Engineering Physics
UNIT 2 QUANTUM MECHANICS
2.1. INTRODUCTION
Quantum is a Latin word, meaning how much? It refers to the discrete units of matter and energy that are
predicted and observed in quantum physics. Even space and time, which appear to be extremely
continuous, have the smallest possible discrete values. It was the phenomenon of discreteness that gave
rise to the name quantum mechanics. The birth of quantum physics is attributed to Max Planck's 1900
paper on black body radiation. Later developments in this field were carried out by Albert Einstein, Neil’s
Bohr, Werner Heisenberg, Erwin Schrodinger, and many others by applying Planck’s quantum theory.
Till the end of 18thcentury the growth of science and technology was based on Newtonian mechanics, laws
of thermodynamics and Maxwell’s laws of electrodynamics, which falls under classical theory. These laws
successfully explain the behavior of macroscopic bodies like planets, stars, bodies having larger mass and
size or visible things, but these classical concepts failed to explain the behavior of atoms and subatomic
particles such as electrons, protons etc. Furthermore, it could not explain: the stability of an atom, discrete
spectra of an atom, Raman effect, black body radiation, photoelectric effect, Compton effect etc. In the
early 20th century, it was discovered that the laws that govern macroscopic objects do not function the
same on microscopic objects and hence this led to the development of quantum mechanics, which is today
regarded as the fundamental theory of nature. Difference between Classical Physics and Quantum Physics
is given as follows:
Table 2.1: Difference between Classical Physics and Quantum Physics
Sr.
Classical Physics Quantum Physics
No.
1 Classical Physics applies to macroscopic bodies Quantum Physics applies to microscopic bodies
or heavy bodies.(i.e. having large no. of atoms or particles. (i.e. atomic particle like electron)
or molecules)
2 Macroscopic bodies follow a definite path. e.g. Microscopic bodies do not follow a definite path
Cricket ball e.g. Electron
3 The interaction of electromagnetic wave with The interaction of electromagnetic wave with a
anybody is studied under classical mechanics particle is studied under quantum mechanics
when the dimension of body is quite larger than when the dimensions of body are comparable to
the wavelength of wave. the wavelength of wave.
4 According to classical wave theory, light is According to quantum physics light radiation is in
electromagnetic wave which is generated by the form of stream of photons. Each radiating
accelerated charges. If charges oscillate with body consists of photon.
constant frequency υ, it produces an
electromagnetic wave of same frequency.
5 Energy of a wave is not related with frequency Energy of wave is related with υ of the light
(υ) of the wave but it is proportional to square radiation i.e. E = hυ
of amplitude of wave. I = Nhυ
I = klAl2
6 Physical quantities such as particle energy are Physical quantities are quantized. They are in
continuously variable and can take any possible discrete units, E=hυ, 2hυ, ….e.g. Consider a
values .e.g. Potential energy of a body falling person climbing a stair case, then his Potential
freely decreases continuously. energy increases in discrete manner, that is in
fixed amount.
, Engineering Physics
2.2. PLANCK’S HYPOTHESIS
Max Planck proposed a new theory on electromagnetic radiation which later known as Quantum theory of
radiation. Two major postulates of this theory are: -
1. Every radiating body consists of large number of atomic oscillators. These atomic oscillators in a
body cannot have any arbitrary amount of energy but they have energy in the form of discrete units of
energy given by
E = nhυ,
Where n is any positive integer, v is frequency of radiation and h is called Planck's constant (h = 6.63 ×
10 -34Js).
2. Atomic oscillators cannot absorb or emit energy of any arbitrary amount but they absorb or emit
energy in indivisible discrete units. The amount of radiant energy in each discrete unit is called a
quantum of energy. Each quantum carries an energy,
E = hυ
Thus, the hypothesis is that radiation energy is emitted or absorbed in a discontinuous manner and in
the form of a quantum is called the Planck’s quantum hypothesis. The energies of the atoms are said to
be quantized. The atoms cannot absorb or emit energy of any arbitrary amount but they absorb or emit
energy in form of discrete units. The amount of radiant energy in each discrete unit is called a quantum of
energy. Each quantum carries an energy,
E = hυ.
