Module 3 Engg Phys.pptx

Quantum nature of radiation
1st evidence from spectrum emitted by a black-body
A black body is a theoretical object that absorbs 100% of the radiation that
hits it. Therefore it reflects no radiation and appears perfectly black.
In practice no material has been found to absorb all incoming radiation .
An object that absorbs all incident radiation, i.e. no reflection - black body
An object that reflects all incident radiation, i.e. no absorption – white body.
What is a black body ?
What is a white body?

A black-body?
(An object that absorbs all incident radiation, i.e. no reflection)
A small hole cut into a cavity is the most popular
and realistic example.
None of the incident radiation escapes
What happens to this radiation?
•The radiation is absorbed in the walls of the cavity
•This causes a heating of the cavity walls
•Atoms in the walls of the cavity will vibrate at frequencies characteristic of
the temperature of the walls
•These atoms then re-radiate the energy at this new characteristic frequency
The emitted "thermal" radiation characterizes the
equilibrium temperature of the black-body

A black-body reaches thermal equilibrium when the incident
radiation power is balanced by the power re-radiated, i.e. if you expose
a black-body to radiation, its temperature rises until the incident and
radiated powers balance.
The maximum wavelength emitted by a black body radiator is
infinite. It also emits a definite amount of energy at each wavelength
for a particular temperature, called black body radiation curves,. All
objects emit radiation above absolute zero.
• The emission from a black-body depends only on its temperature.
• Objects at 300 K radiate in the infrared.
•Objects at 600 - 700 K start to glow.
•At high T, objects may become white hot

Black-body spectrum
Fig : Black body radiation curves showing peak wavelengths at various temperatures
This graph shows how the black
body radiation curves change at
various temperatures.
All these have their peak
wavelengths in the infra-red part
of the spectrum as they are at a
relatively lower temperature
A black-body at room temperature appears
black, as most of the energy it radiates is
infra-red and cannot be perceived by the
human eye.
At higher temperatures, black bodies glow
with increasing intensity and colors that
range from dull red to blindingly brilliant
blue-white as the temperature increases.

As the temperature increases, the
peak wavelength (not peak
intensity) emitted by the black body
decreases.
ie. as the temperature increases
from 1250 to 2000K, the peak
wavelength decreases from
2.5um(2500nm) to 1.5um (1500 nm)

LAVA flow
One can measure the
temperature of the lava
using the colour.

19th century a major problem for physicists : To predict the intensity of
radiation emitted by a black body at a specific wavelength.
Wien’s theory : Predicted the overall form of the curve by treating the
radiation as gas molecules.
Draw back: However, at long wavelengths his theory disagreed with
experimental data.
Rayleigh and Jeans : Later given a formula by considering the radiation
within the black body cavity to be made up of a series of standing waves.
They thought that electromagnetic radiation was emitted by oscillating atoms
in the walls of the black body and this radiation set up a standing wave
between the walls. Their formula stated: as given ……………
Wien, Rayleigh-Jeans and Planck distributions
     
 
/
RJ W P /
4 5 5
8 8
; ;
1
B
T
B
hc k T
k T e hc
u u u
e
 

 
  
  

  


Draw backs : For large wavelengths it fitted the experimental data but it had
major problems at shorter wavelengths.
As the wavelength tended to zero, the curve would tend to infinity.
However we know that there is a peak wavelength for each temperature,
and the energy emitted at either side of this peak dropped.
The Rayleigh-Jeans Law predicted no peak wavelength.

Ultraviolet catastrophe
The formulae failed to explain and account for energy outputs at short
wavelengths (the ultraviolet wavelengths) known as the ultraviolet
catastrophe.
The failure of Rayleigh – Jeans law is known as the ultraviolet catastrophe.

Comparison between Classical and
Quantum viewpoint
There is a good fit at long wavelengths, but at short wavlengths there is a
major disagreement. Rayleigh-Jeans ∞, but Black-body 0.

Max Planck
The law was the first to accurately describe black body radiation, and resolved
the ultraviolet catastrope. It is a breakthrough in modern physics and
quantum theory.
A major breakthrough was made by Max Planck who made a formula that
agreed with experimental data.

