Chapter 05 of the document covers key concepts in quantum physics, including black body radiation, the photoelectric effect, the Compton effect, and the de Broglie hypothesis. It discusses the mathematical framework and applications of quantum mechanics, the principles of radiation, and the quantization of energy, providing insights into various laws and phenomena such as Stefan-Boltzmann law and Planck’s hypothesis. Experimental evidence supporting wave-particle duality is also presented through significant experiments like the Davisson-Germer experiment and the double slit experiment.

1. Quantum Physics & Mechanics
CHAPTER 05

2. Title and Content Layout
• Black body radiation
• Photoelectric effect
• Compton effect
• De-Broglie Hypothesis
• Quantum mechanic postulate

3. Introduction to Quantum Physics
Quantum physics is a branch of physics that describes the physical properties of
nature at the scale of atoms and subatomic particles.
1. Mathematical Framework: Quantum mechanics is built on a mathematical framework that
uses wave functions to describe the state of a particle.
1. Applications: Quantum mechanics has led to the development of numerous
technologies, including lasers, transistors, magnetic resonance imaging (MRI), and
nuclear weapons.
Quantum Theory of Radiation:
It explains that energy is emitted and absorbed in discrete packets called 'quanta,'
which, in the case of electromagnetic radiation, are known as 'photons.'

4. BLACK BODY RADIATION:
A black body is an ideal body that absorbs all radiation incident on it surface regardless of frequency or angle of
incidence. Unlike real objects, a perfect blackbody does not reflect or transmit any radiation; it is a perfect absorber
and emitter. In thermal equilibrium it emits radiation at a specific spectrum that depends on its temperature, known as
black body radiation.
• There is no such body which is 100% blackbody (carbon-graphite is 96%).
Example:
1. Sun at higher temperature radiate or emit all radiation
2. Earth absorb all radiation up to 99%.
(candle is not black body because it emit only yellow)
A good approximation of blackbody is a small hole leading to the inside of hollow
object, shown in figure. Radiation incident on a hole trapped in the cavity.

5. • Light emitted by any object didn’t depend on size and nature of object, but it depend on temperature.
• Any object heated around 4000K it give out radiation in visible region (1st Red) more heated it will
become blue and finally white (all 𝜆 of visible region)
• When light interact with an object surface atom K.E increases because it oscillate more.
𝐾. 𝐸 ↑ ≡ 𝑇𝑒𝑚𝑝 ↑
• E=hf so, 𝑇𝑒𝑚𝑝 ↑ ≡ 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 ↑

6. Classical View and the Blackbody Spectrum
In the late 19th century, physicists attempted to describe the energy distribution of radiation emitted by a blackbody at
different temperatures
1) Stefan Boltzmann Law:
It states that the radiated intensity of a blackbody is directly proportional
to the fourth power of its absolute temperature (T).
Mathematically:
𝐼 𝑇 𝛼 𝑇4
𝐼 𝑇 = 𝜎𝑒𝑇4
Where,
𝜎 = Stefan Boltzmann constant = 5.67 × 10−8
𝑊/𝑚2
𝐾4
e = emissivity
e = 1 for ideal black body
e = 0 for perfect reflector e: define as the ratio of radiation emitted by a surface of a body
to the radiation emitted by blackbody.

7. 2. Wien Displacement Law:
Wien’s Displacement Law is a fundamental principle in thermal radiation that describes how the peak wavelength of radiation
emitted by a blackbody shifts with temperature. This law helps us understand the color changes in heated objects.
Statement:
It states that the wavelength at which the emission of a blackbody is at its
maximum intensity is inversely proportional to the temperature of the blackbody.
Mathematically
𝜆𝑚𝑎𝑥 =
𝑏
𝑇
b = Wien’s displacement constant = 2.897×10-3 m⋅K
3. Rayleigh-Jeans Law:
It states that the intensity of blackbody radiation is inversely proportional to fourth power of the wavelength.
Mathematically
𝐼 𝜆, 𝑇 =
2𝜋𝑘𝑏𝑇
𝜆4
This theory worked well at long wavelengths (low frequencies) but diverged at shorter wavelengths, predicting an infinite intensity
of radiation as the wavelength decreases, especially in the ultraviolet region.

