocity v (comparable with c). Its total energy and momentum is given by

2

0

2

2

02

11

c

v

vmmvpand

cv

cmmcE

(1)

The frequency and the angular frequency of the de-Broglie’s wave are given by

22

0

2

2

0

1

22

1cvh

cmand

cvh

cm

h

E

(2)

Differentiating Eq. 2 with respect to , we get

2/3

2

2

2

0

2

2/322

0

2/122

0

1

2

21

2

121

2

c

vh

cm

d

dor

cc

v

h

cm

c

v

d

d

h

cm

d

d

(3)

The de-Broglie wavelength and the propagation constant k of the wave associated with moving particle are given by

2/3

2

2

0

2

2

2

22/32

0

2/12

2

2/32

0

2/12

0

2

0

0

2/12

1

2

112

12

12

12

12

,

1

221

cvh

m

d

dk

c

v

c

v

cv

h

mor

cv

ccvv

h

m

cvv

d

d

h

m

d

dk

getwevtorespectwithEqaboveatingDifferenti

cvh

vmkand

vm

cvh

p

h

mv

h

According to definition, the group velocity vg of the de-Broglie’s waves associated with the particle is

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ddk

dd

dkdvg

substituting the value of (d/d) and (dk/d) in the above equation, we get

vg = v

Hence, the wave group or wave packet associated with a moving particle travels with the velocity of moving particle. Hence, a moving particle is equivalent to a wave packet or group of waves.

Derivation of de-Broglie’s wavelength expression using group velocity

The de Broglie relation may be derived as follows. If we assume a particle having a kinetic energy

equal to mv2/2 to have a de-Broglie wavelength , we can write

2/2mvh (assuming the energy of the particle to be purely kinetic)

or 2.

2v

h

m (1)

Differentiating with respect to ,

d

d

h

m

d

d.2.

2 (2)

But we have

2

2

12

2

d

d

d

d

d

d

dk

dvvg

(3)

Substituting in Equation (2), we get 2.

dd

d

h

mv

Rewriting this, we have 2

m

h

d

d

Integrating with respect to cm

h

By applying the boundary condition that the wavelength tends to infinity as the velocity tends to zero, we find the constant of integration has to be zero. Hence, we get

p

h

mv

h which is the de Broglie relation.

Numericals

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1 Calculate the number of photons emitted in 3 hrs. by a 60 watt sodium lamp. Given =589.3nm.

2 Light of wave length 4047A0 falls on a photoelectric cell with a sodium cathode. It is found that the photoelectric current ceases when a retarding potential of 1.02 volt is applied. Calculate the work function of the sodium cathode.

3 The most rapidly moving valence electron in metallic sodium at absolute zero temperature, has a kinetic energy 3eV. Show that the de Broglie wavelength is 7A0.

4 Calculate the momentum of an electron possessing the de Broglie wavelength 6.62 x 10-11m.

5 Find the phase and group velocities of an electron whose de-Broglie wavelength is 0.12nm.

6 Calculate the wavelength of a 1kg object whose velocity is 1m/s and compare it with the wavelength of an electron

accelerated by 100 volt (0=6.62 x 10-34 m, e = 1.2 x 10-10m and e = 1.8 x 1023 0)

7 Calculate the de-Broglie wavelength associated with 400 gm cricket ball with a speed of 90Km/hr. (VTU August 2006)

8 Compare the energy of a photon with that of an electron when both are associated with wave length of 0.2nm. (VTU Feb 2006, August 2003, IAS- 1987)

9 Calculate the wavelength associated with electrons whose speed is 0.01 of the speed of light. (VTU August 2004)

10 The velocity of an electron of a Hydrogen atom in the ground state is 2.19 x 106m/s. Calculate the wavelength of the de Broglie waves associated with its motion.

11 Compute the de Broglie wavelength for a neutron moving with one tenth part of the velocity of light.

12 Estimate the potential difference through which a proton is needed to be accelerated so that its de Broglie wavelength becomes equal to 1A0, given that its mass is 1.673 x 10-27kg.