The allowed energy states are called quantum states.
2.2.1. EINSTEIN MODIFICATION TO PLANCK’S HYPOTHESIS
According to Max Planck’s, light is a form of electromagnetic waves i.e. it is a continuous wave and in the
form of quanta. In 1905 Einstein modified Planck’s theory of radiation saying that light inspite of its wave
nature must be composed of energy particles called as photons. Einstein considered that the quantization of
energy which is seen in emission and absorption process is retained as the energy travels through space.
According to Einstein:
1. Light beam is regarded as a stream of photons, which travels with velocity = 3x108 m/s.
2. An electromagnetic wave having frequency ‘ν’ contains identical photons, having energy E= hν. Higher
the frequency of the wave, higher is the energy of each photon.
3. If an electromagnetic wave contains only one photon then it will have energy “hν”, if contains 2
photons, its energy is “2hν”. Therefore, the intensity of monochromatic light beam is related to the
concentration of photons.
∴ I = N hν where, N is no. of photons.
4. When photons encounters matter, they impart all their energy to the particles of matter and vanish.
2.2.2. PROPERTIES OF PHOTON
1. Energy: Energy of photon is determined by its frequency,
𝑐
E = ℎ𝜈 = ℎ 𝜆 … … … … … … … … … … (2.1)
2. Velocity: Like electromagnetic waves, photon always travels with velocity of light ‘c’.
c = 3 x 108 m/s
3. Mass: Rest mass of photon is zero (mo = 0). Since photon can never be at rest and as it travels
with velocity of light ‘c’, its relativistic mass is given by,
𝐸 hν
m = 2= 2 … … … … … … … … … … (2.2)
𝑐 𝑐
4. Linear momentum: Photons possess linear momentum, given by 𝑝 = 𝑚𝑣 = 𝑚𝑐.
, Engineering Physics
𝑚𝑐 2 𝐸 hν ℎ
p = mc =
= = = … … … … … … … … … … (2.3)
𝑐 𝑐 𝑐 𝜆
5. Angular momentum: Angular momentum is also known as spin. Photon has a spin of one unit.
S = 1ℏ
6. Electrical Nature: Photons are electrically neutral. They are neither deflected by electric field
nor by magnetic field.
7. Photons travels in a straight line.
8. The energy of photons can be transferred during an interaction with other particles.
9. Photons can be created or destroyed when radiation is emitted or absorbed.
Q What is Planck’s Quantum Hypothesis.
Q. State the properties of Photon.
Q. Give the characteristics of photon. (2) [Summer-14]
2.3. COMPTON EFFECT
One of the key experiments in quantum mechanics is the Compton Effect, which generally occurs in the X-
ray or the γ-ray region of the electromagnetic spectrum. This effect was first demonstrated in 1923 by
Arthur Compton, for which he received the Nobel Prize in 1927.
STATEMENT:
When a beam of monochromatic X-rays strikes the loosely bound electrons of atoms in graphite target, the
X-rays are scattered in all possible directions. This phenomenon is called as “Compton scattering” or
“Compton effect” The angle between direction of incident and scattered ray is called scattering angle
(ф).
The scattered radiation consists of two components of wavelength-unmodified
and modified. The component having same wavelength as that of incident X-rays is called as unmodified
component and its wavelength is called unmodified wavelength (λ) and the other component having
wavelength slightly higher than incident one is called as modified component and its wavelength is known
as modified wavelength (𝜆′).
The difference between modified wavelength (λ’) and unmodified wavelength
(λ) or the change in wavelength of X-ray photon is known as Compton Shift. It is given by equation,
ℎ
∆𝜆 = 𝜆′ − 𝜆 = (1 − cosΦ)
𝑚𝑜 𝑐
where 𝜆′ = 𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑐𝑎𝑡𝑡𝑒𝑟𝑒𝑑 𝑋 − 𝑟𝑎𝑦 𝑝ℎ𝑜𝑡𝑜𝑛
𝑎𝑛𝑑 𝜆 = 𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑋 − 𝑟𝑎𝑦 𝑝ℎ𝑜𝑡𝑜𝑛
Compton shift increases with increase in scattering angle Φ. Its value depends on scattering angle Φ.