In 1900, Max Planck studied black body radiation from the
standpoint of thermodynamics.
He proposed that the energy of each electromagnetic
oscillator is limited to discrete values and cannot be varied
arbitrarily, contrary to the assumptions of classical physics.
This limitation of energies to discrete values is called the
quantization of energy.
Planck corrected the problem by assuming that only
specific energies hν could be involved. These permitted
energies of an electromagnetic oscillator are integer
multiples of hν
E = nhν ; n = 0, 1, 2, ……….
where h is Planck’s constant. On the basis of this
assumption, Planck was able to derive the Planck’s
Distribution, which fit the experimental data well at all
wavelengths:

Before Planck it was assumed that these could have any value of energy,
but
Planck decided that the energy must go up in discrete amounts (quantised)
because the frequencies of the oscillating electrons could only take certain
values.
As energy is proportional to frequency (E = hf) , where h is the Planck
constant 6.626 x 10-34 Js) if frequency can only take discrete values,
this means that energy is also quantised.
The electrons have a fundamental frequency (like standing waves on a
string) and the frequency can only go up in whole multiples of this
frequency, called the quantum number. This assumption led Planck to
correctly derive his formula.

Limitations of Classical theory (failures)
1. Black body radiation
2. Origin of discrete spectra of atoms (gases)
3. Variation of specific heat of metals and gases
4. Stability of atoms
5. Failed to explain photoelectric effect, Compton effect,
Zeeman effect, Raman effect, etc….

Quantization is the procedure of constraining something from a
continuous set of values (such as the real numbers) to a relatively small
discrete set (such as the integers).
In Physics, quantization is the process of transition from a classical
understanding of physical phenomena to a newer understanding known
as “ quantum mechanics".

Planck’s Radiation Formula
In 1900 Planck explained the experimentally observed distribution of energy in the
spectrum of black body and suggested that the correct results can be obtained if the
energy of oscillating electrons is taken as discrete rather than continuous.
He derived the radiation law by using the following assumptions:
1. A chamber containing blackbody radiations also contains simple harmonic oscillators
of molecular dimensions which can vibrate with all possible frequencies.
2. The frequency of radiation emitted by an oscillator is the same as the frequency of
vibration.
3. An oscillator cannot emit energy in a continuous manner, it can emit energy in the
multiples of a small unit called quantum (photon). If an oscillator is vibrating with a
frequency ‘ν’, it can radiate in quantas of magnitude hν; the oscillator can have
discrete energies values En given by
En= nhν ; where n – an integer; h – Plancks constant (6.625 x 10-34 Joule- sec)
4. The oscillators can emit or absorb radiation energy in packet of hν. (revolutionary
assumption)
This implies that the exchange energy between radiation and matter cannot take
place continuously but are limited to discrete set of values 0, hν, 2hν, 3hν, ……….nhν.

Plancks radiation formula in terms of frequency
Plancks radiation formula in terms of wavelength

The Planck constant compares the sum of energy a photon bears with its
electromagnetic wave frequency. It is named after Max Planck, the physicist.
In quantum mechanics, it is an essential quantity.
What is Planck’s constant in simple terms?

THEORY
• It was thought that light had a wave nature.
Phenomena such as interference, diffraction and polarisation can be
explained only on the basis of wave nature of light.
At the same time, phenomena such as photo electric effect and
compton effect, etc could not be explained by the wave nature of light.
• The theory behind the Compton Effect was to prove that light has a
particle nature also.
Polarized light waves are light waves in which the vibrations occur in a single
plane. The process of transforming unpolarized light into polarized light is known
as polarization.

Compton effect
• Experiment revealing the particle nature of X-ray (radiation, with
wavelength ~ 10-10 nm)
Compton, Arthur Holly (1892-1962), American
physicist and Nobel laureate whose studies of X-
rays led to his discovery in 1922 of the so-called
Compton effect.
The phenomenon in which the wavelength of
scattered X-rays is greater than the
of the incident X-rays is called Compton
The difference in wavelength is known as
Compton shift.
The Compton effect is the change in
of high energy electromagnetic radiation
scatters off electrons.
The discovery of the Compton effect confirmed that electromagnetic
radiation has both wave and particle properties, a central principle of
quantum theory.

The greater the angle of scatter, the more energy is lost by the
photon
)
cos
1
(
'


 


c
m
h
o
Compton scattering is an inelastic scattering of a photon by a free charged
particle, usually an electron.
Compton scattering is an example of inelastic scattering, because the wavelength of
the scattered light is different from the incident radiation
The phenomenon in which the
wavelength of the scattered X-
rays is greater than the
wavelength of the incident X-rays
is called Compton Effect.
The difference in wavelength is
known as Compton shift.
The value of Compton shift is
dependent on the angle of
scattering.

