MMIT, Lohgaon, Pune // Engineering Physics [2019-20] // Unit 3 – Quantum Mechanics
Unit 3: Quantum Mechanics
Syllabus
- De-Broglie hypothesis
- Concept of phase velocity and group velocity (qualitative)
- Heisenberg Uncertainty Principle
- Wave-function and its physical significance
- Schrodinger’s equations: time independent and time dependent
- Application of Schrodinger’s time independent wave equation - Particle enclosed in
infinitely deep potential well (Particle in Rigid Box)
- Particle in Finite potential well (Particle in Non Rigid box) (qualitative)
- Tunneling effect, Tunneling effect examples (principle only): Alpha Decay, Scanning
Tunneling Microscope, Tunnel diode
- Introduction to quantum computing
Prerequisite: Basics of wave particle duality from 11th and 12th standard
3.1 Wave particle duality of radiation
- In classical mechanics, wave and particle are shows different properties and are given
separate treatment.
- A matter particle is identified by the properties such as mass, momentum, kinetic energy,
spin, electric charge, etc.
- A wave is identified by properties such as wavelength, frequency, amplitude, intensity,
energy, etc.
- Electromagnetic radiations (e.g. light) show optical phenomenon such as interference,
diffraction and polarization. These phenomenons require that electromagnetic radiations
must have wave nature.
- However, phenomenon such as photoelectric effect, Compton Effect, emission and
absorption of radiation by matter, black body radiations require that electromagnetic
radiations must have particle nature.
- Thus the electromagnetic radiations have dual characteristics.
- Although particle and wave properties of radiation cannot be observed simultaneously, it is
not possible to separate the particle and wave nature of electromagnetic radiations.
3.2 Wave particle duality of matter / De Broglie’s hypothesis of matter waves
de Broglie hypothesis
Louis de Broglie extended the wave-particle duality of light to the material particles. If a light wave
can show wave-particle duality in some conditions, then particles such as electrons should also act
as waves at some times. This is known as de Broglie hypothesis.
Thus, according to de Broglie hypothesis, a moving particle always has a wave associated with it and
the motion of that particle is guided by the associated wave. The waves associated with particle are
known as matter waves or de Broglie waves.
If a particle of mass m is moving with velocity v, the wavelength of matter waves associated with a
ℎ
particle is inversely proportional to the momentum and is given by = .
𝑝
Page 1 of 20
,MMIT, Lohgaon, Pune // Engineering Physics [2019-20] // Unit 3 – Quantum Mechanics
As a photon travels with velocity c, its momentum can be expressed as
𝐸 ℎ ℎ
𝑝=𝑐 = 𝑐 =
ℎ
Thus, wavelength of the photon is = (1)
𝑝
de Broglie proposed that equation (1) is a universal and is
applicable to photons and material particles also.
A particle of mass m moving with velocity v, has momentum
𝑝 = 𝑚𝑣. Thus the wavelength of de Broglie wave associated
with it can be written as
ℎ ℎ
= =
𝑝 𝑚𝑣
The waves associated with a moving particle are called matter waves or de Broglie waves.
3.3 de Broglie wavelength of a particle in terms of its kinetic energy
Consider a particle of mass m moving with velocity v. Its momentum is p=mv and the de Broglie
wavelength of matter waves associated with it is given by
ℎ ℎ
= 𝑝 = 𝑚𝑣 --- (1)
1
The kinetic energy of the particle is given by 𝐸 = 2 𝑚𝑣 2
1
𝐸 = 2𝑚 𝑚2 𝑣 2
𝑝2
As p=mv 𝐸 = 2𝑚
Or 𝑝2 = 2𝑚𝐸 or 𝑝 = 2𝑚𝐸 --- (2)
Putting the value of p in the equation of matter wave,
ℎ ℎ
=𝑝= 2𝑚𝐸
--- (3)
The above equation gives de Broglie wavelength of a particle of mass m and kinetic energy E. This
equation is true for any particle irrespective of its charge or mass.
3.4 de Broglie wavelength of an electron/proton accelerated by Potential
difference
If a charged particle such as electron or proton of rest mass m0 is accelerated by a potential
difference of V volts, the kinetic energy gained by it is given by
𝐾𝐸 = 𝑒𝑉 (where e is elementary charge)
1
𝑚0 𝑣 2 = 𝑒𝑉 --- (1)
2
1
𝑚02 𝑣 2 = 𝑒𝑉
2𝑚 0
𝑚02 𝑣 2 = 2𝑚0 𝑒𝑉
𝑚0 𝑣 = 2𝑚0 𝑒𝑉 --- (2)
Thus, the de Broglie wavelength associated charged particle is given by
ℎ ℎ
=𝑝=𝑚 --- (3)
0𝑣
ℎ
From equations (2) and (3) = 2𝑚 0 𝑒𝑉
-- (4)
Where, ℎ = 6.63 × 10−34 𝐽𝑠 and 𝑒 = 1.6 × 10−19 𝐶
Page 2 of 20
, MMIT, Lohgaon, Pune // Engineering Physics [2019-20] // Unit 3 – Quantum Mechanics
For an electron
As rest Mass of electron, 𝑚0𝑒 = 9.1 × 10−31 𝑘𝑔, from equation (4)
12.26
= 𝑉
Å
For a proton
Rest mass of proton, 𝑚0𝑝 = 1.673 × 10−27 𝑘𝑔, from equation (4)
0.286
= 𝑉
Å
3.5 Properties of matter waves
According to de-Broglie hypothesis, for a particle mass m moving with velocity v, the wavelength of
ℎ ℎ
matter wave associated with it is given by = = , where h is Plank’s constant. The properties
𝑝 𝑚𝑣
of matter waves can be summarized as below:
(i) Wavelength is inversely proportional to the velocity of particle
The wavelength of matter waves is inversely proportional to the velocity of the particle i.e.