, Engineering Physics
EXPERIMENTAL SETUP:
Fig. 2.1: Illustration of Compton Effect
The experimental set up of Compton to verify the effect is as shown in fig. 2.1. In this experiment a beam
of monochromatic X-rays from an X-ray source passing through collimator is allowed to fall on block of
light elements (low atomic no.) like graphite (or Carbon or Boron) target element. After interaction with
the atoms in the target, X-rays are scattered in all possible directions. The intensity of the scattered X-rays
was measured using detector as a function of wavelength of X-rays at different scattering angles with the
help of detector which is movable on measuring circular scale of Bragg’s Spectrometer. The results found
are discussed below:
VARIATION OF COMPTON SHIFT AS A FUNCTION OF SCATTERING ANGLE:
Case i] When Φ = 0
ℎ
∆𝜆 = (1 − 𝑐𝑜𝑠𝛷)
𝑚𝑜 𝑐
ℎ
∆𝜆 = (1 − 𝑐𝑜𝑠0)
𝑚𝑜 𝑐
ℎ
= (1 − 1)
𝑚𝑜 𝑐
= 0 (minimum value of Compton
Shift)
Case ii] When Φ = 90o
ℎ
∆𝜆 = (1 − 𝑐𝑜𝑠90)
𝑚𝑜 𝑐
6.626 × 10−34
= (1)
9.11 × 10−31 × 3 × 108
= 0.0242𝐴𝑜
The numerical value of Compton shift ∆λ for Ф= 900 is
called Compton wavelength (λc).
Case iii] When Φ = 180o
ℎ
∆𝜆 = (1 − 𝑐𝑜𝑠180)
𝑚𝑜 𝑐 Fig. 2.2: The variation of intensity of
X-rays as a function of wavelength at
different angles.
UNIT 2 QUANTUM MECHANICS
2.1. INTRODUCTION
Quantum is a Latin word, meaning how much? It refers to the discrete units of matter and energy that are
predicted and observed in quantum physics. Even space and time, which appear to be extremely
continuous, have the smallest possible discrete values. It was the phenomenon of discreteness that gave
rise to the name quantum mechanics. The birth of quantum physics is attributed to Max Planck's 1900
paper on black body radiation. Later developments in this field were carried out by Albert Einstein, Neil’s
Bohr, Werner Heisenberg, Erwin Schrodinger, and many others by applying Planck’s quantum theory.
Till the end of 18thcentury the growth of science and technology was based on Newtonian mechanics, laws
of thermodynamics and Maxwell’s laws of electrodynamics, which falls under classical theory. These laws
successfully explain the behavior of macroscopic bodies like planets, stars, bodies having larger mass and
size or visible things, but these classical concepts failed to explain the behavior of atoms and subatomic
particles such as electrons, protons etc. Furthermore, it could not explain: the stability of an atom, discrete
spectra of an atom, Raman effect, black body radiation, photoelectric effect, Compton effect etc. In the
early 20th century, it was discovered that the laws that govern macroscopic objects do not function the
same on microscopic objects and hence this led to the development of quantum mechanics, which is today
regarded as the fundamental theory of nature. Difference between Classical Physics and Quantum Physics
is given as follows:
Table 2.1: Difference between Classical Physics and Quantum Physics
Sr.
Classical Physics Quantum Physics
No.
1 Classical Physics applies to macroscopic bodies Quantum Physics applies to microscopic bodies
or heavy bodies.(i.e. having large no. of atoms or particles. (i.e. atomic particle like electron)
or molecules)
2 Macroscopic bodies follow a definite path. e.g. Microscopic bodies do not follow a definite path
Cricket ball e.g. Electron
3 The interaction of electromagnetic wave with The interaction of electromagnetic wave with a
anybody is studied under classical mechanics particle is studied under quantum mechanics
when the dimension of body is quite larger than when the dimensions of body are comparable to
the wavelength of wave. the wavelength of wave.