Experimental Demonstration of Compton Effect

A beam of X-rays falls on the target (graphite).
Detector capable of moving in an arc, measures the energy of the scattered
X-rays at various angles of θ.
The graphs show the Compton experimental results.
For each angle other than 0o, two peaks appear.
Wavelength of 1st peak matches with the original (incident ) wavelength. This
corresponds to the photon that gets scattered from the tightly bound electrons of
the graphite. Hence they don’t lose their energy.
Wavelength of the second peak corresponds to the scattered photon due to
loosely bound electron in the graphite, where they lose some amount of energy.
Compton performed the experiment with gamma rays and also found that there
was no change in Compton shift, proving that Compton shift does not depend on
the incident wavelength.
WRITE - DERIVATION DERIVED IN CLASS ON THE BOARD

photon + electron photon + electron

Beam of x-ray with sharp wavelength 
falls on graphite target. For various angle
q the scattered x-ray is measured as a
function of their wavelength
Unexplained by classical
wave theory for radiation
No shift of wavelength is
predicted in wave theory
of light
Although initially the incident beam
consists of only a single well-defined
wavelength () the scattered x-rays have
intensity peaks at two wavelength (’ in
addition), where ’ > .
Wavelength
shift

Wave nature
We regard EM waves as WAVES because under suitable circumstances they
exhibit
1) Diffraction
2) Interference
3) Polarization
Particle nature
Similarly other circumstances they behave as though they consist of stream
of PARTICLES
1) Photoelectric effect
2) Compton effect.
Thus radiation or light waves has dual nature (both wave
and particle nature)

The properties of blackbody radiation, the radiation emitted by hot objects,
could not be explained with classical physics.
Max Planck postulated that energy was quantized and could be emitted or
absorbed only in integral multiples of a small unit of energy, known as a
quantum.
The energy of a quantum is proportional to the frequency of the radiation; the
proportionality constant h is a fundamental constant (Planck’s constant).
Albert Einstein used Planck’s concept of the quantization of energy to explain
the photoelectric effect, the ejection of electrons from certain metals when
exposed to light.
Einstein postulated the existence of what today we call photons, particles of
light with a particular energy, E = hν.
Both energy and matter have fundamental building blocks: quanta and atoms,
respectively.
Idea of Quantization (Planck and Einstein)

The photoelectric effect was the observation that an electric current can be
generated if light of certain frequencies hits the surface of a metal object, but that
light of only certain colors can do this.
In fact, for each type of metal there was a maximum wavelength that could eject an
electron, and any light with a longer wavelength would have no effect, even if you
increased its intensity.
Einstein recognized that this could be explained by Plank's equation where one
could define a "particle of light" or photon as having a fundamental energy of
in an electric field, it can kick an electron off the object and a current can flow.
Einstein was able to use Plank's quantizations hypothesis to explain the
photoelectric effect, where a "particle of light," a photon, had a characteristic
energy described by Plank's constant
Ephoton=hν
Photoelectric Effect

As can be showed in figure, a minimum energy of 2.0 eV is required to eject a
photon off of potassium, and so red light would not work, while green and purple
would.
Fig. : Potassium requires 2.0eV to eject an electron, and a photon of red light
(700nm) only has 1.77eV and so no matter intense the red light is, it can not kick of
an electron.
Green and purple light have more energy, and when you shine them on the
potassium you can kick off an electron.

Louis de Broglie
• 1892 – 1987
• French physicist
• Originally studied history
• Was awarded the Nobel
Prize in 1929 for his
prediction of the wave
nature of electrons
de Broglie (1924) made a great unifying, speculative hypothesis that just as
radiation has particle-like properties, electrons and other material particles
possess wave-like properties.
For free material particles, de Broglie assumed that the associated wave also
has a frequency and wavelength related to its energy E and momentum p

The concept of dual nature of radiation prompted Louis de Broglie to suggest the
idea of matter waves.
In 1924, his suggestion was based on the following facts
1. Nature loves symmetry (nature is symmetrical)
2. The entire universe consists of radiation (energy) and matter only
Therefore the two physical entities ie. matter and energy must be symmetrical .
ie. to say, if radiant energy has dual characterization, matter must also have dual
(particle like and wave like) nature.
So a moving particle is associated with a wave known as matter wave or pilot
wave or de Broglie wave.
(symmetry : two halves of an object are exact mirror images of each other)
λ = h/p