1
𝑣. Thus, when a particle is at rest, its velocity is zero and . Smaller is the value of v
longer is the wavelength of the matter waves associated with it. It means that matter waves
are detectable only for moving particles.
(ii) Wavelength is inversely proportional to the mass of particle
1
The wavelength of matter waves is inversely proportional to the mass of the particle i.e. 𝑚 .
Lighter is the particle, smaller is the value of m and hence longer is the wavelength of the
matter waves associated with it. Therefore, wave behavior of micro-particles will be significant
whereas waves associated with macro-bodies can never be detected.
(iii) Matter waves are related with probability of finding the particle
If a photon is considered as a particle then corresponding electromagnetic wave is de Broglie
wave for that photon. Similarly, atomic particles are associated with matter waves, which do
not have similarity to any known waves such as electromagnetic or sound waves. de Broglie
waves are associated with locating the probability of particle and hence also known as
probability waves.
(iv) Matter waves are not electromagnetic waves
Any types of waves are produced by oscillations of certain quantity. When a charged particle
oscillates, it produces electromagnetic waves. Matter waves are produced by the motion of
the particles and do not depend on the charge of the particle. Therefore, matter waves are not
electromagnetic waves.
(v) Matter waves are not mechanical waves
Mechanical waves, such as sound waves, are produced by the vibrations of particles of the
medium though which they travel. A particle can always travel in the vacuum and hence
matter waves are associated with it even in vacuum. As matter waves do not require any
medium for propagation, they are not mechanical waves.
3.6 Phase velocity and group velocity
Phase velocity (or wave velocity)
The phase velocity of a wave is the rate at which the
phase of the wave propagates in space. Phase
velocity of a single wave is the velocity with which a
definite phase point (of either crest or trough) of the
wave propagates in the medium.
Page 3 of 20
Unit 3: Quantum Mechanics
Syllabus
- De-Broglie hypothesis
- Concept of phase velocity and group velocity (qualitative)
- Heisenberg Uncertainty Principle
- Wave-function and its physical significance
- Schrodinger’s equations: time independent and time dependent
- Application of Schrodinger’s time independent wave equation - Particle enclosed in
infinitely deep potential well (Particle in Rigid Box)
- Particle in Finite potential well (Particle in Non Rigid box) (qualitative)
- Tunneling effect, Tunneling effect examples (principle only): Alpha Decay, Scanning
Tunneling Microscope, Tunnel diode
- Introduction to quantum computing
Prerequisite: Basics of wave particle duality from 11th and 12th standard
3.1 Wave particle duality of radiation
- In classical mechanics, wave and particle are shows different properties and are given
separate treatment.
- A matter particle is identified by the properties such as mass, momentum, kinetic energy,
spin, electric charge, etc.
- A wave is identified by properties such as wavelength, frequency, amplitude, intensity,
energy, etc.
- Electromagnetic radiations (e.g. light) show optical phenomenon such as interference,
diffraction and polarization. These phenomenons require that electromagnetic radiations
must have wave nature.
- However, phenomenon such as photoelectric effect, Compton Effect, emission and
absorption of radiation by matter, black body radiations require that electromagnetic
radiations must have particle nature.
- Thus the electromagnetic radiations have dual characteristics.
- Although particle and wave properties of radiation cannot be observed simultaneously, it is
not possible to separate the particle and wave nature of electromagnetic radiations.
3.2 Wave particle duality of matter / De Broglie’s hypothesis of matter waves
de Broglie hypothesis
Louis de Broglie extended the wave-particle duality of light to the material particles. If a light wave
can show wave-particle duality in some conditions, then particles such as electrons should also act
as waves at some times. This is known as de Broglie hypothesis.
Thus, according to de Broglie hypothesis, a moving particle always has a wave associated with it and
the motion of that particle is guided by the associated wave. The waves associated with particle are
known as matter waves or de Broglie waves.
If a particle of mass m is moving with velocity v, the wavelength of matter waves associated with a
ℎ
particle is inversely proportional to the momentum and is given by = .
𝑝
Page 1 of 20
,MMIT, Lohgaon, Pune // Engineering Physics [2019-20] // Unit 3 – Quantum Mechanics
As a photon travels with velocity c, its momentum can be expressed as
𝐸 ℎ ℎ
𝑝=𝑐 = 𝑐 =
ℎ
Thus, wavelength of the photon is = (1)
𝑝
de Broglie proposed that equation (1) is a universal and is
applicable to photons and material particles also.