8. Ultraviolet Catastrophe:
The failure of classical physics to describe the behavior of blackbody radiation at short wavelengths was a significant problem.
Experimental observations showed that at lower wavelength, the intensity sharply decreases rather than increasing infinitely. This
discrepancy between theory and observation became known as the ultraviolet catastrophe.
4. Planck’s Hypothesis:
Max Planck was the first to introduce the idea of quantized energy to explain blackbody radiation. He proposed that the energy of
electromagnetic waves is not continuous but emitted in discrete packets, or quanta. This revolutionary idea explained the observed
black body radiation spectrum.
𝐼 𝜆, 𝑇 =
2ℎ𝑐2
𝜆5
1
𝑒
ℎ𝑐
𝜆𝐾𝑇 − 1
𝐼 𝑓, 𝑇 =
2ℎ𝑓3
𝑐2
1
𝑒
ℎ𝑓
𝐾𝑇 − 1

9. He gave two assumption
1) The energy of an oscillator can have only certain discrete value 𝐸𝑛.
𝐸 = 𝑛ℎ𝑓
2) Oscillator emit or absorb energy when making a transition from one quantum state to another.
Δ𝐸 = 𝐸𝑓 − 𝐸𝑖
(a) Point (1):
At shorter wavelengths (𝜆), the separation between quantum energy levels
is large. This results in a lower probability of transitions between these
states. Consequently, the intensity of radiation is low.
(b) Point (2):
At the peak of the curve (intermediate wavelength), the energy levels are
closer together, and transitions have a high probability. This leads to the
highest intensity of radiation.
(c) Point (3):
At longer wavelengths (𝜆), the energy difference becomes smaller, but the
probability of transitions and the energy associated with them decrease. As
a result, the intensity gradually decreases.

10. PHOTOELECTRIC EFFECT:
• Work function:
The work function of a material is the minimum amount of energy required to eject an electron from its surface. It is measured in
electron volts (eV).
• Threshold frequency:
The threshold frequency of a material is the minimum frequency of light (or other electromagnetic radiation) needed to eject an
electron from its surface. It is measured in hertz (Hz).
The photoelectric effect is the emission of electrons from a material when it absorbs electromagnetic radiation (light). When a
photon's energy is greater than or equal to the work function of the material, it can eject an electron. These ejected electrons are
called photoelectrons.

11. Experimental Result:
1. At constant potential difference, number of photons emitted from cathode is
directly proportional to intensity of radiation.
2. For a constant intensity, the photocurrent varies with the potential difference V
and reaches a constant value beyond which further increase in potential has no
effect; instead, a negative potential (retarding potential) reduces the current until
it stops. Point where photocurrent become zero known as cut-off or stopping
potential.
𝐾. 𝐸max = 𝑒𝑉
3. The maximum kinetic energy of photoelectrons depends on the light frequency,
not intensity.
𝐸 = ℎ𝑓
4. Each substance has a threshold frequency, below which no photoelectrons are
emitted.
5. The time delay between radiation incidence and photoelectron emission is
extremely small, less than 10-9 seconds.

12. Einstein Analysis:
1) Einstein suggested that the electromagnetic radiation field is quantized into particles called photons. Each photon has the
energy quantum of
𝑬 = 𝒉𝒇 =
𝒉𝒄
𝝀
Where, f is the frequency of light and h is the Plank’s constant.
2) The Photon travels at the speed of light in a vacuum, and it’s wavelength is given by
𝝀 =
𝒄
𝒇
3) Conservation of energy:
𝑬𝒏𝒆𝒓𝒈𝒚 𝒃𝒆𝒇𝒐𝒓𝒆 𝑷𝒉𝒐𝒕𝒐𝒏 = 𝑬𝒏𝒆𝒓𝒈𝒚 𝒂𝒇𝒕𝒆𝒓 (𝒑𝒉𝒐𝒕𝒐𝒏)
𝒉𝒇 = 𝝓 + 𝒌. 𝑬 (𝒆𝒍𝒆𝒄𝒕𝒓𝒐𝒏)
Where, 𝝓 is the work function of the metal
𝒉𝒇 = 𝝓 +
𝟏
𝟐
𝒎𝒗𝟐
𝒉𝒇 = 𝒉𝒇𝒐 +
𝟏
𝟐
𝒎𝒗𝟐

13. NUMERICAL
Q1: A 430 nm Violet light incident on a calcium photoelectrode with a work
function of 2.71 eV. Find the energy of the incident photons and the maximum K.E
of ejected electron.
Q2: When an 180 nm light is used in an experiment to a metal, the measured
photocurrent drops to zero at potential 0.8V. Determine work function of metal
and its cutoff frequency.