Properties of MATTER WAVES.
1. Lighter is the particle, greater is the wavelength associated with it.
2. Smaller is the velocity of the particle, greater is the wavelength associated with
it.
3. Matter waves are not electromagnetic waves.
4. Matter waves are generated by the motion of particles. If the particles are at
rest, then there is no meaning of matter waves associated with them.
5. The wavelength of the matter waves are independent of charges on the
particles, but depends upon the velocity of particle.
6. These waves can travel faster than the velocity of light.
7. The velocity of the matter waves is not constant as that of E-M radiation. It
depends on the velocity of the material particle.
8. The only function of the wave is to pilot or to guide the matter particles as
shown and hence it is called as pilot wave.
9. The matter wave is not a physical phenomenon. It is rather a symbolic
representation of what we know about the particle. It is a wave of probability.

Davisson and Germer Experiment
This experiment performed in 1927 confirmed the de Broglie hypothesis.
Ie. To prove the hypothesis that particles of matter such as electrons have
wave like properties.
The experiment not only played a major role in verifying the de Broglie
hypothesis and demonstrated the wave-particle duality, but also was an
important historical development in the establishment of quantum mechanics
and Schrodinger equation

The Thought Behind the
Experimental Setup
• The basic thought behind the
Davisson and Germer
experiment was that the waves
reflected from two different
atomic layers of a Ni crystal
will have a fixed phase
difference. After reflection,
these waves will interfere
either constructively or
destructively. Hence producing
a diffraction pattern.
• In the Davisson and Germer
experiment waves were used
in place of electrons. These
electrons formed a diffraction
pattern. The dual nature of
matter was thus verified. We
can relate the de Broglie
equation and the Bragg’s law
as shown below:

Davisson and Germer
Experiment

• Electrons were directed onto nickel
crystals
• Accelerating voltage is used to control
electron energy: E = |e|V
• The scattering angle and intensity
(electron current) are detected
– φ is the scattering angle
https://www.youtube.com/watch?v=Ho7K27B_Uu8

Observations of the Davisson and Germer Experiment
• The detector used here can only detect the presence of an electron in the
form of a particle. As a result, the detector receives the electrons in the form
of an electronic current.
Results of the Davisson and Germer Experiment
• From the Davisson and Germer experiment, we get a value for the
scattering angle θ and a corresponding value of the potential difference V
at which the scattering of electrons is maximum. Thus these two values
from the data collected by Davisson and Germer, when used in equation
(1) and (2) give the same values for λ. Therefore, this establishes the de
Broglie’s wave-particle duality and verifies his equation as shown below:

The maximum interference is due to constructive interference , a
phenomenon confined only to waves.

Davisson and Germer Experiment
• Observations:
– Current vs accelerating
voltage has a maximum,
i.e. the highest number
of electrons is scattered
in a specific direction
– This can’t be explained
by particle-like nature of
electrons  electrons
scattered on crystals
behave as waves
For φ ~ 50° the maximum is at ~54V

The peak
indicates the
wave nature of
electrons.
When accelerated at different voltages……………..

i) de Broglie wavelength verification
2d sinΦ = nλ
As ‘d’ was known to be 2.15Ao
from x-ray diffraction
measurements, Davisson and
Germer calculated , λ to be
λ= (2.15 Ao) sin (50o) = 1.65 Ao
Result is in excellent agreement with the de Broglie
formula
ii) Checking with Braggs law
For a single layer of atoms, Braggs law
turns to be

The more precisely the position is
determined, the less precisely the
momentum is known in this instant,
and vice versa.
--Heisenberg, uncertainty paper, 1927

In quantum physics, the Heisenberg uncertainty principle states that certain pairs
of physical properties like position and momentum cannot both be known to
arbitrary precision. I.e.. More precisely one property is known, the less precisely
the other can be known.
It is impossible to measure simultaneously both position and velocity of a
microscopic particle with any degree of accuracy or certainty.
In quantum mechanics a particle is described by a wave. The position is where
the wave is concentrated and the momentum is the wavelength. The position is
uncertain to the degree that the wave is spread out and the momentum is
uncertain to the degree that the wavelength is ill- defined.
“It is impossible to know both the exact position and exact momentum of an
object at the same time”.