13 Calcualte the de Broglie wavelength associated with an electron with a kinetic energy of 2000eV. (VTU March 2006)

14 Evaluate de Broglie wavelength of Helium nucleus that is accelerated through 500V.

15 A particle of mass 0.5 Mev/c2 has kinetic energy 100eV. Find its deBroglie wavelength, where c is the velocity of light. (VTU Jan 2007, IAS 1978)

16 Compare the momentum, the total energy, and the kinetic energy of an electron with a deBroglie wavelength of 1A0, with that of a photon with same wavelength.

17 Calculate the de Broglie wavelength of a proton whose kinetic energy is equal to the rest energy of the electron. Mass of proton is 1836 times of electron.

18 A fast moving neutron is found to have an associated de Broglie wavelength of 2 x 10-12m. Find its kinetic energy and the phase and group velocities of the de Broglie waves ignoring the relativistic change in mass. Mass of neutron= 1.675 x 10-27kg.

19 If an electron has a de Broglie wavelength of 2 nm, find its kinetic energy and group velocity, given that it has a rest mass energy of 511 kev.

20 A Particle of mass 0.65 MeV/c2 has kinetic energy 80eV. Find the de Broglie wavelength, group velocity and phase velocity of the de Broglie wave. (VTU Model Question Paper)

21 An electron has a wavelength of 1.66 x 10-10m. Find the kinetic energy, phase velocity and group velocity of the de Broglie wave. (VTU August 1999)

22 Calculate the wavelength associated with an electron having kinetic energy 100eV. (VTU July 2002)

23 Calculate the wavelength associated with an electron raised through a potential difference of 2 kV. (VTU Feb 2002)

Questions

1 What is a black body? Explain the nature of the black body radiation?

2 Explain Wien’s law and Rayleigh-Jeans law. Mention their drawbacks to explain the Lummer and Priengshiem experimental result?

3 Show how Rayliegh-Jeans law’s drawbacks can be overcome using Planck’s law of radiation?

4 Describe the Ultraviolet catastrophe.

5 State and explain Planck’s law of radiation. Show that it reduces to Wien’s law and Rayleigh-Jeans law under certain conditions?

6 Explain Planck’s quantum hypothesis. What is Planck’s radiation law?

7 Explain Einstein’s theory of Photoelectric effect. Mention two factors controlling photoelectric effect.

8 Give a qualitative account of Compton effect.

9 Explain the duality of matter waves from the inferences drawn from photoelectric effect and Davisson-Germer Experiment.

10 Explain the dual nature of matter and arrive at the concept of matter.

11 State de Brolige hypothesis. Show that the de Brolige wavelength for an electron accelerated by a potential difference

V volts is = 1.226 /v nm for non-relativistic case.

12 Explain the characteristics of matter wave.

13 Describe Davisson and Germer’s experiment and explain how it established the proof for wave nature of electrons?

14 Discuss Phase velocity and Group velocity? And obtain the expressions for both

15 Explain group velocity and phase velocity and derive the relation between them.

16 Deduce an expression for de Broglie wavelength in terms group velocity.

17 Derive expressions for phase velocity and group velocity on the basis of superposition of waves.

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18 Show that the group velocity of de Brolige waves is equal to the velocity of the particle with which the waves are associated.

For Reference only (Not in the VTU Syllabus)

The discovery of Compton scattering of x-rays provides direct support that light consists of point like quanta of energy called photons.

The experiment:A graphite target was bombarded with monochromatic x-rays and the

wavelength of the scattered radiation was measured with a rotating crystal spectrometer. The

intensity was determined by a movable ionization chamber that generated a current

proportional to the x-ray intensity. Compton measured the dependence of scattered x-ray

intensity on wavelength at three different scattering angles of 45o, 90o, and 135o. The

experimental intensity vs. wavelength plots observed by Compton for the above three

scattering angles (See Fig. below) show two peaks, one at the wavelength of the incident

x-rays and the other at a longer wavelength '. The functional dependence of ' on the

scattering angle and was predicted by Compton to be:

' - = (h/mec)[ 1- cos ] =0 [ 1- cos ].

The factor 0=h/mec, also known as Compton wavelength can be calculated to be equal to

0.00243 nm.

being much heavier compared to the electron, produces negligible wavelength shift.

Conclusion:Compton effect gives conclusive evidence in support of the corpuscular character of electromagnetic radiation.

Wave-Particle Duality

j / Bullets pass through a double slit.

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k / A water wave passes through a double slit.

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