14. Compton Effect
It is a phenomenon in which high energy electromagnetic waves (X-ray) are observed to be shifted to a longer
wavelength after being scattered off of the electron in material.
• This wavelength shift cannot be explained using classical wave theory.
• It provide evidence for the particle nature of light supporting the concept of photon and quantum theory.
• The resulting change in the 𝜆 of the radiation is a direct result of the conservation of energy and momentum in
the collision between photon and an electron.
Explanation:
➢In the target material, the valence electron is loosely bound in atoms
and behave like a free electron.
➢The incident beam of photon collides with a valence electron and
transfer some part of its energy and momentum to the target electron
And leaves as a scattered photon.

15. The collision between the photon and electron obeys 2 principles;
1) The conservation of linear momentum:
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑜𝑓 𝛾 = 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑜𝑓 𝑒−𝑎𝑓𝑡𝑒𝑟 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛 + 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑜𝑓 𝛾 𝑎𝑓𝑡𝑒𝑟 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛
2) Conservation of total relativistic energy:
𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑝ℎ𝑜𝑡𝑜𝑛 𝑎𝑛𝑑 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑏𝑒𝑓𝑜𝑟𝑒 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛 = 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑝ℎ𝑜𝑡𝑜𝑛 𝑎𝑛𝑑 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑎𝑓𝑡𝑒𝑟 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛
ℎ𝑓𝑜 + 𝑚𝑜𝑐2 = ℎ𝑓 + 𝓇𝑚𝑐2
Using this principle, the following relationship can be derived:
Δ𝜆 =
ℎ
𝑚𝑜𝑐
1 − cos 𝜃
Here, Δ𝜆 = 𝜆𝑓 − 𝜆𝑖; Compton shift
𝐶𝑜𝑚𝑝𝑡𝑜𝑛 𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ =
ℎ
𝑚𝑜𝑐
=
6.63×10−34
(9.11×10−31)(3×108)
𝜆𝑐 = 2.43 𝑝𝑚

16. NUMERICAL
Q3: An 71 pm X-ray is incident on a calcite target. Find the wavelength of X-ray scattered at a 30°
angle. What is the largest shift that can be expected in this experiment.
Q4: An 71 pm X-ray is incident on a calcite target. Find:
1. Wavelength of scattered x-ray at 60°
2. Angle in which smallest change in wavelength expected.
Q5: An X-ray photon has wavelength at 41.6 pm. Calculate the photon’s
a) Energy
b) Frequency
c) Momentum

17. Wave Nature of Matter/ De-Broglie Hypothesis
• De Broglie suggested that analogous to how light exhibits particle like behavior, microscopic
particles like electrons, protons and even larger macroscopic objects can exhibit wave like
behavior.
• Under certain conditions matter is seen to show wavelike behavior while in others particle like
behavior can be seen, this is known as wave-particle duality of matter.
• To describe these ‘matter waves’ de Broglie defined a wavelength known as the de Broglie
wavelength. He suggested that the relationship proposed by Einstein for the momentum and
wavelength of a photon should also apply to matter.
𝑝 =
ℎ
𝜆
So, rearranging for wavelength we have:
𝜆 =
ℎ
𝑝
𝜆 =
ℎ
𝑚𝑣

18. • Experimental evidence of de Broglie hypothesis
o Davisson-Germer Experiment (1927):
In this experiment, electrons were fired at a nickel crystal, producing a diffraction pattern.
The pattern, similar to light waves interacting with a diffraction grating, confirmed that
electrons exhibit wave-like behavior, as predicted by de Broglie. The observed wavelength
matched de Broglie's theoretical prediction.
o Double slit experiment with electrons:
When electrons are fired one by one through two slits, they create an interference pattern
on a detection screen. If electrons behaved only like particles, they would pass through one
slit or the other, forming two distinct bands behind the slits. However, the experiment
shows an interference pattern of alternating light and dark regions, typical of wave
interference. This behavior cannot be explained by classical particle theory but agrees with
de Broglie’s hypothesis that electrons and other particles have wave-like properties.

19. Q: Why is the wavelike behavior of only microscopic objects observed?
The wave-like nature of macroscopic objects is not observed because their de Broglie
wavelengths are extremely small, far smaller than light or other wave phenomena, and negligible
compared to their dimensions. As a result, their wave properties are undetectable by our senses
or instruments.
Q6: Calculate the de Broglie Wavelength of the following
a. An electron moving with a speed of 2 × 106
𝑚/𝑠. (Ans: 0.364 nm)
b. A tennis ball (55 g) travelling at 35 m/s (Ans: 3.44x10-34 m)
c. A proton travelling with velocity 10% the speed of light. (Ans: 13.3 fm)
d. A car of mass 1200 kg travelling at 30 m/s (Ans: 1.851x10-38 m)
NUMERICAL