A particle of mass m moving with velocity v, has momentum
𝑝 = 𝑚𝑣. Thus the wavelength of de Broglie wave associated
with it can be written as
ℎ ℎ
= =
𝑝 𝑚𝑣
The waves associated with a moving particle are called matter waves or de Broglie waves.
3.3 de Broglie wavelength of a particle in terms of its kinetic energy
Consider a particle of mass m moving with velocity v. Its momentum is p=mv and the de Broglie
wavelength of matter waves associated with it is given by
ℎ ℎ
= 𝑝 = 𝑚𝑣 --- (1)
1
The kinetic energy of the particle is given by 𝐸 = 2 𝑚𝑣 2
1
𝐸 = 2𝑚 𝑚2 𝑣 2
𝑝2
As p=mv 𝐸 = 2𝑚
Or 𝑝2 = 2𝑚𝐸 or 𝑝 = 2𝑚𝐸 --- (2)
Putting the value of p in the equation of matter wave,
ℎ ℎ
=𝑝= 2𝑚𝐸
--- (3)
The above equation gives de Broglie wavelength of a particle of mass m and kinetic energy E. This
equation is true for any particle irrespective of its charge or mass.
3.4 de Broglie wavelength of an electron/proton accelerated by Potential
difference
If a charged particle such as electron or proton of rest mass m0 is accelerated by a potential
difference of V volts, the kinetic energy gained by it is given by
𝐾𝐸 = 𝑒𝑉 (where e is elementary charge)
1
𝑚0 𝑣 2 = 𝑒𝑉 --- (1)
2
1
𝑚02 𝑣 2 = 𝑒𝑉
2𝑚 0
𝑚02 𝑣 2 = 2𝑚0 𝑒𝑉
𝑚0 𝑣 = 2𝑚0 𝑒𝑉 --- (2)
Thus, the de Broglie wavelength associated charged particle is given by
ℎ ℎ
=𝑝=𝑚 --- (3)
0𝑣
ℎ
From equations (2) and (3) = 2𝑚 0 𝑒𝑉
-- (4)
Where, ℎ = 6.63 × 10−34 𝐽𝑠 and 𝑒 = 1.6 × 10−19 𝐶
Page 2 of 20
, MMIT, Lohgaon, Pune // Engineering Physics [2019-20] // Unit 3 – Quantum Mechanics
For an electron
As rest Mass of electron, 𝑚0𝑒 = 9.1 × 10−31 𝑘𝑔, from equation (4)
12.26
= 𝑉
Å
For a proton
Rest mass of proton, 𝑚0𝑝 = 1.673 × 10−27 𝑘𝑔, from equation (4)
0.286
= 𝑉
Å
3.5 Properties of matter waves
According to de-Broglie hypothesis, for a particle mass m moving with velocity v, the wavelength of
ℎ ℎ
matter wave associated with it is given by = = , where h is Plank’s constant. The properties
𝑝 𝑚𝑣
of matter waves can be summarized as below:
(i) Wavelength is inversely proportional to the velocity of particle
The wavelength of matter waves is inversely proportional to the velocity of the particle i.e.
1
𝑣. Thus, when a particle is at rest, its velocity is zero and . Smaller is the value of v
longer is the wavelength of the matter waves associated with it. It means that matter waves
are detectable only for moving particles.
(ii) Wavelength is inversely proportional to the mass of particle
1
The wavelength of matter waves is inversely proportional to the mass of the particle i.e. 𝑚 .
Lighter is the particle, smaller is the value of m and hence longer is the wavelength of the
matter waves associated with it. Therefore, wave behavior of micro-particles will be significant
whereas waves associated with macro-bodies can never be detected.
(iii) Matter waves are related with probability of finding the particle
If a photon is considered as a particle then corresponding electromagnetic wave is de Broglie
wave for that photon. Similarly, atomic particles are associated with matter waves, which do
not have similarity to any known waves such as electromagnetic or sound waves. de Broglie
waves are associated with locating the probability of particle and hence also known as
probability waves.
(iv) Matter waves are not electromagnetic waves
Any types of waves are produced by oscillations of certain quantity. When a charged particle
oscillates, it produces electromagnetic waves. Matter waves are produced by the motion of
the particles and do not depend on the charge of the particle. Therefore, matter waves are not
electromagnetic waves.
(v) Matter waves are not mechanical waves
Mechanical waves, such as sound waves, are produced by the vibrations of particles of the
medium though which they travel. A particle can always travel in the vacuum and hence
matter waves are associated with it even in vacuum. As matter waves do not require any
medium for propagation, they are not mechanical waves.
3.6 Phase velocity and group velocity
Phase velocity (or wave velocity)
The phase velocity of a wave is the rate at which the
phase of the wave propagates in space. Phase
velocity of a single wave is the velocity with which a
definite phase point (of either crest or trough) of the
wave propagates in the medium.
Page 3 of 20