4 According to classical wave theory, light is According to quantum physics light radiation is in
electromagnetic wave which is generated by the form of stream of photons. Each radiating
accelerated charges. If charges oscillate with body consists of photon.
constant frequency υ, it produces an
electromagnetic wave of same frequency.
5 Energy of a wave is not related with frequency Energy of wave is related with υ of the light
(υ) of the wave but it is proportional to square radiation i.e. E = hυ
of amplitude of wave. I = Nhυ
I = klAl2
6 Physical quantities such as particle energy are Physical quantities are quantized. They are in
continuously variable and can take any possible discrete units, E=hυ, 2hυ, ….e.g. Consider a
values .e.g. Potential energy of a body falling person climbing a stair case, then his Potential
freely decreases continuously. energy increases in discrete manner, that is in
fixed amount.
, Engineering Physics
2.2. PLANCK’S HYPOTHESIS
Max Planck proposed a new theory on electromagnetic radiation which later known as Quantum theory of
radiation. Two major postulates of this theory are: -
1. Every radiating body consists of large number of atomic oscillators. These atomic oscillators in a
body cannot have any arbitrary amount of energy but they have energy in the form of discrete units of
energy given by
E = nhυ,
Where n is any positive integer, v is frequency of radiation and h is called Planck's constant (h = 6.63 ×
10 -34Js).
2. Atomic oscillators cannot absorb or emit energy of any arbitrary amount but they absorb or emit
energy in indivisible discrete units. The amount of radiant energy in each discrete unit is called a
quantum of energy. Each quantum carries an energy,
E = hυ
Thus, the hypothesis is that radiation energy is emitted or absorbed in a discontinuous manner and in
the form of a quantum is called the Planck’s quantum hypothesis. The energies of the atoms are said to
be quantized. The atoms cannot absorb or emit energy of any arbitrary amount but they absorb or emit
energy in form of discrete units. The amount of radiant energy in each discrete unit is called a quantum of
energy. Each quantum carries an energy,
E = hυ.
The allowed energy states are called quantum states.
2.2.1. EINSTEIN MODIFICATION TO PLANCK’S HYPOTHESIS
According to Max Planck’s, light is a form of electromagnetic waves i.e. it is a continuous wave and in the
form of quanta. In 1905 Einstein modified Planck’s theory of radiation saying that light inspite of its wave
nature must be composed of energy particles called as photons. Einstein considered that the quantization of
energy which is seen in emission and absorption process is retained as the energy travels through space.
According to Einstein:
1. Light beam is regarded as a stream of photons, which travels with velocity = 3x108 m/s.
2. An electromagnetic wave having frequency ‘ν’ contains identical photons, having energy E= hν. Higher
the frequency of the wave, higher is the energy of each photon.
3. If an electromagnetic wave contains only one photon then it will have energy “hν”, if contains 2
photons, its energy is “2hν”. Therefore, the intensity of monochromatic light beam is related to the
concentration of photons.
∴ I = N hν where, N is no. of photons.
4. When photons encounters matter, they impart all their energy to the particles of matter and vanish.
2.2.2. PROPERTIES OF PHOTON
1. Energy: Energy of photon is determined by its frequency,
𝑐
E = ℎ𝜈 = ℎ 𝜆 … … … … … … … … … … (2.1)
2. Velocity: Like electromagnetic waves, photon always travels with velocity of light ‘c’.
c = 3 x 108 m/s
3. Mass: Rest mass of photon is zero (mo = 0). Since photon can never be at rest and as it travels
with velocity of light ‘c’, its relativistic mass is given by,
𝐸 hν
m = 2= 2 … … … … … … … … … … (2.2)
𝑐 𝑐
4. Linear momentum: Photons possess linear momentum, given by 𝑝 = 𝑚𝑣 = 𝑚𝑐.
, Engineering Physics
𝑚𝑐 2 𝐸 hν ℎ
p = mc =
= = = … … … … … … … … … … (2.3)
𝑐 𝑐 𝑐 𝜆
5. Angular momentum: Angular momentum is also known as spin. Photon has a spin of one unit.
S = 1ℏ
6. Electrical Nature: Photons are electrically neutral. They are neither deflected by electric field
nor by magnetic field.