The Heisenberg uncertainty principle states that in an simultaneous
determination of a pair of physical quantities which describes the motion
of an atomic particle , the product of uncertainties is equal to

As a result it is impossible to design an experiment to prove the wave and particle
nature of matter at the same time.
If one measures the position accurately, then the measurement of momentum
becomes incorrect and vice versa.
or
The principle can also be written as
or the errors in the measurement of energy and time respectively
Wave diagram refer (single and group waves)

Physical significance of Ψ
1. It relates the particle and wave nature of matter statistically.
2. It is a complex quantity and hence we cannot measure it. It shall be expressed of
the form
where ‘a’ and ‘b’ are real functions of variables (x,y,z,t)
The complex conjugate of Ψ is denoted by Ψ* by changing ‘i” to ‘-i’
ib
a
t
z
y
x 

 )
,
,
,
(
ib
a
t
z
y
x 


)
,
,
,
(
2
2
)
,
,
,
(
)
,
,
,
( b
a
t
z
y
x
t
z
y
x 


 


2
)
,
,
,
(
)
,
,
,
(
)
,
,
,
( t
z
y
x
t
z
y
x
t
z
y
x 


 
2
2
2
)
,
,
,
( b
a
t
z
y
x 


2
)
,
,
,
( t
z
y
x

(1)
(2)
multiplying (1) and (2)
(3)
The product (LHS) is denoted by P
Ie. P =
(4)
3. Its square is a measure of the probability of finding the particle at a particular
position . It cannot predict the exact location of the particle
is called probability density

dv
t
z
y
x
t
z
y
x
Pdv )
,
,
,
(
)
,
,
,
( 



dv
t
z
y
x
2
)
,
,
,
(


dx
t
x
t
x
Pdx
2
)
,
(
)
,
( 



dx
t
x
Px
2
)
,
(

 



dv
t
z
y
x
P
2
)
,
,
,
(

 



1
)
,
,
,
(
2




dv
t
z
y
x


4. For the motion of a particle, the probability of finding the particle in a
volume element ‘dv’ is
For the motion of the particle in one dimension,
probability per unit
distance
Therefore , the total probability is
For 3-dimensional space
5. The normalising condition for the wave function for the motion of
a particle in 3-dimension is
Where Ψ is the function of space coordinates (x,y,z)
(7)









0
)
( dx
x
j
i
6. If the wave function does not satisfy the normalization condition eqn (7), then
it must be multiplied by a constant factor called normalization factor.
For eg.
 
 N
dx
2
1

N
( )
The normalization factor will be and the normalized wave function
will be N
1

N
1
7. The wave function Ψ must approach zero as ‘x’ tends to ±α.
8. The wave function satisfying the following condition is said to be an
orthogonal wave function
The orthogonality condition guarantee the non-interference of the
wave function representing different states.
9. Degeneracy
If Ψi and Ψj corresponds to the same energy “E” then this is called as degenerate
state.

 



1
dv
j
i
10. Orthonormal set: The normalization and orthoganility conditions may be
combined as follows
j
i 
j
i 
0
 (10)
Wave function of the form eqn (10) is said to be orthogonal wave function.
11. Ψ must be finite for all values of x,y,z.
12. Ψ must be well behaved i.e single valued and continuous every where.
13. Ψ must be continious in all regions except in those region where the
potential energy V(x,y,z) = α

E < V
The particle/electron can tunnel/
penetrate even if the energy of the
particle E, is less than the energy of
the potential barrier V.

Also Refer
https://www.youtube.com/watch?v=BcgG3Cp8QQY
https://www.youtube.com/watch?v=wNEqRq6NyUw
http://www.iap.tuwien.ac.at/www/surface/stm_gallery/animated_stm
http://www.hk-phy.org/atomic_world/stm/stm04_e.html#working_principle
http://virtual.itg.uiuc.edu/training/AFM_tutorial/

Application: Tunneling Microscopy
 Due to the quantum effect of “barrier
penetration,” the electron density of a
material extends beyond its surface:
material STM tip
STM tip
material
~ 1 nm
x Metal
tip
One can exploit this
to measure the
electron density on a
material’s surface:
Na atoms
on metal:
Real STM tip
STM images
DNA Double
Helix:
http://www.quantum-physics.polytechnique.fr/en/

In the above slide, the material is applied with negative potential and the tip is
applied with positive potential. Hence the electron tunnels from the electron
cloud of the material to the tip surfaces.
It can be vice versa also…………..
Tip can be with negative potential and the sample can be applied with positive
potential……. In that case tunneling will take from right side to left side.
1. Constant height mode
2. Constant current mode

Module 3 Engg Phys.pptx

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