7. Photons travels in a straight line.
8. The energy of photons can be transferred during an interaction with other particles.
9. Photons can be created or destroyed when radiation is emitted or absorbed.
Q What is Planck’s Quantum Hypothesis.
Q. State the properties of Photon.
Q. Give the characteristics of photon. (2) [Summer-14]
2.3. COMPTON EFFECT
One of the key experiments in quantum mechanics is the Compton Effect, which generally occurs in the X-
ray or the γ-ray region of the electromagnetic spectrum. This effect was first demonstrated in 1923 by
Arthur Compton, for which he received the Nobel Prize in 1927.
STATEMENT:
When a beam of monochromatic X-rays strikes the loosely bound electrons of atoms in graphite target, the
X-rays are scattered in all possible directions. This phenomenon is called as “Compton scattering” or
“Compton effect” The angle between direction of incident and scattered ray is called scattering angle
(ф).
The scattered radiation consists of two components of wavelength-unmodified
and modified. The component having same wavelength as that of incident X-rays is called as unmodified
component and its wavelength is called unmodified wavelength (λ) and the other component having
wavelength slightly higher than incident one is called as modified component and its wavelength is known
as modified wavelength (𝜆′).
The difference between modified wavelength (λ’) and unmodified wavelength
(λ) or the change in wavelength of X-ray photon is known as Compton Shift. It is given by equation,
ℎ
∆𝜆 = 𝜆′ − 𝜆 = (1 − cosΦ)
𝑚𝑜 𝑐
where 𝜆′ = 𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑐𝑎𝑡𝑡𝑒𝑟𝑒𝑑 𝑋 − 𝑟𝑎𝑦 𝑝ℎ𝑜𝑡𝑜𝑛
𝑎𝑛𝑑 𝜆 = 𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑋 − 𝑟𝑎𝑦 𝑝ℎ𝑜𝑡𝑜𝑛
Compton shift increases with increase in scattering angle Φ. Its value depends on scattering angle Φ.
, Engineering Physics
EXPERIMENTAL SETUP:
Fig. 2.1: Illustration of Compton Effect
The experimental set up of Compton to verify the effect is as shown in fig. 2.1. In this experiment a beam
of monochromatic X-rays from an X-ray source passing through collimator is allowed to fall on block of
light elements (low atomic no.) like graphite (or Carbon or Boron) target element. After interaction with
the atoms in the target, X-rays are scattered in all possible directions. The intensity of the scattered X-rays
was measured using detector as a function of wavelength of X-rays at different scattering angles with the
help of detector which is movable on measuring circular scale of Bragg’s Spectrometer. The results found
are discussed below:
VARIATION OF COMPTON SHIFT AS A FUNCTION OF SCATTERING ANGLE:
Case i] When Φ = 0
ℎ
∆𝜆 = (1 − 𝑐𝑜𝑠𝛷)
𝑚𝑜 𝑐
ℎ
∆𝜆 = (1 − 𝑐𝑜𝑠0)
𝑚𝑜 𝑐
ℎ
= (1 − 1)
𝑚𝑜 𝑐
= 0 (minimum value of Compton
Shift)
Case ii] When Φ = 90o
ℎ
∆𝜆 = (1 − 𝑐𝑜𝑠90)
𝑚𝑜 𝑐
6.626 × 10−34
= (1)
9.11 × 10−31 × 3 × 108
= 0.0242𝐴𝑜
The numerical value of Compton shift ∆λ for Ф= 900 is
called Compton wavelength (λc).
Case iii] When Φ = 180o
ℎ
∆𝜆 = (1 − 𝑐𝑜𝑠180)
𝑚𝑜 𝑐 Fig. 2.2: The variation of intensity of
X-rays as a function of wavelength at
different angles.
